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[42] Baldick, R., and Kahn, E., “Transmission Planning Issues in a CompetitiveEconomic Environment,” IEEE Transactions on Power Systems, vol. 8, no. 4,November 1993, pp. 1497-1503.

[43] Vojdani, A. F., Imparato, C. F., Saini, N. K., Wollenberg, B. F., and Happ, H. H.,“Transmission Access Issues,” IEEE Transactions on Power Systems, vol. 11, no.1, February 1996, pp. 41-51.

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[47] Hughes, W. R., and Felak, R., “Bridging the gap between theory and practice oftransmission pricing,” in Electricity Transmission Pricing and Technology,Einhorn, M., and Siddiqi, R., eds., Kluwer Academic Publishers, Boston, pp. 25-58, 1996.

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[37] Rudnick, H., Palma, R., Cura, E., and Silva, C., “Economically AdaptedTransmission Systems in Open Access Schemes -Application of GeneticAlgorithms,” IEEE Transactions on Power Systems, vol. 11, No. 3, pp. 1427-1440,August 1996.

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[15] Dusonchet, Y. P., and El-Abiad A.H., “Transmission Planning Using DiscreteDynamic Optimization,” IEEE Transactions on Power Apparatus and Systems, vol.PAS-92, pp. 1358-1371, 1973.

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95

6.3. Concluding Remarks

The electric utilities are experiencing radical changes with the upcoming deregulation

process. Traditional methods to solve power systems problems need to be updated and

adapted to the new times coming ahead. New software environments are urgently needed

to understand player’s interaction and to model new market structures in the future. Isola-

tionist operating and planning policies will soon become obsolete, and the need for a

decentralized but coordinated structures is clear.

To address these issues, the approach provided in this thesis can be applied to a new

deregulated environment to simulate not only the interaction between the players, but also

to implement innovative policies with the help of our multiagent environment. A decen-

tralized approach to power systems problems, like the one we propose, will have to com-

bine individual decision making with a coordinated environment. Communication among

the players and fast algorithms to solve large scale problems will be a must.

We also believe that Ptolemy, with its communications capability, modularity, graphical

and external interfaces, and object-oriented structure is an ideal software to model new

power system paradigms in the future.

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Chapter 6

Conclusions and Future Work

6.1. Contributions

Through the previous chapters we have outlined a new method to simulate coalitions of

players in the “transmission planning game” and to allocate the expansion costs that any

expansion creates. The approach that we have followed to solve the problem combined a

multiagent environment and a cooperative game theory framework to form coalitions and

allocate costs respectively. To test our theoretical formulation of the transmission plan-

ning problem, we have also implemented a new software environment to solve this and

other problems in the power systems and optimization areas.

6.2. Future Research

Future work will add the Bilateral Shapley Value coalition scheme and other coalition for-

mation schemes to the existing IDEAS [6] framework; in particular kernel transfer

schemes (see Chapter 2, Section 8). A more detailed study of the coalition stability prop-

erties of our Bilateral Shapley Value recursive cost allocation procedure is forthcoming.

At the same time, new algorithms are being added to DistOpt, and new applications of

DistOpt in decentralized planning are considered. On the other hand, a genetic algorithm

implementation of individual agents’ expansion planning using [68] is forthcoming, and

dynamic (more than one year) expansions are under study.

We are currently integrating the coalition formation and cost allocation algorithms in the

Ptolemy environment. The interested reader can see some of the capabilities of this graph-

ical software framework in power systems applications [73, 74].

93

Applications to the optimal power flow problem in [74] show the effectiveness of the

software package.

92

Figure 5-9 : IEEE-57 Bus Case Subproblem 2 Maximum Error


5.7. Conclusions

We set out a research infrastructure with the goal of tackling optimization problems aris-

ing from power system planning and operations. A generic structure for these problems

based on decomposition and co-ordination for power system applications was con-

structed. The methods are designed to be compatible with vastly enhanced computing

power offered by current technology of parallel and distributed computing. The software

implementation utilizes recent advances in object-oriented software that is flexible, exten-

sible, and conducive to cooperative development.

The object-oriented methodology is especially suited for the emerging design approach in

power industry. The distributed optimization research environment is a power tool to han-

dle a variety of projects, to develop innovative solution approaches, and to solve critical

emerging problems in power systems.

91

solve the problem can be less if we divide the problem in very small subsystems, because

the problems are smaller now and they take less time to solve. For an asynchronous exe-

cution, the objective function approaches its final value faster than with the synchronous

execution, but, to grant convergence, parameter -see (5-2)- has to be significantly low

[74].


ρ

90

Figure 5-6 : IEEE-57 Bus Case Subproblem 2 Objective Function


Figure 5-8, Figure 5-9, and Figure 5-10 show the value of the error: maximum discrep-

ancy between the values of neighboring quantities [ ] -see (5-1)- ver-

sus the computational time. Note that no matter how many subsystems the problem is

divided into, the APP always ensures convergence to a solution, although more sub-

systems usually require more APP iterations, for the same problem. However, the time to

error maxi

= Θi

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5.6. An Optimal Power Flow Application

We study the performance in executing the optimization functions for the IEEE-57 bus

network. In particular, we study the Optimal Power Flow (OPF) problem, for which a full-

fledged model is considered (with no model simplifications), solved without any P-Q

decomposition. For evaluating the performance, the following aspects are considered:

• value of objective function

• maximum discrepancy between the values of the duplicated variables

The algorithm we adopt for solving the various optimization problems is a first-order one,

namely SGRA [66], for which an adequate interface is provided in the Solver palette. For

space reasons, we only consider a three subsystems example of the IEEE-57 bus case,

with a synchronous execution, and an Arrow-Hurwicz core function. The computational

time, i.e., the number of total APP iterations is 300; the number of iterations of the SGRA

algorithm depends on each case. In Figure 5-5, Figure 5-6, and Figure 5-7, the value of

the objective function for the three subsystems versus the computational time is reported.


88

5.5. DistOpt Simulation

A simulation case is run through a universe; this is built starting with the specification of

the number of subproblems by selecting the appropriate block from the mathematical

building palette. Then, the choice of a particular solver has to be made for each subprob-

lem. This is done by grabbing the icon representing that algorithm from the Solvers pal-

ette and dragging it to the appropriate ports of the subproblem block [73]. Then,

according to the needs of the information exchange of the subproblems, input/output ports

of the icons representing neighboring subproblems have to be graphically connected. A

universe for the simulation of the solution of a problem split into three subproblems is

shown in Figure 5-4.

The simulation is then started (the universe is run), and the user is requested via Tcl/Tk

window interface what kind of core function K to use (Arrow-Hurwicz, Uzawa-like, or

other supplied), and the value of some parameters influencing the convergence (computed

values are suggested). Parameters such as synchronization technique may be easily

entered and changed with pop-up menus.

The evaluation of alternative ways of solving the subproblems with different algorithms is

easily carried out by connecting the icon of the desired algorithm to the appropriate port

of the subproblem block. The behavior of the communication system that cares for the

data exchange between subproblems can be easily simulated by using the blocks already

developed for simulating communication networks [75]. The ease of such operation

comes from the modularity, flexibility, and extendibility of the Ptolemy environment, with

its block-oriented graphical construction of the applications.

87

Figure 5-3 : APP_3 Galaxy

Figure 5-4 : Final3 Universe

86

APP_# Galaxy takes care of the initialization of the solution (by reading the informa-

tion contained in the user-supplied file, and by sending to Measures galaxy and to

SubProblem star the appropriate information on the methods of the optimization prob-

lem to be solved, as well as on the splitting of the problem). APP_# chooses the core

function and the value of several parameters of the computation scheme, as well.

Measures galaxy models the front-end between the computing facility and the real

world. It receives initialization information from APP_#, and reads data from a file, rep-

resenting the data/measurements from which the subproblem has to be solved. It has a

state, the name of the input file (which has a default value), that can be modified by the

user.

SubProblem star, which receives initialization information and data from Measures,

updates the local copy of the Lagrange multipliers associated with the coupling con-

straints (the prices), interfaces with the solution algorithms, and exchange information

with the neighboring subproblems.

In Figure 5-2 and Figure 5-3, we can observe more details of the Measures and APP_3

galaxies.

Figure 5-2 : Measures Galaxy

The overall picture of an instance of DistOpt universe that solves an optimization problem

by splitting it into three subproblems can be seen in Figure 5-4.

85

once they have the information from the other solvers (and their own updates), they will

incorporate new information immediately. Asynchronous methods and partially synchro-

nous methods are also provided in this version of DistOpt.

The modularity of design in the Ptolemy environment allows the implementation of

instances of the DistOpt universe for the user to built her own examples. All the blocks

(galaxies in Ptolemy) pertaining to the DistOpt environment are contained in the corre-

sponding Ptolemy palette as shown in Figure 5-1.

Figure 5-1 : DistOpt Palette

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and IEEE*bus_dat respectively. In any particular application, the user simply has to

change the input file, without relinking, in order to solve a different problem (though it

has to be of the same type). The user can define her own application by adding it to

Methods.cc and recompiling again.

The second level of abstraction represents the decomposition-coordination approach, and

it uses the Auxiliary Problem Principle (APP) scheme. Now, the program duplicates the

variables that belong to at least two subproblems and calculates the number of duplica-

tions (couplings) for the whole optimization problem. The APP scheme is used to imple-

ment the decomposition-coordination algorithm. The user is prompted via Tcl/Tk

interface to decide which core function she desires to use to solve the problem. The core

function is a function that “mimics” the objective function (see [70]) and it is built-in the

APP algorithm. After that, the user is prompted to choose parameter’s values to ensure

convergence of the algorithm. The program calculates these values, but the user can

change them interactively via Tcl/Tk script. In particular, an accurate evaluation of is

calculated in the spectral_radius.cc file (see [70] for details). If the new value is

out of bounds, the program will pop up a warning message before continuing.

The third level of abstraction represents the interface with the solvers. There exists a

library of different solvers that can be used for different applications. If the problem is lin-

ear, the available solvers are: DECFSQPSolver.pl, DELSGRG2Solver.pl, and

DESGRASolver.pl. If the problem is of quadratic nature: DEQProgSolver.pl

(based on Schittkowski’s QLD), DEQPMATLABSolver.pl, DECFSQPSolver.pl,

DELSGRG2Solver.pl, and DESGRASolver.pl. To solve OPF problems: DECF-

SQPSolver.pl, DELSGRG2Solver.pl, and DESGRASolver.pl. There is one

more solver, although in experimental phase. It is called DEMIQPMATLABSolver.pl:

It is a Mixed Integer Quadratic Programming solver based on the MATLAB script

setconst_qp.m, and (sometimes) cannot take advantage of decomposition schemes,

because convergence is not always guaranteed. It can be tested with the files prob-

lem*_MIQP.dat provided in this distribution.

Also, the synchronization method to exchange information among subproblems is up to

the user. By default, the solvers will exchange information in a synchronous mode, i.e.,

τ

83

5.4. DistOpt Structure

DistOpt [21, 73, 74] is a distributed optimization application built in the Ptolemy environ-

ment as a universe. In DistOpt, blocks are modeled in the event-driven model of computa-

tion Discrete Event (DE) of Ptolemy. DE stars act as event-processing units, which

receive particles from the outside, process them, and generate output events after a user-

given latency. A particle carries a time stamp, generated by the block which produced the

particle, and represents an event corresponding to a change in a system state. The DE

scheduler processes events in chronological order, until the global time reaches a user-

specified ‘stop-time’. A star is executed (fired) whenever a new event appears at the input

portholes; after execution, output events may be generated at the output portholes of the

star. The user is referred to Chapter 3: Running DistOpt of [21] for a detailed description

of the DistOpt universe user interface. In this Section, we explain the overall structure that

supports the software implementation.

There are three levels of abstraction implemented in the overall design of DistOpt. The

first level of abstraction represents the mathematical formulation of the optimization prob-

lem. The user has to provide objective function and constraint coefficients in one input

file. The file contains the number of variables, number of equality and inequality con-

straints, and the coefficients of all the constraints and objective functions. The number of

variables and constraints that belong to each subproblem (and their respective indexes) are

also included. It can happen that some constraints in a particular subproblem contain vari-

ables from some other subproblems. It is possible because the user has decided to split the

number of variables, but she also splits the constraint set, and some of the variables may

need to be duplicated. Three applications are available by simply changing the file Meth-

ods.cc, and relinking. In particular: linear programming, quadratic programming, and

optimal power flow problems (non-linear quadratic programming application in power

systems). The input files are of the type problem*_LP.dat, problem*_quad.dat,

1. Ptolemy Version 0.6, released on April 1996, incorporates a new syntax manager: Tycho, that can be used as a texteditor, or as a GUI within Ptolemy. There are other features in the new release, like the use of itcl2.0 (an object-ori-ented version of Tcl/Tk), and new Tcl/MATLAB and Tcl/Mathematica interfaces for scripted runs and numeric/symbolic parameter calculations.

82

extensible. Users can create new component models, new design process managers, and

even entirely new programming environments. The core of Ptolemy is an object-oriented

toolkit and library that makes few assumptions about the system to be modeled; rather,

standard interfaces are provided for generic objects and more specialized, application-

specific objects are derived from these. In short, Ptolemy can be used to model complex

heterogeneous systems.

Ptolemy achieves its goals using the principle of polymorphism. The Ptolemy kernel,

written in C++, defines the basic classes that allow the components of the system to func-

tion together; application-specific objects are derived from these. Information hiding and

data abstraction are key to the design, so that is easy to modify; a key goal is that the sys-

tem be extensible in many dimensions without the need to modify the kernel.

Each model of computation in Ptolemy is called a Domain, an extensible library of func-

tional blocks. In Ptolemy, an overall description of a system is decomposed into software

modules called Blocks, the basic units of computation. Graphically, a block is an icon

with terminals, corresponding to its input/output Portholes. Blocks may have State

variables, which serve as user-settable parameters or as named variables that blocks can

modify and users can monitor. The blocks are invoked at run time in an order determined

by a Scheduler, and exchange data among themselves as they execute. The scheduler

determines the operational semantics of a network of blocks, so schedulers are different in

each domain. There are two types of blocks: the Star and the Galaxy. The star is ele-

mental, in the sense that is implemented by user-provided code. A galaxy is a block which

internally contains stars as well as possibly other galaxies. The galaxy is thus a construct

for producing a hierarchical description of the simulation. A Universe is a complete

program, or application. Data passes between blocks in discrete units called Particles,

which may be of many types -integer, real, complex, fixed-point, matrix, or a general

structure (a data packet, for example).

To get started, you can use the Ptolemy interactive graphical interface (pigi) or the

Ptolemy interpreter (ptcl)1. The interpreter has a textual command language based on

Tcl. Both interfaces are extensible using Tcl, and the graphical interface is extensible

using the associated X-Window toolkit Tk.

81

In this last scheme, the updating of and can be done in parallel. Equation (5-5)

always holds if:

(5-8)

The convergence results hold for the synchronous execution of the different computations

the algorithm is made of; if a processor is assigned the tasks regarding a given subprob-

lem, namely the solution of the minimization problem and the updating of the local copy

of the prices, at the iteration k it would need the value k-1 of the neighboring variables to

solve the problem, and the value k of the same variables to update the prices.

The main advantages of an APP approach are the following. First, the APP allows one to

replace the master problem by a sequence of auxiliary problems and, taking this opportu-

nity, to give particular desirable features to the latter problems (well conditioning, decom-

posability, etc.). Also, the APP approach not only allows one to give general proofs of

convergence for a large class of algorithms (Newton, Projected Gradient, Uzawa, Arrow-

Hurwicz, etc.), but it gives systematic guidelines for designing new algorithms. As far as

decomposition-coordination is concerned, sequential decomposition (see [69]) seems to

be useful for solving problems of a size larger than that one can deal with using the largest

computer facilities available nowadays. Parallel decomposition may be a way of achiev-

ing fast computation in on-line situations (for example, process control).

5.3. Ptolemy Environment

Ptolemy [22], programmed in C++, is an extensible object-oriented environment intended

as a general framework in which system specification, simulation, and design are possi-

ble. Several environments have been built, including dataflow-oriented graphical pro-

gramming for signal processing, discrete-event modeling of communication networks,

register-transfer-level circuit design, synthesis environments for embedded software, and

design assistants for hardware/software codesign. The Ptolemy system is fundamentally

pvk 1+

pvk ρ v

k 1+u

k 1+–( )+=

pu pv

pu0

pv0

–=

80

It is apparent that the two minimization problems in (5-4) can be solved in parallel, pro-

vided that the values of the duplicated variables and the multipliers of the previous itera-

tion are known.

Let us look in deeper detail into the coordinator’s task, represented in (5-4) by the updat-

ing of the Lagrange multipliers p associated with the coupling constraints . Note that

this updating is the solution of the coordinator’s problem represented by the maximization

of the dual function of the minimization problem. For the particular form of the coupling

constraints , the coordinator’s task can be duplicated in such a way that the minimiza-

tion subproblems to be solved in subspaces U and V have exactly the same form. To do so,

let vectors and be defined such that:

(5-5)

Combining (5-5) and (5-4), the two-level iterative scheme (5-4) can be written as:

(5-6)

and then

(5-7)

pk 1+

pk ρΘ u

k 1+v

k 1+,( )+ pk ρ u

k 1+v

k 1+–( )+= =

Θ

Θ

pu pv

p pu pv–= =

minu U∈

K1 u( ) εJ'u uk

vk,( ) K'1 u

k( ) u,–⟨ ⟩ ε puk

c uk

vk

–( ) u,+⟨ ⟩ uk 1+⇒+ +

minv V∈

K2 v( ) εJ'v uk

vk,( ) K'2 v

k( ) v,–⟨ ⟩ ε pkv

– c uk

vk

–( ) v–,+⟨ ⟩ vk 1+⇒+ +

puk 1+

puk ρΘ u

k 1+v

k 1+,( )+ puk ρ u

k 1+v

k 1+–( )+= =

pvk 1+

pvk ρΘ u

k 1+v

k 1+,( )– pvk ρ u

k 1+v

k 1+–( )–= =

minu U∈

K1 u( ) εJ'u uk

vk,( ) K'1 u

k( ) u,–⟨ ⟩ ε puk

c uk

vk

–( ) u,+⟨ ⟩ uk 1+⇒+ +

minv V∈

K2 v( ) εJ'v uk

vk,( ) K'2 v

k( ) v,–⟨ ⟩ ε pvk

c vk

uk

–( ) v,+⟨ ⟩ vk 1+⇒+ +

puk 1+

puk ρ u

k 1+v

k 1+–( )+=

79

adding convex/concave terms which can be chosen to be separable. Under some condi-

tions, the new problem (after applying the APP) has the same saddle-point as the original

problem. A description of an iterative algorithm to find the saddle point of this new prob-

lem can be found in [70].

By choosing the APP terms to be decomposable, the central step of this iterative algo-

rithm decomposes into independent parallel minimizations, followed by Lagrange multi-

plier updates which involve only local information transfers. After each step, the

processing elements exchange coordination information and re-solve the subproblems.

The idea is that several steps of this distributed iterative algorithm may produce a solution

faster than a more traditional centralized algorithm.

It is useful to our purposes to explicit the iterative scheme form (5-2), such that (5-1) can

be put in the following equivalent form:

(5-3)

subject to ,

where v is the duplication of some of the variables u of the original problem; this duplica-

tion can be made according to the wanted splitting of the problem into subproblems. It is

noteworthy that the technique of variable duplication generates couplings between vari-

ables of different subsystems that are always linear equalities, and then affine; in other

words, by this technique, we are always able to formulate problems where the decomposi-

tion is made such that the coupling constraints are exactly of the type the APP requires.

With a core function K additive with respect to u and v [ ], the

application of the scheme (5-2) to problem (5-3) yields:

(5-4)

minu U∈ v V∈,

J u v,( )

Θ u v,( ) u v– 0= =

K u v,( ) K1 u( ) K2 v( )+=

minu U∈

K1 u( ) εJ’u uk

vk,( ) K’1 u

k( ) u,–⟨ ⟩ ε pk

cΘ uk

vk,( ) u,+⟨ ⟩ u

k 1+⇒+ +

minv V∈

K2 v( ) εJ’v uk

vk,( ) K’2 v

k( ) v,–⟨ ⟩ ε pk

cΘ uk

vk,( ) v,+⟨ ⟩ v

k 1+⇒+ +

78

solved with the following two-level iterative scheme [69] ( stands for scalar prod-

uct):

(5-2)

where the auxiliary function K, named the core function, can be given particular charac-

teristics to allow a parallel solution of the auxiliary minimization problem; in particular, K

has to be additive with respect to the chosen variable decomposition. The global conver-

gence of the iterative scheme is granted upon the following conditions:

• J is convex, with derivative Lipschitz with constant A;

• is affine, with Lipschitz constant ;

• K is strongly convex, with constant b, and Lipschitz derivative;

•

• .

The restriction to the convex case allows us to have necessary and sufficient optimality

conditions, and also to obtain proofs for global convergence (i.e., starting from any initial

guess). Of course, anybody can use the algorithm in practical non-convex cases, but then

the necessary convergence conditions are only met in the limit.

Before using the APP, we duplicate some (or all) of the problem variables, so as to

decompose the constraints into purely local subsets. This produces an artificial decompo-

sition and an increase in the number of variables. In order to make sure the solutions of

the new problem and the original one coincide, we introduce consistency constraints

which force all duplicated copies of a variable to be equal at the solution. These equality

constraints are the only non-local constraints in the resulting transformed problem. Then

we apply the Auxiliary Problem Principle to the problem of finding a saddle-point of the

Augmented Lagrangian [72] of the transformed optimization problem. Informally, the

APP involves linearizing any non-separable terms in the Augmented Lagrangian, and

° °,⟨ ⟩

minu U∈

K u( ) εJ’ uk( ) K’ u

k( ) u,–⟨ ⟩ ε pk

cΘ uk( ) Θ u( ),+⟨ ⟩ u

k 1+⇒+ +

pk 1+

pk ρΘ u

k 1+( )+=

J’

Θ τ

0 ρ 2c< <

0 ε b A cτ2+( )⁄< <

77

By contrast, DistOpt uses a particular methodology called the Auxiliary Problem Princi-

ple [69] (APP), that consists of decomposing a large problem into several subproblems,

where each one can be solved using any optimization technique (decomposition), given

that some of the subproblems share common variables and need to exchange their values

during the iterative solving process (coordination). The advantage of using the APP

method shows when dealing with large problems where the speed of convergence and the

computation time can be extremely scattered, and often very large. DistOpt takes advan-

tage of the convergence properties [70] that the APP has, and splits a large problem into

many subproblems that are easier to solve and faster to converge to the optimum. The last

but not least advantage that DistOpt offers is the versatility in the decomposition phase,

where the user is free to use many optimization techniques from the solver’s palette. This

very fact makes DistOpt a very powerful software environment.

5.2. The Auxiliary Problem Principle and Problem Transformation

As explained in the preceding Section, the Auxiliary Problem Principle (APP) allows one

to find the solution of a problem (minimization problem, saddle point problem, etc.) by

solving a sequence of auxiliary problems. There is a wide range of possible choices for

these problems, so that one can choose suitable algorithms to make them easier to solve.

The frame is that of convex mathematical programming, although several applications of

the APP [71] have extended the convex range to non-convex scenarios. The general struc-

ture of an optimization problem can be expressed as follows:

(5-1)

subject to ,

where u is the vector of optimization variables, J(u) is convex, U is a closed convex feasi-

bility set for the quantities, representing typically both equality and inequality constraints,

is affine, and J represents the criterion of the optimization problem. (5-1) can be

minu U∈

J u( )

Θ u( ) 0=

Θ u( )

76

Chapter 5 DistOpt: A Distributed Optimization Software Environment

5.1. Introduction

DistOpt [21, 73, 74] is a distributed software environment for solving large scale optimi-

zation problems developed at the Department of Electrical Engineering and Computer

Sciences, University of California, Berkeley. At the present time, the software tools that

have been integrated in DistOpt for solving such problems include: Ptolemy [22]: A plat-

form for heterogeneous simulation and prototypes, CFSQP [64]: A C code for solving

constrained optimization problems, a C version of QLD [65]: A FORTRAN Code for

Quadratic Programming, SGRA [66]: A Sequential Gradient and Restoration Algorithm,

LSGRG2 1[67]: A Generalized Reduced Gradient software, and MATLAB 2: An envi-

ronment for high-performance numerical computations. The main objective of DistOpt is

to solve a variety of big size distributed optimization problems that can arise in several

fields, from network planning to power systems or transportation networks.

Traditionally, large scale optimization problems have been solved using classical tech-

niques. Newton’s method and its variations still stand as the main methodologies to solve

any nonlinear problem and the Simplex method is still widely applied in linear program-

ming. In the last few years, some other techniques have emerged, like Karmarkar’s

method, Interior Point method, or Genetic Algorithms, although recent advances are more

devoted to improve the speed of convergence than to develop new methods. The interested

reader is referred to [68] for a Genetic Algorithm implementation in the Ptolemy environ-

ment.

1. Copyright 1979-1986 by Winward Technologies, Inc., and Optimal Methods, Inc.

2. MATLAB is a trademark of The MathWorks, Inc.

75

egy that a player follows. For instance, agent 6, knowing his value in the game, will team

either with 2 or with 4 to reduce her own final cost.

We can recall that in our decentralized environment the offers are only bilateral, building

up to larger coalitions. By building only 2 member coalitions we are missing some combi-

nations that may be more beneficial to the agents. Therefore, not all combinations are con-

sidered, like in a SV centralized environment, like in a SV centralized environment, but

the coalition formation algorithm terminates in polynomial time.

It can be inferred that the skill of the agents’ skill to choose the right coalition is crucial to

get the largest slice of the allocation pie. By relinquishing his right to choose the right

coalition, an agent may end up with a value that does not reflect his true power in the

game.

Therefore, a decentralized approach like the one we propose, will reflect agents’

strengths, and it will also set a scheme for negotiation. Finally, were we stopping the

negotiation process, we could do it at any time, and calculate the BSVs for the current

scheme. This in turn provides a new approach in power transmission planning for self-

organizing regional cooperations within reasonable time.

74

Figure 4-4 : Core, Nucleolus, and Shapley Value Calculations

Comparing this result with the three decentralized simulations of the previous Section, we

can observe that there is a clear mismatch between agents 3 and 6 allocations. The reason

is that, when using pure SVs to allocate values, we consider all agents’ combinations: 3

becomes a valuable agent for some coalitions: 3,5, 1,3,5, and 1,2,3,6; but 6 is a

worthless agent: 2,5,6, 4,5,61. On the other hand, SVs are suited for superadditive

environments, which is not the case here. By contrast, using BSVs, only the best combina-

tions will form, and the concepts of “valuable” and “worthless” will depend on the strat-

1. See Table 4-1; a valuable player for a coalition refers to her positive marginal contribution.

73

72

information amongst coordinator and agents, neither among the agents themselves. The

coordinator’s role becomes more important in a centralized approach, then. He not only

calculates joint values, but also Shapley values for all the agents, given all possible combi-

nations, as shown in Table 3-2. Were we implementing that scheme, these would be the

allocation values: (16.85,-45.86,54.73,-19.41,-42.48,-93.81). Figure 4-3 shows it graphi-

cally.

Figure 4-3 : BSV versus SV Allocations

Note that the game that we are simulating does not have a core in general, so that the only

measure available to the modeler is the Shapley value, whose calculation was performed

in Mathematica , as it can be seen in Figure 4-4. The reader can observe the core, nucle-

olus, and Shapley value calculations in the same figure, as calculated using the software

program provided in [29].

71

Figure 4-2 : Overall Coalition Formation and Cost Allocation Procedure

A second simulation was run, since player 6 is indifferent whether to team with 2 or 4 at

the beginning of the game. The coalition process was: [1,2,3-5,4-6] -> [1-3-

5,2-4-6] -> [1-2-3-4-5-6]. The allocated values were given by: (21.25,-49.375,6.25,-

55,-33.75,-19.375).

The third simulation did not started from a grand coalition scheme, its order was: [1,2-

6,3,4,5] -> [1-2-6,3,4,5] -> [1-2-3-6,4,5]. The allocation was

(still total of -130 monetary units): (22.5,-48.75,15,-60,-40,-18.75).

For all simulations, agents 1 and 3 receive monetary units for their contribution to the wel-

fare of the system, while the remaining agents must contribute, although less that what

they would pay on their own. This is a clear incentive for all players to play this game.

4.5. Discussion

The main advantage that a centralized approach offers to the expansion game is its sim-

plicity. Using a central planner, we do not need a synchronized algorithm to exchange

70

Figure 4-1 : Cost Allocation Procedure

69

calculate value allocations via BSVs.: BIM (or Backward Induction Method). The process

is as follows:

ROUND 1: founders: 1-2-3-5-6 and 4 for 1-2-3-4-5-6, the grand coalition

BIM:

BIM:

ROUND 2: founders: 1 and 2-3-5-6 for 1-2-3-5-6

BIM:

BIM:

ROUND 3: founders: 2-6 and 3-5 for 2-3-5-6

BIM:

BIM:

ROUND 4: founders: 2 and 6 for 2-6; 3 and 5 for 3-5

BIM:

BIM:

BIM:

BIM:

The process ends here, because the values for individual agents are found: (12.5,-

49.375,10.625,-55,-29.375,-19.375). Note that only the first round uses BSVs in their

“classical” sense, i.e., in the second and subsequent steps of the BIM, the total value to

split is given by the previous BSV, instead of the value. Figure 4-1 and Figure 4-2 depict

the entire cost allocation and coalition formation processes for this example.

bsv 12356 4, 12356( ) 1 2v 12356( ) 1 2⁄+⁄ v 123456( ) v 4( )–( ) 75–= =

bsv 12356 4, 12356( ) 1 2v 4( ) 1 2⁄+⁄ v 123456( ) v 12356( )–( ) 55–= =

bsv 1 2356, 1( ) 1 2v 1( ) 1 2⁄+⁄ bsv 1 2356, 1 2356,( ) v 2356( )–( ) 12.5= =

bsv 1 2356, 2356( ) 1 2v 2356( ) 1 2⁄+⁄ bsv 1 2356, 1 2356,( ) v 1( )–( ) 87.5–= =

bsv 26 35, 26( ) 1 2v 26( ) 1 2⁄+⁄ bsv 26 35, 26 35,( ) v 35( )–( ) 68.75–= =

bsv 26 35, 35( ) 1 2v 35( ) 1 2⁄+⁄ bsv 26 35, 26 35,( ) v 26( )–( ) 18.75–= =

bsv 2 6, 2( ) 1 2v 2( ) 1 2⁄+⁄ bsv 2 6, 2 6,( ) v 6( )–( ) 49.375–= =

bsv 2 6, 6( ) 1 2v 6( ) 1 2⁄+⁄ bsv 2 6, 2 6,( ) v 2( )–( ) 19.375–= =

bsv 3 5, 2( ) 1 2v 3( ) 1 2⁄+⁄ bsv 3 5, 3 5,( ) v 5( )–( ) 10.625= =

bsv 2 6, 6( ) 1 2v 6( ) 1 2⁄+⁄ bsv 2 6, 2 6,( ) v 2( )–( ) 29.375–= =

68

However, there is no proof that the individual rationality condition expressed in

Equation 4-10 implies individual rationality of single agents in .

Note that Equation 4-7 is always met when the grand coalition is formed, since the super-

additive cover is also equal to the coalition value (cost) in that case. If it is the case that the

grand coalition is not formed, Equation 4-7 is also met, because final coalitions are indif-

ferent whether to team or stay solo: thus the maximum (minimum) possible value (cost) is

obtained, and it is also equal to . Note that in this particular case, Equation 4-9

is equal to zero. The reader can check this assertion for the cases: [1-2-3-4-6,5] and

[1-2-3-6,4,5]. Therefore, we can guarantee for a general environment that the pro-

posed algorithm meets the SOF that any cooperative fair cost allocation game has to sat-

isfy. For comparison with other definitions of SOFs, the reader is referred to Section 5.1.2.

of [24].

It is also possible to determine coalition stability using the “propensity to disrupt” concept

provided in [2]. Gately defined this concept as the ratio between how much a set of play-

ers in a coalition would lose if player refused to cooperate to how much would lose if

she refused to cooperate. For a bilateral coalition, the ratio is equal to 1 for both players,

as seen in (4-9). For any step of the algorithm, the players are “fairly” rewarded, since

their respective propensities to disrupt are equal.

4.4. Simulation Results

We have run three simulations for the Garver 6 bus test case implementing the BSV recur-

sive cost allocation procedure. In this Section, we explain in detail the first simulation, for

which the final resulting coalition is the grand coalition: 1,2,3,4,5,6, and the coalition

creation order is as follows: [1,2-6,3-5,4] in the first round, [1,2-3-5-6,4]

in the second, [1-2-3-5-6,4] in the third, and [1-2-3-4-5-6] in the fourth and last

round. Once the grand coalition is created we follow our backward induction algorithm to

C

v 123456( )

i i

67

where is the superadditive cover of a coalition:

(4-8)

That is, coalition divides itself up into subcoalitions in such a way that the sum of the

values of the subcoalitions is a maximum. Clearly, for superadditive games, the superad-

ditive cover of a coalition is the value of the coalition.

Equation 4-6 states that any negotiation group can command as a fair share at least the

value it would obtain were it to bind itself to a coalition. Equation 4-7 defines the entity

being divided into fair shares as the maximum joint payoff available in the game, i.e., the

superadditive cover of the grand coalition.”

In our case, the backward induction algorithm to allocate values (costs) is more specific,

because a coalition structure , and its sequence of bilateral assign-

ments (of coalition founders) is given

for each coalition formation round, i.e., the history of bilateral assignments for each coali-

tion structure is known. In particular, our coalition formation algorithm allows only bilat-

eral coalitions with individually rational founders in each round. Thus, given two players

and as founders of in some coalition structure (with

) it holds that:

(4-9)

It is guaranteed that at any step, agents and , as founders of the created coalition, are

individually rational, and the subgame is superadditive, as shown in Equation 4-9. Thus,

given a partition with a bilateral assignment , we have that:

for each , : (4-10)

v N( )

v N( ) maxP

v Pj( )j 1=

P

∑

=

P

CS C1 … Cn, , =

ce ceCS C1( ) … ceCS Cn( ), ,( ) ceCS Ci( ), 2= =

a b C CS ceCS,( )

ceCS C( ) a b, =

bsv a b, a( ) v a( )– bsv a b, b( ) v b( )–12--- v a b,( ) v a( )– v b( )–( ) 0≥= =

a b

P P1 P2, = ce

Pj P∈ Ck ce Pj( )∈ bsv Pj Ck( ) v Ck( )≥

66

4.3. Coalition Stability

Following the decentralized coalition formation algorithm shown in the previous Section,

it remains to be proved that the resulting coalition(s) is (are) stable. To say that a coalition

is stable means that none of its members desires to leave the group to obtain a better pay-

off. In our particular 6 bus example, there are only three sets of coalitions that can divide

the total cost in an optimal way:

1) Grand coalition: [1-2-3-4-5-6],

2) [1-2-3-4-6,5], and

3) [1-2-3-6,4,5]

All of them divide a total value (cost) of (-)130, but the negotiation process in this game is

not unique.

The stability method proposed by Aumann and Myerson [63] is applicable for some

cases; however it does not provide a general rule for stability. It is particularly valuable in

cases where the games present specific properties, as shown in Chapter 5 of [63].

For a more general criterion, we use a cooperative standard of fairness (SOF), as pre-

sented in Chapter 5 of [24]. Quoting Kahan and Rapoport, “a SOF for a game is a vector

valued function with elements , defined for each par-

tition of the players into negotiation groups . is a

nonempty negotiation group in the partition , and designates the fair share of

that group according to the rules generating the SOF. The SOF function is

assumed to satisfy two conditions:

for any and all (4-6)

for all (4-7)

t Ψ ψ1 P( ) ψ2 P( ) … ψt P( ), , ,( )=

P P1 P2 … Pt, , ,( )= n t 1 t n≤ ≤( ) Pj

P ψj P( )

Pj Ψ P( )

ψj P( ) v Pj( )≥ P Pj P∈

ψ1 P( ) ψ2 P( ) … ψt P( )+ + + v N( )= P

65

for each do

for each

do bcu ;

for each do

if then begin recutil:= ; break;

end;

bcu(S: Coalition, bsv: real);

begin

if then begin

:= ;

for each do

begin := ;

if then bcu

else := ;

end

end

else := ;

end;

Two examples with the 6 bus example are provided in the Simulation Results Section of

this Chapter.

C coaltuple 1[ ] BCLC∈∈

S bsvC S( ),( ) coaltuple 2[ ]∈

S bsvC S( ),( )

ai u,( ) ARUL∈

ai ax= u

S 1>

v S( ) bsv

K FS∈

ru bsvS K( )

K 1> K ru,( )

ARUL ARUL K ru,( ) ∪

ARUL ARUL S bsv,( ) ∪

64

Table 4-1. Coalition Expansion Costs

This recursive division of Bilateral Shapley Values within each coalition can be written as

follows:

1. :=

2. :=

3. :=

where is a set containing the history of coalition structures and monetary profits,

is the set of founders, and is the set of agent’s utility distribution.

Agent of coalition entity in a bilateral coalition

receives a monetary reward of the joint utility for :

:= recutil (4-5)

based on the functions

recutil(C: Coalition Structure, : SetCoalTuple, : Agent): real;

begin := ;

5,6 183 2,3,4,6 120

1,2,6 60 2,3,5,6 100

1,3,5 20 3,4,5,6 161

1,4,6 60 1,2,3,4,6 90

1,5,6 183 1,2,3,5,6 80

2,3,6 60 1,2,4,5,6 272

2,4,6 150 2,3,4,5,6 160

1,2,3,4,5,6 130

Coalition Cost Coalition Cost

BCLC C C1 bsvCkC1( ),( ) C2 bsvCk

C2( ),( ), ,( ) C C∈;

FS FS ceC S( ) S1 S2, = =

ARUL ai u,( ) u ℜ ai A∈,∈;

BCL F

ARUL

ax Ci Ck C ceC Ck( ),∈ Ci Cj, =

rux ℜ∈ bsvCkCi( ) Ci

rux C BCLC ax,,( )

BCLC a

ARUL ∅

63

Starting from the grand coalition, we divide the team into the two founding members: [1-

2-3-5-6,4], and split the total value of -130. Let be the value allocated to

agent using the Bilateral Shapley Value rationale:

(4-1)

(4-2)

where is the value of coalition , as per Table 4-1, when we reverse signs. Going

backwards one more step, we find the following values for the next subdivision of [1-2-

3-5-6] into their founders [1] and [2-3-5-6]:

(4-3)

(4-4)

The process is followed until the values for individual agents are found: (12.5, -49.375,

10.625, -55, -29.375, -19.375), adding up to -130 total final value. Note that in the second

and subsequent steps of the backward algorithm, the total value to split is given by the

previous Bilateral Shapley Value, as shown in Equation 4-3 and Equation 4-4, by using

instead of . This backward process can be used as a gen-

eral calculation scheme for every agent utility in its coalition.


2 90 2,5,6 334

4 60 3,5,6 101

5 40 4,5,6 304

6 60 1,2,3,6 30

2,6 90 1,2,4,6 120

3,5 40 1,2,5,6 273

4,6 60 1,4,5,6 243

bsv i j, i( )

i

bsv 12356 4, 2356( ) 1 2⁄ v 12356( ) 1 2⁄ v 12356 4,( ) v 4( )–( )+=

bsv 12356 4, 4( ) 1 2⁄ v 4( ) 1 2⁄ v 12356 4,( ) v 12356( )–( )+=

v i( ) i

bsv 1 2356, 1( ) 1 2⁄ v 1( ) 1 2⁄ bsv 1 2356, 1 2356,( ) v 12356( )–( )+=

bsv 1 2356, 2356( ) 1 2⁄ v 2356( ) 1 2⁄ bsv 1 2356, 1 2356,( ) v 1( )–( )+=

bsv 1 2356, 2356( ) v 1 2356,( )

62

The solution that we suggest will make use of bilateral agreements based on Shapley val-

ues, that are beneficial to the individuals, but we will also consider the “history” of the

coalition formation algorithm shown in the previous Chapter in order to reward or penal-

ize players that benefit or harm the overall system. In our game, a valuable player will be

rewarded accordingly to the benefit that he adds to the system and vice versa, a worthless

player will be penalized. This Chapter is devoted to finding such an allocation scheme.

4.2. The Backward Induction Algorithm

Defining a coalition formation scheme is not enough to solve the transmission expansion

game. Even if all agents agree on the final coalition arrangements, the issue of allocating

the total cost among them remains yet unsolved. To address this issue we propose a solu-

tion for a given cooperative game in general environments1 using

BSVs. Our method is based on a special algorithmic calculation of each agent’s utility

which is not necessarily , if , and is a founder of in

the coalition structure . In the following, we will denote coalitions with more than one

agent as (multi-parties) players.

To illustrate the algorithm let us return to our familiar 6 bus example test case. We know

from the previous Chapter, that the grand coalition was formed in the first simulation,

where we assumed that whenever an agent or multi-parties player was undecided, he

tossed a fair coin to decide. The resulting final coalition for the first simulation was the

grand coalition: 1,2,3,4,5,6, with the following formation stages: [1,2-6,3-5,4]

in the first round, [1,2-3-5-6,4] in the second, [1-2-3-5-6,4] in the third, and

[1-2-3-4-5-6] in the fourth and last round. Now, we will follow a backward-induction

algorithm method to calculate value (cost) allocations via BSVs.

1. General environments allow for both, superadditive as well as subadditive games. A game issuperadditive iff : ; it is subadditive iff it is not superaddi-tive.

C x1 … xn, ,( ),( ) A v,( )

A v,( )C∀ C1 C2 P A( )⊆∪= v C( ) v C1( ) v C2( )+≥

xi i 1 … m, , ∈( ) bsv ai Ck∈ Ck C

CS

61

Chapter 4 A Cost Allocation Method: Backward Induction Algorithm

4.1. Introduction

While cost allocation is an interesting problem, its relation to game theory is not obvious.

What does the division of costs (benefits) has to do with game theory? The answer is that

cost allocation is a kind of game in which costs (and benefits) are shared among different

parts of an organization. The organization wants an allocation mechanism that is efficient,

equitable, and provides appropriate incentives to its various parts. Cooperative game the-

ory provides the tools for analyzing these issues. Moreover, cooperative game theory and

cost allocation are closely intertwined in practice [60]. Some central ideas in cooperative

game theory have been presented in Chapter 2: core, kernel, and Shapley value standing

out as the most relevant solution concepts. However, new solutions are required in a

decentralized environment where the players are independent decision makers whose

decisions have important effects on the rest of them.

A decentralized framework, where the entities cooperate with the net effect of all individ-

ual actions converging to the best overall result, is the answer. A Multiagent System

(MAS) is our proposed solution, as introduced in Chapter 3, Section 3 of this dissertation.

However, the answer is not obvious, as Billera et al. point out in [61]; they propose a

unique solution, somewhat in the Shapley value fashion. We do not postulate such a solu-

tion in general, because the new power systems environment will not be centralized, and a

more sophisticated solution is required to guarantee a global optimum with decentralized

decision makers.

60

players attain via negotiation with BSVs. Furthermore, we also consider the players in a

network structure, so that a link between some players is certainly easier to build than

with other players. In other words, the grid and its players are intertwined, and one

player’s decision affects the other ones, and so their willingness to build a “link” with this

player. Our framework is the Multiagent System (MAS) environment, the coalitions are

based on rules that are based on power system criteria, and the decentralization (and coor-

dination) of all decisions fit the new deregulated environment in power systems planning.

59

if with 4:

4: solo: ;

if with 12356:

Status: 12356 prefers 4; 4 prefers 12356.

Coalitions after round 4: 12356 teams with 4.

FINAL RESULT: Grand coalition 123456 is formed!

A second simulation was run, since player 6 was indifferent to team with 2 or 4 at the

beginning. The sequential process was: [1, 2, 3-5, 4-6] -> [1-3-5, 2-4-6] -> [1-

2-3-4-5-6]. Note that not all simulations drove the coalition to a grand coalition scheme;

here is one example: [1, 2-6, 3, 4, 5] -> [1-2-6, 3, 4, 5] -> [1-2-3-6, 4, 5]. This was

a case where the sum of the costs of the final coalitions were also equal to the grand coali-

tion total cost.

3.8. Discussion

On the whole, the theories on coalition formation work well, but for very specific exam-

ples. They all follow game theory prescriptions of rationality to some extent at least.

Kahan and Rapoport [24] provided a critical review of ten experiments of three-person

cooperative games played in characteristic function form.

Interesting results are also provided by dynamic coalition formation models, as seen in

Chapter 2, Section 8. In particular, Komorita’s Equal Excess Theory is the only one that

introduces dynamics to the game, unifying both coalition formation and cost allocation

aspects in the same framework.

In the light of these results, for the transmission expansion problem, we propose that the

dynamics of the coalition formation must be determined by individual benefits that the

bsv 12356 4, 12356( ) 1 2⁄ v 12356( ) 1 2⁄ v 12356( ) v 4( )–( )+ 75–= =

v 4( ) 60–=

bsv 12356 4, 4( ) 1 2⁄ v 4( ) 1 2⁄ v 12356 4,( ) v 12356( )–( )+ 55–= =

58

if with 35:

35: solo: ;

if with 26: ;

if with 1:

4: solo: ;

if with 26:

Status: 1 prefers 35; 26 is undecided between 1 and 35 (tossing a fair coin, he prefers 35);

35 prefers 26; 4 is undecided between solo and 26 (tossing a fair coin, he prefers 26).


ROUND 3: players: 1, 2356, and 4

1: solo: ;

if with 2356:

2356: solo: ;

if with 1:

4: solo: ;

if with 2356:

Status: 1 prefers 2356; 2356 prefers 1; 4 is undecided between solo and 2356 (tossing a

fair coin, he prefers 2356).


ROUND 4: players: 12356 and 4

12356: solo: ;

bsv 26 35, 26( ) 1 2⁄ v 26( ) 1 2⁄ v 26 35,( ) v 35( )–( )+ 75–= =

v 35( ) 40–=

bsv 26 35, 35( ) 1 2⁄ v 35( ) 1 2⁄ v 26 35,( ) v 26( )–( )+ 25–= =

bsv 1 35, 35( ) 1 2⁄ v 35( ) 1 2⁄ v 1 35,( ) v 1( )–( )+ 30–= =

v 4( ) 60–=

bsv 2 46, 4( ) 1 2⁄ v 4( ) 1 2⁄ v 2 46,( ) v 26( )–( )+ 60–= =

v 1( ) 0=

bsv 1 2356, 1( ) 1 2⁄ v 1( ) 1 2⁄ v 1 2356,( ) v 2356( )–( )+ 10= =

v 2356( ) 100–=

bsv 1 2356, 2356( ) 1 2⁄ v 2356( ) 1 2⁄ v 1 2356,( ) v 1( )–( )+ 90–= =

v 4( ) 60–=

bsv 2356 4, 4( ) 1 2⁄ v 4( ) 1 2⁄ v 23456( ) v 2356( )–( )+ 60–= =

v 12356( ) 80–=

57

ROUND 1: players: 1, 2, 3, 4, 5, and 6

1: no choice

2: solo: ; if with 6:



5: solo: ; if with 3: ;

if with 6:

6: solo: ; if with 2: ;

if with 4: ;

if with 5:

Status: 1 has no choice; 2 prefers 6; 3 is undecided between solo and 5 (tossing a fair coin,

he prefers 5); 4 prefers 6; 5 is undecided between solo and 3 (tossing a fair coin, he pre-

fers 3); 6 is undecided between 2 and 4 (tossing a fair coin, he prefers 2).

Coalitions after round 1: 2 teams with 6; 3 teams with 5.

ROUND 2: players: 1, 26, 35, and 4

1: solo: ;

if with 26: ;

if with 35:

26: solo: ;

if with 1: ;

if with 4: ;

v 2( ) 90–= bsv 2 6, 2( ) 1 2⁄ v 2( ) 1 2⁄ v 2 6,( ) v 6( )–( )+ 60–= =

v 3( ) 0= bsv 3 5, 3( ) 1 2⁄ v 3( ) 1 2⁄ v 3 5,( ) v 5( )–( )+ 0= =

v 4( ) 60–= bsv 4 6, 4( ) 1 2⁄ v 4( ) 1 2⁄ v 4 6,( ) v 6( )–( )+ 15–= =

v 5( ) 40–= bsv 3 5, 5( ) 1 2⁄ v 5( ) 1 2⁄ v 3 5,( ) v 3( )–( )+ 40–= =

bsv 5 6, 5( ) 1 2⁄ v 5( ) 1 2⁄ v 5 6,( ) v 6( )–( )+ 81.5–= =

v 6( ) 60–= bsv 2 6, 6( ) 1 2⁄ v 6( ) 1 2⁄ v 2 6,( ) v 2( )–( )+ 30–= =

bsv 4 6, 6( ) 1 2⁄ v 6( ) 1 2⁄ v 4 6,( ) v 4( )–( )+ 30–= =

bsv 5 6, 6( ) 1 2⁄ v 6( ) 1 2⁄ v 5 6,( ) v 5( )–( )+ 101.5–= =

v 0( ) 0=

bsv 1 26, 1( ) 1 2⁄ v 1( ) 1 2⁄ v 1 26,( ) v 26( )–( )+ 15= =

bsv 1 35, 1( ) 1 2⁄ v 1( ) 1 2⁄ v 1 35,( ) v 35( )–( )+ 10= =

v 26( ) 90–=

bsv 1 26, 26( ) 1 2⁄ v 26( ) 1 2⁄ v 1 26,( ) v 1( )–( )+ 75–= =

bsv 26 4, 26( ) 1 2⁄ v 26( ) 1 2⁄ v 26 4,( ) v 4( )–( )+ 90–= =

56

Table 3-2. Coalition Expansion Costs1

The first simulation starts with the individual agents 1 to 6 creating lists of preferences.

We assume for the first two simulations that whenever a tie occurs between going solo or

teaming, teaming is preferred. The final resulting coalition is 1,2,3,4,5,6, and the order

of coalition formation is the following: [1, 2-6, 3-5, 4] in the first round, [1, 2-3-5-

6, 4] in the second, [1-2-3-5-6, 4] in the third, and [1-2-3-4-5-6] in the fourth and

last round. The step-by-step run of the algorithm is as follows:

1. E.g., the value (cost) for the grand coalition is -130 (130).

1. If a coalition is not included in Table 3-2, it means that its cost is zero.


2 90 2,5,6 334

4 60 3,5,6 101

5 40 4,5,6 304

6 60 1,2,3,6 30

2,6 90 1,2,4,6 120

3,5 40 1,2,5,6 273

4,6 60 1,4,5,6 243

5,6 183 2,3,4,6 120

1,2,6 60 2,3,5,6 100

1,3,5 20 3,4,5,6 161

1,4,6 60 1,2,3,4,6 90

1,5,6 183 1,2,3,5,6 80

2,3,6 60 1,2,4,5,6 272

2,4,6 150 2,3,4,5,6 160

1,2,3,4,5,6 130

55

Figure 3-4 : Overall Coalition Formation Framework

Finally, even if common resources are shared (the lines), it does not mean that all agents

can access them. In the case of multi-parties players, for example, load at bus 2 and gener-

ation at bus 6 forming a single agent, we disregard all lines connecting agent 2-6 to other

agents. For single agent expansion plans, the agent pays in full for all the lines that it

needs to meet its own load at a minimum cost.

3.7. Simulation Results

We have run a coalition formation simulation for the Garver 6 Bus test case implementing

the Bilateral Negotiation algorithm. Cost functions, as per Section 3.4, are shown in

Table 3-2. Note that when using SVs and BSVs, values are negative to reflect positive

expansion costs in monetary units. Note that the respective coalition values of this (subad-

ditive) game are negative, reflecting the utility of a coalition in a cost-oriented view1.

54

tions. These lists will change whenever step 3.6.3. is called again, with new multi-parties

players acting as agents1.

3.6.4. Bilateral Negotiation Phase

Each agent looks at the head of his ordered list, and extends an offer to the preferred part-

ner. The offer consists of sending the partner’s BSV: the value that he would attain for col-

laborating with the sender. If it happens that the sender receives also a message from the

preferred partner, and they both find it is beneficial to join, they do. They create a multi-

parties player that will behave like one agent from then on. Every other agent is also

informed, for him to erase the members of the multi-parties players from his own prefer-

ence list.

The coalition formation process is repeated by all agents and multi-parties players, start-

ing from Step 3.6.2, until no more coalitions are possible. If no coalition is possible at one

particular step, the agents look at the second best partner; if still not possible, to the third,

etc., until reaching the end of the list. Figure 3-4 shows the overall coalition formation

framework.

There are several features of this negotiation algorithm that are only relevant to expansion

planning. First, this is a general (non superadditive) environment, thus the grand coalition

will not necessarily be formed. Second, previous negotiation algorithms used a utilitarian

coalition building scheme, where the agents have to satisfy their tasks via cooperation,

thus increasing their benefits. The utilitarian coalition formation in the transmission

expansion domain is done in a cost-oriented view2.

1. This means that the agents in the coalition will vote for a representative agent that will be the spokesper-son for this coalition in the future.

2. Related work can be found in the DAI area for task-oriented domains with cost functions by Rosenscheinand Zlotkin.

53

that agents 1 and 3 are self-sufficient, because the load is met by the existing bus genera-

tion. However, agents 2, 4, and 5 need to use extra lines to meet their own demand.

Finally, agent 6 needs to be attached to the network, not to become isolated. These facts

are reflected in their self-values. In the case of different initial settings, agents 3 and 5

being a single initial agent for instance, costs would change, reflecting different initial

conditions.

3.6.2. Communication Phase

Once each agent has calculated its own self-cost, it is time to determine the joint cost that

he will have when cooperating with another agent. Unfortunately, an agent is not neces-

sarily aware of the environment surrounding him, thus a coordinator is needed to gather

this information. This is certainly the case of transmission expansion planning, where the

entire network is not known completely by any player. In order to calculate the BSVs, the

agents need to send all their proposed line addition(s) to this central figure, and then,

receive the adequate number of new lines for requested coalitions, via messages sent by

the coordinator to all of them. It is possible that two agents reach an agreement that is sat-

isfactory to both of them, but detrimental to the security of the system. That is why the

role of an independent coordinator is needed to check network reliability and quality of

service. Usually, a power flow subject to security constraints should be enough. On the

other hand, agents freely exchange self-costs with each other.

3.6.3. BSV Calculation Phase

Now the agents know their own self-values, and every possible cost with other coalition

partners via Step 3.6.2. After getting these messages from the coordinator, they proceed to

calculate BSVs if teaming with another agent. Then, the agents determine individually

rational1 lists of preferred agents: ordered list of local agents’s BSVs for two-entity coali-

1. Individual rationality means that the agent wants to have a new value that is, at least, as good as the onethat he could attain alone.

52

lition formation that they provided is also useful in the power transmission planning envi-

ronment. Thus, to formulate our problem using BSVs, let us define our framework first.

Let be a coalition structure on a given set of agents ,

where is a (bilateral) coalition of disjoint (n-agent) coalitions and

. The Bilateral Shapley Value for some coalition in a bilateral coalition is

defined as := . Both coalitions and are

called founders of , and denotes the self-value of coalition 1.

It can be seen that the founders will get half of their local contribution, and the other half

stemming from cooperative work with the other entity. The second term of the BSV

expression reflects the strength of each agent concerning his contribution, therefore avoid-

ing the “free-rider2” problem, so common in transmission expansion value allocation

schemes. The BSV is a particular case of the Shapley value concept, because they create a

fair distribution of resources among two agents only.

Our coalition formation method is based on the approach followed by Klusch and She-

hory in [7]. A summary of the steps in the coalition formation process is given in the fol-

lowing subsections.

3.6.1. Self-Calculation Phase

Each individual agent gathers information to determine its self-cost. Calculation of the

self-cost determines the monetary cost of line expansion for individual agents, following

the three axioms from the previous Section. It is possible that some of the players are

unwilling to use new lines on their own, and this fact is reflected by a cost of zero. For the

6 Bus Garver example, where each individual agent is attached to a bus, we can observe

1. Note that , and := .

2. The “free-rider” concept addresses the issue of new agents that take advantage of the work done by theexisting ones without paying them any compensation.

CS P A( )⊆ A a1 … am, , =

Ci Cj A⊆∪ Ci Cj

n 0≥( ) Ci C

bsv Ci Cj, Ci( )12---v Ci( )

12--- v C( ) v Cj( )–( )+ Ci Cj

C v C( ) C

bsv C ∅, C( ) v C( )= v ∅( ) 0

51

ley value is a cooperative game theory solution concept that calculates a fair division of a

common utility (money, resources, etc.) among the members of a coalition. It can be

defined as the weighted average of marginal contributions of a member to all possible

coalitions in which it may participate. It assumes that the game is superadditive (see Sec-

tion 2.3), and that the grand coalition is formed. The reader is referred to Section 2.6 and

[33, 58] for a detailed explanation of necessary conditions to calculate a meaningful Shap-

ley value.

The mathematical expression of the Shapley value, , is given as follows:

(3-4)

where

i = player

s = coalition of players

q = size of a coalition

n = total number of players

v(q) = characteristic function (cost savings) associated with coalition q.

c(q) = number of coalitions of size q containing the designated player i, given by,

(3-5)

In order to avoid the exponential complexity of a Shapley value calculation, Ketchpel

introduced the so-called Bilateral Shapley Value (BSV) [5]1. Klusch and Shehory [6, 7,

62] adapted this approach for a completely decentralized and bilateral negotiation process

among rational information agents using these values. In particular, the algorithm for coa-

1. Similar work has been done by Kraus and Shehory [59]. In contrast to our work, they did not provide asolution for cooperative games and a given coalition structure with other than 2-agents coalitions.

φi

φi1n---

1c s( )----------

q 1=

n

∑= v s( ) v s i–( )–[ ]i s∈

∑

c q( ) n 1–( )!n q–( )! q 1–( )!--------------------------------------=

50

Figure 3-3 : Two Examples of Autonomous Coalitions

For the 6 bus case, the set of feasible autonomous coalitions is as follows:

1 agent: 2, 4, 5, and 6

2 agents: 2,6, 3,5, 4,6, and 5,6

3 agents: 1,2,6, 1,3,5, 1,4,6, 1,5,6, 2,3,6, 2,4,6, 2,5,6, 3,5,6, and

4,5,6

4 agents: 1,2,3,6, 1,2,4,6, 1,2,5,6, 1,4,5,6, 2,3,4,6, 2,3,5,6, and 3,4,5,6

5 agents: 1,2,3,4,6, 1,2,3,5,6, 1,2,4,5,6, and 2,3,4,5,6

grand coalition: 1,2,3,4,5,6

3.6. Four Phases of the Coalition Formation Algorithm

The use of decision techniques to analyze DAI problems, like the one posed in the previ-

ous Section, started in the present decade. However, the Shapley value (SV) is known

since the seminal work by Lloyd Shapley (see Chapter 2, Section 6, and [33]). The Shap-

6 4

180

UNITS120240240

6 4

2

240

180

100

UNITS120240240

49

although these would be the “atomic” agents -the minimal ones. We also assume that any

set of generation units and loads attached to the same bus belong to a single agent. For our

familiar 6 bus example, we have a maximum of 6 agents, corresponding to the 6 genera-

tion/load entities attached to the buses.

A coalition of agents is a set of agents consisting of: at least one generator, one load, and

one transmission line. There are three axioms that a coalition has to satisfy:

1.- Generator(s) must meet the demand, i.e., the total generator output in the coalition

must be greater than or equal to the load.

2.- New and existing line(s) thermal limits can not be exceeded when running a power

flow for the new coalition.

3.- There must be one or more transmission lines, either existing or possible candidates,

connecting all the buses in the coalition.

These three axioms create what we call autonomous coalitions, because they can try

their own expansion plans without having to necessarily negotiate with any other similar

entity. Figure 3-3 shows some examples of feasible autonomous coalitions for the 6 bus

case. It can be anticipated from the set of axioms that some of the possible coalitions may

be ruled out. In particular, if there is a single bus in the system, with either a generator, a

load, or both, it can not be considered a coalition, because it lacks at least one transmis-

sion line. Nevertheless, we will assume that single agents will pay for line investments if

necessary, so that it is not necessary for them to own transmission lines at the beginning of

the expansion game. Also, if two buses meet the first axiom, and there are no line candi-

dates that can connect them, the second axiom is violated. Finally, if not all buses are in

the coalition connected to each other, the third axiom is violated.

48

tion rescheduling (generator outputs 1, 3, and 6 are 50, 165, and 545 MW respectively).

The optimal solution has a cost of 200 monetary units, and circuit additions are:

circuits, circuit, and circuits. Should we allow generation reschedul-

ing, i.e. the real power generation ranging from 0 to the maximum generation available

(150, 360, and 600 MW respectively), the optimal cost would be 1301 monetary units, and

circuit additions would be: circuits, and circuits. In the remainder of

the thesis we will assume that generation rescheduling is possible, disregarding dynamic

expansion plans for more than one year shot, as is the case in some models [56, 57]. Note

that these solutions correspond to what is defined in cooperative game theory as grand

coalition structure.

In the next Section we will analyze the rules of the transmission expansion game and the

construction of feasible coalitions.

3.5. Coalitions in Expansion Planning

In this Section we are going to define what is the “game” that is played in a transmission

expansion process (from a cooperative game theory standpoint), who are the players

(agents2), and what is a coalition in expansion planning.

The purpose of the game is the expansion of the transmission network, with the minimum

possible cost, subject to constraints given in (3-1), (3-2), and (3-3), and with a “fair” allo-

cation of the total cost among the agents.

An agent in the game can be either a generator, a load, or an independent third party (for

example, an independent company who owns transmission lines) that is physically

attached to a bus. A typical agent in this context is regarded as an independent entity: a

customer or group of customer loads, a generator or a set of generators, or a combination

of both. For simplicity, we do not consider for now fractional bus loads or generators,

1. Note that 130 is not the best possible result; the optimum value is 110, as shown in [54]. This deficiencyis due to the simplicity of the model, that does not explore all possible line combinations.

2. We will follow a Multi-Agent System (MAS) terminology, where players are called agents.

n26 4=

n35 1= n46 2=

n26 3= n35 2=

47

into account the effect of the power transmission cost, the line candidate with the largest

power flow is most effective in the expanded network1.

Figure 3-2 : Six Bus Problem

We have coded the transmission expansion problem in the Scilab computing environment

[55], and the solution obtained is the same as the one provided by Garver without genera-

1. See [53], pp. 394-400, for a very detailed explanation.

Bus 6 Bus 4

Bus 2

Bus 1Bus 5

Bus 3

240

180

80

40

3 UNITS

120

240

240

4 UNITS

30

30

30

60

4 UNITS

60

60

120

120

80

100

100

100

100

100

240

LEGEND

Existing line

Possible line addition

46

(3-1)

(3-2)

(3-3)

Table 3-1. 6 Bus System Circuit Data

where is the cost of adding line to the network, is the active power (in p.u.) flow-

ing through the added line , is the number of possible new lines, is the matrix

whose elements are the imaginary parts of the nodal admittance matrix of the existing net-

work, is the phase angle vector, is the transpose of the node-branch connection

matrix, is the flow vector for possible lines, is the nodal injection power for the

overall network, is a diagonal matrix whose elements are branch admittances, and

is the branch active power vector. The minimization algorithm is run recursively until

there are no overloads in the system. Since the objective function, Equation 3-1, has taken

Bus:from/to

Cost(units)

Suscept.(1/ )

Capacity(MW)

1/2 40 2.50 100

1/4 60 1.67 80

1/5 20 5.00 100

2/3 20 5.00 100

2/4 40 2.50 100

2/6 30 3.33 100

3/5 20 5.00 100

4/6 30 3.33 100

5/6 61 1.64 78

min12--- cjPj

2

j 1=

M

∑

BΘ KTPD+ P=

BLAΘ PL≤

Ω

cj j Pj

j M B

Θ KT

PD P

BL PL

45

Finally, Sheblé [51] is using a genetic algorithm to simulate combinations of simple strat-

egies for a small number of agents that act as traders in a ‘buy-sell’ game for bulk power

transactions. He is able to model power pool arrangements, not only bilateral trading.

Currently, he is extending his trading model to include futures trading and retail as well as

wholesale contracts.

3.4. Simplified Network Expansion Model

The objective of a mathematical model of a transmission expansion planning problem is

minimizing the capital and operating costs associated with the system expansion over the

planning horizon. The constraints associated with this model are the physical and eco-

nomical constraints that are important when attempting to expand a utility system at min-

imum cost, and yet meet all economic and demand restrictions that are placed upon the

system.

To formulate our problem, we will follow the simple model that Garver [14] used to solve

a six bus (a bus is equivalent to a node in communication networks) system, as shown in

Figure 3-2. Note that the dotted lines represent possible line additions, and the straight

lines are existing lines connecting buses. The buses in the network are numbered from 1 to

6: buses 1 and 3 have both generation and load supplies; 2, 4, and 5 are pure loads, and 6

is a new generation bus that needs to be connected to the network. The generation upper

limits for buses 1, 3, and 6, and line flows upper limits for existing lines are also shown.

The reader is referred to [14, 53] for further details.

When ranking possible additions to the system, we follow the heuristic approach sug-

gested by [18, 53], such that the original linear programming problem is transformed into

a quadratic problem subject to linear constraints. The general formulation of the expan-

sion problem can be expressed as:

44

the dynamics of the coordination and communication, it is a mere snapshot of a MAS. As

we can see in the figure, first the agents develop their own plans autonomously. Then, they

transfer their individual plans to each other: either via a coordination agent or a black-

board system. As a result of the coordination process, the individual plans are reconciled;

each agent adapts its own plan because of the existence of other agents with their own

plans. The result can be seen as a set of plans being integrated into one global plan;

although individual plans still exist.

Several software packages have been developed to simulate MASs. In particular, there are

two that deserve our attention. IDEAS [6] is an interactive development environment for

the specification and simulation of agent systems that is currently applied to simulate coa-

lition formation in transmission planning. The other package is called SWARM [50].

SWARM is a toolkit for the study of complex adaptive systems using MAS discrete simu-

lation. The basic unit of a SWARM simulation is an agent. Simulations consist of groups

of many interacting agents, called “swarms”, which may, themselves, be grouped into

more comprehensive swarms. The SWARM software package is currently used in a power

systems research project initiated by Ramesh in 1996, as cited in Wildberger [51].

Ramesh is investigating a multiagent approach to contingency analysis. He is building

individual agent solvers to solve the problem (still modeled as constrained nonlinear

flow); each of the agents attacks the solution in a different way. He expects that coopera-

tion and competition will achieve faster convergence than an agent alone.

Wildberger [52] suggests not only using the MAS model to solve specific power system

problems, but also to simulate the entire North American power grid. He has two pur-

poses: simulate the network as a MAS for real-time distributed control, and perform

‘what-if’ studies and computer experiments to get insight into the evolution of the power

industry under various forms of organization, including various degrees of deregulation,

competition, and unbundling of services.

43

3) Coordinating Plans of Autonomous Agents

We will assume that the agents are intelligent, autonomous (each agent wants to solve its

individual problem first) problem solvers. An autonomous agent has its own goals, inten-

tions, capabilities and knowledge. The agents will also be able to incorporate the actions

of other agents as part of their plans (multiagent planning). They will broadcast their plans

(always telling the truth) before execution in order to allow other agents greater access to

their anticipated future behavior.

Figure 3-1 : Coordinating Plans for Agents

The scheme of coordinating individual plans in a MAS is illustrated in Figure 3-1, as

shown in von Martial’s book [49]. This figure is somewhat simplified, because it neglects

AGENTS make individual, partial PLANS

negative and positive

relationships

Reaction/Proposal

Communication/negotiation

Plan

modification

Coordinated Plans

until free of conflictsand chances for

beneficial combinations

until consensus

42

transmission investment requirements. Under some simplifying assumptions, their

scheme reduces to a modified MW-mile rule. Tsukamoto et al. [3] have proposed a meth-

odology to allocate the cost of transmission network facilities if there are wheeling trans-

actions according to the transmission usage pattern. They incorporate MW-mile method

considering economies of scale.

3.3. Multiagent Environments in Power Systems

Any intelligent behavior is supposed to have the ability to coordinate and communicate.

Coordination is a way of adapting to the environment. In the area of Distributed Artificial

Intelligence (DAI), there is a branch called Multiagent Planning that studies intelligent

behavior of autonomous entities called agents within specific environments. An agent is a

fuzzy notion that can designate either a physical entity (human, computer, etc.) or a for-

mal entity (process, program). A Multiagent System (MAS) is a structure given by an

environment together with a set of artificial agents able to act on an environment and pos-

sibly to collaborate with each other and/or external agents. In any multiagent environ-

ment, the agents coordinate by modifying their intentions; the reason to coordinate are the

intentions of other agents. Communication is the way by which information relevant for

coordination is exchanged. The agent communicates their intentions, which then will be

negotiated among them. The agents can interact with each other at two different levels:

indirectly through the plans they intend to execute, and directly by communicating with

each other. As a consequence of this, we are mainly concerned with (see [49]):

1) Planning in Distributed Systems

2) Communication and Negotiation among Autonomous Agents: either to communicate

plans, or to reach agreements via negotiation.

3. The MW-mile method allocates costs of transmission plant according to the product of line length andMW flow, thereby incorporating distance in the rate.

41

In California, the rules for expansion of the grid are structured around the following broad

points (see the CPUC homepage at http://www.cpuc.ca.gov):

- The Independent System Operator (ISO) should not build or own transmission facilities,

to assure its independence. He should not take a financial interest, or be part of generation,

transmission or distribution. Thus, the owner of the transmission grid has the ultimate

obligation to build.

- The rules for transmission system expansion will depend upon whether the expansion

project is less expensive than the cost of other alternatives, or, alternatively, if it is driven

solely by reliability needs. In the first case, a marginal cost-based usage charge resulting

from congestion will send appropriate signals for transmission expansion, enabling the

expansion that is economically beneficial. In the latter case, the Transmission Owner (TO)

is obligated to build if market participants decide it is justified and are willing to pay.

To implement these broad principles, several procedures have been proposed:

- Determination of need of expansion, either for economic or reliability reasons

- State Approval and RTGs coordination

- Obligation to Build

- Cost Recovery Approval

7) Cost Allocation and Recovery: The question of how to allocate line investment costs

among the participants in a “fair” way is still open. Several allocation rules have been pro-

posed in the literature [47], such as postage stamp1, contract path2, short term marginal

costs, long term marginal costs, and MW-mile3. In particular, Marangon Lima et al. [48]

have proposed that each participant share has to be proportional to its impact on system

1. The postage stamp rates are a flat amount per kW.

2. The contract path method is based upon the assumption that a wheeling transaction is confined to flowalong a specified electrically continuous path through the wheeling company’s transmission system.

40

6) Responsibilities and Obligations: Under the current regulatory environment, building

of the transmission facilities is the obligation of utilities that provide electricity to custom-

ers in that region. In the future, RTGs will be set up to determine the best alternatives for

transmission additions, but this issue is still under debate. Some proponents of regulatory

reforms, like Smith et al. [44, 45], suggest that “the transmission network would be

owned jointly by retail power merchants, commercial and industrial customers, and, per-

haps, by the generating companies (case of New Zealand). The network as a joint venture

would be an operating company, run as shared resource cost center, not as a profit center.

It would be constituted as a competitively ruled property right system defined by the gov-

ernment, and not as ordinary shared ownership corporation. The property right rules

would have the objective of providing incentives for prudent investment, maintaining

competition as follows. Capital and maintenance cost for the transmission system would

be shared by the owners in proportion to their installed capacities to withdraw power

from, or to inject into, the network. Large customers, consortia of small customers, and

new generators would also have access to the grid, subject to the payment of their direct

connection costs and their share of grid capital and maintenance costs. Capacity expan-

sion of any part of the grid would be the responsibility of any user or consortium of users

willing to make the investment. In turn, the user(s) would obtain rights to the increased

capacities made possible by the expansion. Users who are not part of the capacity expan-

sion will have no rights to block the expansion or demand compensation if the result is to

shift the supply or demand for power in favor of others. But they will be free to join the

joint venture, to share the creation of more favorably located new grid rights. Network

capacity rights can be brought, sold, rented or leased subject only to antitrust laws that

apply to any other industry.” Smith et al. have even run simulations of their own auction

algorithm [45]; some other authors, like Wilson, propose a different auction mechanism

[46].

39

2) Externalities: externalities occur when the actions of one group of economic agents

have impacts on parties who are not directly participating in the given activity and who

are not compensated for its effects. They can be negative, where the impacts upon

involved third parties result in higher costs and losses. One example of negative external-

ity is the cost increase created by unintended power flows unrelated to a ‘contract path’.

Externalities can also be positive, where a benefit is produced for parties not involved in a

particular transaction. In a similar fashion, a synergy can be defined as the non-linear

effect of two or more projects or factors, where the sum of their benefits considered sepa-

rately is not equal to the benefits of all projects considered together.

3) Economies of scale: an economy of scale describes situations where quantity counts:

where the average cost of producing an item falls as the number produced increases. The

more of an item that a company makes, the less each item costs to make on average. One

typical example of economies of scale is a new projected transmission line triggered by a

specific generation project. Since the incremental costs of additional transmission capac-

ity are low, it may make economic sense to invest in additional capacity beyond the cur-

rent need of the specific project.

4) Lumpiness of investment: Traditionally, line investments have not been systematically

applied to the overall transmission network. In the past, certain areas of the network were

upgraded by the line owners with an absolute control over time and location of the invest-

ment. However, in a future open access environment, investments will be more complex:

line ownership issues and investment side effects will arise.

5) Reliability under TOA: Utilities will not be able to build facilities using the same crite-

ria for wheeling transactions as has been done before. With TOA, more simultaneous

transactions are expected to be going on across the transmission network. New techniques

must be developed to evaluate simultaneous transfer and to measure its impact on the reli-

ability of the entire grid.

38

casted competitive electric utility industry, the diverse parties using the transmission

system will be subject to new challenges. Even new figures, like the power marketer, an

entity that will make money on transmission transactions, guaranteeing security of the

grid, are expected to appear. The new competitive environment will have generators look-

ing for their own individual economic interests, while sharing an open access network.

The development of transmission will be seen differently by each of the participants, and

while a particular capacity expansion may be of interest to one party, there will be others

that may be against it. To alleviate that problem, setting up of Regional Transmission

Groups (RTGs) and sharing of information among not only other utilities, but also other

interest groups, like Independent Power Producers (IPPs), Exempt Wholesale Generators

(EWGs) or customer groups, has been suggested.

At the same time, Transmission Open Access (TOA) is already changing the traditional

concepts and approaches to providing electric service. TOA refers to the regulatory con-

struct (rights, obligations, operational procedures, economic conditions) enabling two or

more parties to use a transmission network that belongs totally or in part to another party

or parties.

As a result of all these issues, conflicts over transmission planning are due to happen, and

they can be typified according to their origin as follows [42, 43]:

1) Radial and network line expansion: radial connections involve the initial connection of

two participants where there was no prior interconnection, or the strengthening of a corri-

dor between two participants. Network projects involve the reinforcement of the power

grid. Capacity is usually added over an extended period in complex patterns between

many individual pairs of nodes in the network. A line can have both radial and network

characteristics, and that may complicate the classification of any expansion projects.

37

The commission will also adopt the establishment of a wholesale pool called the Power

Exchange (PX). This PX, and independent entity separate from the ISO, will provide a

market for electric power with hourly and half-hourly prices published to electric consum-

ers and other participants such as investors and power marketers. This marketplace will be

reliable, fast and based upon simple trading algorithms, but recognizing security con-

straints. Purchasing from and selling to the PX will be voluntary except for investor-

owned utilities who will bid all their generation output to the PX.

In the new market organization, California three largest IOU’s are to be divided into affil-

iated generation companies (gencos), which will be unregulated, and distribution compa-

nies (distcos) that will remain regulated. Ownership of transmission assets will pass to the

distribution companies, although control of these assets will be held by the ISO. The gen-

cos will be required to sell, and distcos required to purchase, all of their power through the

PX. Any other power producers or customers may also participate in the PX on a volun-

tary basis. The PX is considered one of potentially many “schedule coordinators” compil-

ing load balanced power transactions that would then be submitted to the ISO.

Eventually, the commission foresees the figure of an Independent System Planner (ISP)

that will effect transmission enhancements for the good of the system. As wheeling and

other forms of competition arise, however, transmission planning decisions, access poli-

cies, and prices for transmission services will be subject to more scrutiny both by regula-

tors and by competing parties vying for access to limited capacity. Therefore, conflicts

over transmission planning will arise, and they are the subject of our next Section.

3.2. Transmission Planning under the Open Access Paradigm

Traditionally, transmission system planning has been performed by the vertically inte-

grated firm, with limited regulatory review of transmission capacity additions. In the fore-

36

a problem. The CMLT model defines the notion of multilateral trade as a generalization

of a bilateral trade. “A multilateral trade is a trade involving two or more parties in which

the sum of generation minus the sum of consumption losses is equal to its share of losses.

The party that arranges the trade is called a broker. The broker may be a generator or

consumer involved in the trade but may also be an unrelated third party. But, in order to

relieve the congestion or to ensure security of operation, it is essential to have coordi-

nated trades involving three or more parties [10].” Wu et al. have proved in [10] that even

without a central authority, participants in the coordinated multilateral trade can reach

either an optimal point or a trade that is feasible and profitable.

In the United States, and, in particular, California, the new market structure has been sub-

ject to some preliminary ruling policies, as can be seen in California Public Utilities Deci-

sion 95-12-063 of Dec. 20, 1995, and in the modifications specified in D.96-01-009, of

Jan. 10, 19961. Under the new plan, the investor-owned electric utility industry in Califor-

nia will be restructured to allow for wholesale and retail competition beginning in 1998.

An independent system operator will operate the transmission systems that at present are

owned and operated by the three largest investor owned utilities (IOU’s) in the state:

PG&E, Southern California Edison Co., and San Diego Gas & Electric Co. The new regu-

lation foresees that the ISO will act as an independent, statewide transmission system

operator that will control and operate the state’s transmission system. The ISO is sup-

posed to ensure the feasibility of all the aggregated scheduled transactions and, if neces-

sary, ration transmission access in a “non-discriminatory” way. Although the utilities will

retain ownership of their transmission facilities, the ISO will have operational control of

the facilities.

1. For more information regarding the basic California plan, see the California Public Utilities Commission(CPUC) homepage at http://www.cpuc.ca.gov.

35

In the poolco model, the ISO performs all the system coordination functions plus the

operation of a centralized spot market for electricity. The claim of the poolco proponents

is that a competitive market equilibrium can be achieved, but it has been subject to contro-

versy.

The pool-based transmission bidding model explicitly incorporates the network external-

ity impacts into the competitive trading mechanism. In this model, Chao and Peck have

designed a market mechanism for electric power transmission that consists of tradable

transmission capacity rights and a trading rule that induces a socially optimal allocation.

Their approach allocates transmission rights based on contract paths, but sets explicit

trading rules which recognize the externality impacts. According to this model, a trans-

mission capacity right entitles its owner to the right to send a unit of power through a spe-

cific transmission line in an specific direction, and to collect economic rents associated

with the transmission line according to the trading rule. A fixed set of transmission capac-

ity rights are issued for each link (i,j), and these rights are tradable. This entitlement

mechanism only needs to define transactions between the base node n and every other

node in the network.

The trading rule consists of a set of coefficients , where

the value represents the quantity of transmission capacity rights on line (i,j) that a

trader needs to acquire in order to inject one additional unit of power at node k in the net-

work, and deliver unit of power at base node n1. The dynamic trading process

is (Lyapunov) stable and converges to a competitive equilibrium.

The coordinated multilateral trade model (CMLT) completely separates the power system

operation from the trading market functions. Economic efficiency is obtained through dis-

tributed decisions, and neither centralization (poolco) nor feasibility (bilateral trades) are

1. Let describe the system marginal transmission loss attributable to the injection of power at node k.

B t( ) βijk

t( ) 1 i j k n≤, ,≤⁄ =

βijk

1 λk t( )–( )

λ k t( )

34

responsible for ensuring that schedules for use of the transmission system are feasible,

balancing electricity demand and supply, operating the transmission system in real time,

and ensuring that all standards of transmission service are satisfied. The ISO should not

have any financial interest in the source of generation in order to guarantee open access to

the transmission grid.

This method provides economic incentives to find the most economic arrangements but its

fundamental flaw is the lack of coordination among the independent trades; this can lead

to a violation of transmission network constraints [10]. Facing that problem, the propo-

nents of bilateral trading propose to grant the ISO with the authority to determine the safe

level of generation and the power to curtail contracts.

The poolco model coordinates all electricity transactions in a single official spot market

called the “poolco”. Generators sell power in this spot market1 and distribution companies

purchase it from this market. All traders submit hourly bids for marginal cost of supply or

marginal value of demand2 and the ISO (the poolco also performs the functions of an

ISO) dispatches the market so as to minimize the cost of electricity supply, as reflected by

the bids, subject to security constraints, and sets nodal prices at the marginal bid price for

generation under optimal dispatch. To hedge fluctuations in the nodal price differences,

the poolco model defines Transmission Congestion Contracts (TCCs): contracts that pay

the right holder the price difference between the nodes specified by that right. Similarly,

Contracts for Differences (CFD) enable buyers and sellers to enter into a “financial bilat-

eral contract” that mitigates the risk of price variation. CFDs are financial contracts where

sellers compensate buyers when the spot price exceeds the contract price, and buyers

compensate sellers when the spot price is below the contract price.

1. A spot market is a market where sellers have their commodities on hand, and the goods are deliveredimmediately: on the spot.

2. [41] is an excellent reference to understand spot pricing in electric markets.

33

generators look for their own individual economic interests, while sharing an open access

network, is very different. The development of transmission is seen differently by each of

the participants, and while a particular capacity expansion may be of interest to one party,

they will be others that may be against it [37]. Transmission planning in a competitive

economic environment is clearly an emerging complex issue [42].

Because electric utilities are interconnected via the transmission network, together they

can provide improved reliability and lower overall generation costs. With the current

thrust towards a competitive generation market with new independent market entrants, the

correct “pricing” of transmission is crucial in providing the right signals to the market-

place.

Pioneering work on transmission pricing was done by Schweppe et al. Their conclusion

was that the short term value (i.e. price) of transmission services across a line is the differ-

ence of spot prices between the two nodes connected by that line [41]. The result illus-

trated that alternative geographic-based pricing methodologies such as the postage stamp

rates1 do not provide the correct market signals.

Among the many proposals that have been put forward, aiming at a new transmission

pricing scheme, four models have emerged as possible new electric market structures:

bilateral contracting [38], poolco [13, 39], pool-based transmission bidding [12], and

coordinated multilateral trading [10]. Whether one or a mix of these market models

will prevail as the future model is still uncertain.

Bilateral trading model is based on the principle that free market competition is a route to

economic efficiency. In this model, suppliers and consumers independently arrange con-

tracts setting the amount of generation and consumption under specific supply contracts.

The model introduces the key concept of Independent System Operator (ISO). The ISO is

1. The postage stamp rate method assumes that the entire transmission system is used in a wheeling trans-action, irrespective of the actual transmission facilities that carry the wheel.

32

Chapter 3 A Coalition Formation Algorithm in Transmission Planning

3.1. New Policies in a Deregulated Environment

The electricity industry is undergoing a deregulation process where the next regulatory

policies and industry future structure are still under debate1. From a traditional regulated

monopoly, the electric utility industry is shifting to a newer model, where competition is

allowed. Unbundling generation, transmission, and distribution for effective and efficient

competition, aims to break the current monopoly, but the way in which the electricity sys-

tem blocks (generators, separate grid owners, jointly owned power pool, and consumers)

will interact with each other is still under discussion. There are three main reasons that are

largely responsible for the difficulties of implementing competition [40]:

• Unlike other network industries, such as natural gas and telecommunications, the flows

of electricity across the network of interconnected power lines cannot be directed

• Electricity is a unique commodity in that it must be produced largely upon demand so

that blackouts don’t occur

• Development of efficient markets for competitive power will require regulatory

reforms and coordination by both state and federal regulatory agencies

In the traditional vertically integrated and regulated electric utility, generation and trans-

mission expansion plans are coordinated so that the network does not restrict optimal dis-

patch, the network itself is not overloaded, and specific desirable technical conditions are

met. The environment in an increasing competitive deregulated electrical sector, where

1. [40] is an excellent introduction to the new deregulated environment.

31

Transfer schemes cannot be regarded as complete theories of coalition formation for n-

person games in characteristic form, but they provide some insight of the dynamics of the

bargaining process. One shortcoming of this approach is the fact that transfer schemes

proceed from payoff vector to payoff vector within the same coalition structure, and that

seriously limits the practical applicability of these techniques.

2.9. Summary

We have introduced the basic tools of n-person cooperative game theory throughout this

Chapter. The terms payoff vector, coalition structure, characteristic function and imputa-

tion have been defined and applied in static solution concepts such as: core, stable set,

bargaining set, kernel, nucleolus and Shapley value.

Towards finding a general theory of coalition formation, we have presented the classical

equal excess theory model, and several transfer schemes for the bargaining set, kernel and

core.

In the next two chapters we will combine an innovative coalition formation scheme (bor-

rowed from Distributed Artificial Intelligence) with a new recursive cost allocation imple-

mentation, and we will show its application for Power Systems Transmission Planning.

30

excesses -see (2-17)- and maximum surpluses -see (2-18)-. If is the maximum surplus

of player over player , then a demand function can be defined as:

, if , (2-32)

,otherwise

where and are members of the same coalition. To insure convergence, the transfer is

limited by the demand, i.e., .

The transfer scheme for the core is due to Wu (1977) [36], and it is a variation of the tech-

nique used by Stearns in 1968. She proved that a sequence of payoff vectors will always

converge to the core if two conditions are set:

1) If a payoff is not coalition rational (i.e., has no positive excess), it needs a correction.

Given a payoff vector at a stage , if it is objected by some coalition with positive

excess with respect to it: , the members of can negotiate a new payoff vector

that insures that by defining an S-correction such that:

, for , (2-33)

otherwise.

2) If a payoff is not group rational -see (2-3)-, because, for example the claims for payoff

are too high, an N-correction is equally applied to each player’s payoff:

(2-34)

where is the total number of players and is the set containing all players.

With these two corrections, Wu proved that starting from an arbitrary payoff vector, this

could be altered in subsequent stages by applying corrections as needed, to converge to an

element of the core if the latter is nonempty.

skl

k l

dkl min skl slk–( ) 2 xl,⁄[ ]= skl slk>

dkl 0=

k l

0 α dk≤<

xj

j S

e S x,( ) 0> S

xj 1+

xj 1+

S( ) v S( )>

xij 1+

xij e S x

j,( )s

------------------+= i S∈

xij 1+

xij

=

xij 1+

xij e N x

j,( )n

-------------------–=

n N

29

Although there are no coalition formation theories that can be empirically tested so far,

the existence of transfer schemes is known since 1968 (Stearns [35]). Theses schemes can

be used to construct sequences of payoff configurations for n-person games.

A transfer scheme can be defined as a sequence of proposals, where players start negotia-

tions at some arbitrary allocation of a given coalition structure, and proceed sequentially

to transfer money from one player to another according to some “formula”.

A transfer scheme can be expressed therefore as a sequence of payoff configurations

in which the payoff vectors can change, but the coalition

structure remains the same. The change in payoff vectors is due to the transfer, of size

, from player to player at stage ; given some , , the next mem-

ber of the sequence is given by

,

, (2-31)

for all .

Transfer schemes can be applied to several of the static payoff allocation theories we have

seen, in particular to the bargaining set, kernel, and the core. The transfer scheme for the

bargaining set is based on the idea of objection -see (2-15)- and counterobjection -see (2-

16)- such that for a payoff allocation: if within a coalition , a player has a justified

objection against another member of the same coalition, then player makes the (mini-

mum) necessary transfer to a member of (not necessarily player ) so that player no

longer has a justified objection against player . Stearns proved that this sequence will

converge to a payoff configuration that is in the bargaining set.

For the kernel, there are different conditions for the transfer scheme, although converging

to the kernel as proved again by Searns (1968). Now, the transfer scheme employs

x0

S;( ) x1

S;( ) … xj

S;( ) …, , , , x

S

α l k j xj

S,( ) j 0 1 …, ,=

xj 1+

S,( )

xkj 1+

xkj α+=

xlj 1+

xlj α–=

xij 1+

xij

= i k l,≠

S k

l l

S k k

l

28

Based on these expectations, each player builds a priority list and coalitions are formed

whenever expectations are jointly maximized for all players in the coalition. If no coali-

tion is formed, the expectations in round r are used for the next round, r+1. For a player i

and coalition S, the highest expectation is given by:

(2-29)

Now, to create a new expectation for round r+1, each player claims its best expectation

from round r, and the remainder of the sum of these claims is divided into equal parts, i.e.:

. (2-30)

Then, each player has a new expectation for each coalition he is a member for the next

round of negotiations.

Unlike most of the theories we examined earlier, equal excess makes predictions about

coalition structures by ranking coalitional preferences based on player’s expectations for a

given round. However, several problems that arise are that some of the definitions are

ambiguous, there is no algorithmic formula for given only the game and round

number, and the construction of all possible alternative coalitions is not clear, except for

very simple games. That is why Komorita’s theory is a good step towards a general theory

of coalition formation, but the ambiguity of some terms makes it not very useful.

2.8. Dynamic Coalition Formation Models: Transfer Schemes

The previous theoretical models, except for Komorita’s Equal Excess Theory, were static

in nature. They did not address the negotiation dynamics among the players. They all

assumed that the final coalition or coalitions were already created and then allocated the

total payoff according to different rules.

Ar

i S,( )

Ar

i S,( ) maxT S≠

Er

i T,( )[ ]=

Er 1+

i S,( ) Ar

i S,( ) v S( ) Ar

j S,( )j S∈∑– s⁄+=

Er

i S,( )

27

2.7. Equal Excess Theory Model

So far we have been concerned with models that predict the allocation of payoffs among

the members of a coalition once it has been formed. Historically, cooperative game theory

has studied this aspect in more detail, until approximately twenty years ago.

A newer field of study emerged in the last decades, and it was called coalition-prediction

theory. From the practical point of view, the question now was which particular coalitions

are going to form. The strong requirement that was imposed historically on the character-

istic function was its superadditivity property -see equation (2-2)-. However, in real life,

the grand coalition, which was the implicit result of a superadditive game, is not always

formed, and there is a need to address the question of which particular coalitions will

form.

Among the theories of coalition formation, Equal Excess Theory (Komorita, 1979) [34]

stands out because it also deals with payoff division. At the beginning of the game, all

players expect equal shares in any coalition that they are part of, and they also want to

maximize their own benefit. For the next rounds, the theory predicts that each player

expects a share equal to the payoff in the best available coalition the player may poten-

tially join. As the negotiation proceeds, player’s expectations change because of the dif-

ferent coalitions that they may join. Any excess that is left over after the players receive

what they expect is divided equally among them. More formally, assume that each player

enters the game assuming equal share of each coalition he belongs to, and define a round

as the rth discrete stage of the bargaining process where each player has

an expectation of what he will obtain from each coalition he is member of. This expecta-

tion is called subsequently. Komorita also assumed that

and .

r 0 1 …, ,( )=

Er

i S,( ) Er

i S,( )i S∈∑ v S( )= r∀

S N⊆∀

26

(2-27)

where

(2-28)

and is the number of players in , and for , .

For the already familiar three person game, the resulting Shapley value is: 50, 45, 40,

very similar to the nucleolus solution.

The interpretation of equation (2-27) is that player i’s reward should be the expected

amount that player i adds to the coalition made up of the players who are present when he

or she arrives, given that equation (2-28) is the probability that when player i arrives, the

players in the subset S will be present. With his derivation of the value of a game, Shapley

presented a combinatorial procedure for playing it. In his procedure, a single players starts

the game alone. Then, players are added, one at a time, until n players have been admitted

and the grand coalition is formed. The order of arrival is given by a chance mechanism, as

shown in (2-28), that considers all permutations of the players to be equiprobable. Each

player, after joining the coalition, receives the full amount that his entry has added to the

coalition, as shown in (2-27).

The Shapley value has some attractive features, one of which is the fact that it yields a

unique point for every game and coalition structure. The axiomatic approach based on

somewhat weak assumptions provides a good framework and relatively easy calculations.

However, it does not reflect the strengths of the players in the game, like the core or the

bargaining set, and it is not suitable for decentralized games. Nevertheless, an extension

of it, the bilateral Shapley value, is a viable alternative, as will be seen in the next Chapter.

xi pn S( ) v S i ∪( ) v S( )–[ ]S i S∉∀∑=

pn S( ) S ! n S– 1–( )!n!

---------------------------------------=

S S n 1≥ n! n n 1–( )…2 1( )= 0! 1=( )

25

; (2-22)

(2-23)

(2-24)

(2-25)

(2-26)

2.6. Shapley Value

The Shapley value is an a priori worth to each player of playing a game that is a function

of the characteristic function and any coalition structure . For any characteristic func-

tion, Shapley (1953) [33] showed that there is a unique payoff vector

satisfying the following axioms:

Axiom 1: Relabeling of players interchanges the player’s payoffs. In other words, the

value of a game does not depend on the “names” of the players.

Axiom 2: The sum of the worth of each player of every coalition in a given coalition struc-

ture equals the value of the coalition, i.e., group rationality: .

Axiom 3: If holds for all coalitions , then the Shapley value has

: if player adds no value to the coalition, he receives a payoff of 0 from the Shap-

ley value.

Axiom 4: Let x be the Shapley value vector for game , and let y be the Shapley value vec-

tor for game . Then, the Shapley value vector for the game is the vector .

If Axioms 1-4 are valid, Shapley proved the following result:

The payoff (or reward) of the ith player is given by

emin x i( )≤ i 1 2 3, ,=

emin x 1( ) x 2( ) v 12( )–+≤

emin x 1( ) x 3( ) v 13( )–+≤

emin x 2( ) x 3( ) v 23( )–+≤

x 1( ) x 2( ) x 3( )+ + v 123( )=

v S

x x1 x2 … xn, , ,( )=

xi

i 1=

i n=

∑ v N( )=

v S i –( ) v S( )= S

xi 0= i

v

v v v+( ) x y+

xi( )

24

The nucleolus is a solution concept introduced by Schmeidler in 1969 [32]. It is a concept

related to the kernel, because it also uses the idea of an “excess”. It combines the idea of

stability of the core and the equity of the Shapley value, as we will see in the next Section.

One advantage of the nucleolus is that every game has one and only one nucleolus, and

unless the core is empty, the nucleolus is in the core.

The basic concept is to find an imputation that makes the most unhappy member of a

potential coalition happier than the most unhappy member of a potential coalition under

any other imputation; in other words, the nucleolus is the imputation for which the maxi-

mal excess is minimized. It can be expressed as

, where , over all coalitions S (2-21)

The nucleolus for the three person example shown in Figure 2-1 is the point: 55,45,35,

and it is inside the core, as it can be seen in Figure 2-2. The corresponding equations are

shown below:

Figure 2-2 : The Nucleolus of a Three Person Game.

Maximize subject to:

max emin emin min xi v S( )–i S∈∑

=

emin

23

(2-17)

i.e., it is the amount by which the worth of a coalition exceeds its preferred payoff, or, in

other words, the total amount that the prospective members of coalition S collectively gain

or lose (depending on the sign of the excess) if they withdraw the imputation to form

coalition S.

Now, consider two distinct players k and l within a coalition S, and all alternative coali-

tions that include k but exclude l. Each of these coalitions will have an excess given by

equation (2-17); choose the largest of these excesses and call it the maximum surplus of

player k over player l, i.e.:

(2-18)

The next step is a comparison of the maximum surpluses of player k over player l and vice

versa. If both

and (2-19)

player k is said to outweigh player l with respect to for coalition S. If neither player out-

weighs the other, the two are in equilibrium. Equilibrium conditions exist between the two

players if any of the following three conditions is satisfied:

or and or and (2-20)

Note that equilibrium is defined only for pairs of players who are members of the same

coalition.

The definition of kernel follows immediately from the above; the kernel is the set of all

imputations such that any two players are in equilibrium for any S. In short, the

kernel solution reaches equality of “strengths” by all pairs of players within a common

coalition.

e S x,( ) v S( ) xi

i S∈∑–=

x

skl maxS k S∈ l S∈,

e S x,( )=

skl slk> xl v l( )>

x

skl slk= skl slk> xl v l( )= skl slk< xk v k( )=

x k l S∈,

22

; and , for all ; ; ; (2-15)

In the objection against l, player k can gain more in the new coalition Y, without l,

and his claim is reasonable because is possible through , and each member of coa-

lition Y gets at least as much as what he could obtain in the original imputation .

Now, being an objection of player k against player l, as defined by equation (2-15),

for a coalition Z, and an allocation of of its value to its members, the pair is

called a counterobjection to the objection if:

; for all ; and for all ; ; (2-16)

By making a counterobjection to player’s k objection, player l claims that he can maintain

al least a payoff equal to by giving each of the other members of the coalition Z at least

their payoff. Moreover, if any member of Z is also a member of coalition Y proposed in

the objection, that player gets at least what he could have received in . An objection

is said to be justified if no counterobjection to it exists. Otherwise, it is said to be unjusti-

fied. If we also define that a bargaining point of a game has the property that for each pair

of players i, j, any objection of i against j can be met by a counterobjection by j against i,

the bargaining set is the set of all bargaining points.

2.5. Excess Theories: Kernel and Nucleolus

The kernel as a solution concept was introduced by Davis and Maschler in 1965 [31], and

it is based on pairwise comparisons of all players in a game in terms of the “excess” pay-

off that one of the players could have by forming a coalition that excludes the other.

Therefore, we need to define first what is the excess of a coalition. By the excess of a coa-

lition S at an imputation we mean:

yi

i Y∈∑ v Y( )= yk xk> yi xi≥ i Y∈ k l S∈, k Y∈ l Y∉

y Y;( )

y v Y( )

x

y Y;( )

z v Z( ) z Z,( )

y Y;( )

zi

i Z∈∑ v Z( )= zi xi≥ i Z∈ zi yi≥ i Y Z∩( )∈ l Z∈ k Z∉

x

y Y;( )

x

21

tion concept more generally applicable than the core. That proposal is called the von

Neumann-Morgenstern solution or the stable set.

The stable set is based in the concept of dominance, that is explained as follows. One

imputation is said to dominate another if there is a subset of players who all prefer the first

to the second and can enforce it by forming a coalition. Mathematically, imputation

is said to dominate imputation if there exists

some subset S of players for which the following two inequalities are satisfied:

for every ; (2-13)

for . (2-14)

The first inequality states that every player in the subset S prefers the payoff offered by the

imputation ; the second inequality states that members of the subset S have the power to

form the coalition and they are in a position to enforce their preference, because the value

of the game to the coalition is at least as high as the sum of the payoffs with imputation .

Turning back to the original stable set solution by von Neumann and Morgenstern (1947),

a solution of the game has to satisfy two conditions:

• Internal stability: None of the imputations in the stable set dominates any other one

inside the set.

• External stability: There is at least one member of the stable set that dominates any

other imputation outside the set.

The bargaining set is a concept originally due to Aumann and Maschler (1964) [30]. To

properly define this concept, we need to introduce the terms objection and counterobjec-

tion, similarly to the concept of domination in stable sets. Let S be a coalition containing

at least two players k and l: given an imputation that does not satisfy player k, for a coa-

lition Y and an allocation of of its own value among its members, the pair

is called an objection of k against l via coalition Y if:

y y1 y2 … yn, , ,( )= x x1 x2 … xn, , ,( )=

yi xi> i S∈

yi v S( )≤∑ i S∈

y

y

x

y v Y( ) y Y;( )

20

payoffs to all of its members. However, there are many games without a core, coreless

games, or empty core games, precisely because (2-7) is not satisfied.

Figure 2-1 : The Core of a Three Person Game.

When the core is nonempty, the cooperative demands of every coalition can be granted,

but when the core is empty, at least one coalition will be dissatisfied.

Shapley and Shubik (1973) noted that a game with a nonempty core is sociologically neu-

tral, i.e. every cooperative demand by every coalition can be granted, and there is no need

to resolve conflicts. On the other hand, in a coreless game, the coalitions are too strong for

any mechanism to satisfy every coalitional demand. However, a core set with too many

elements is not desirable, and it has little predictive power [24].

Imputations in the core, where they exist, have a certain stability, because no player or

subset of players has any incentive to leave the grand coalition. But since many games

have empty cores, the core fails to provide a general solution for n-person games in char-

acteristic form1. Von Neumann and Morgenstern (1944) [27] proposed a different solu-

1. Any game can be expressed either in extensive (spanning tree) form or strategic form (matrix), moresuited for noncooperative n-person games, or in characteristic form (using Characteristic functions), as isshown in this Chapter.

19

More precisely, the core, introduced by Gillies in 1953 [28], is the set of imputations sat-

isfying individual, collective, and coalitional rationality:

(Group rationality) (2-5)

(for each ) (Individual rationality), and (2-6)

for all , for all (Coalitional rationality) (2-7)

We can also define the core as the set of all undominated imputations. For a simple super-

additive 3-person game: 1, 2and 3, this may be expressed as the following set of inequali-

ties (let in this particular case):

, , , (2-8)

, (2-9)

, (2-10)

. (2-11)

(2-12)

Equation (2-12) is equivalent to the familiar Pareto-optimum, which requires that no

member’s share can be increased without decreasing the share to another member. Giving

numerical values to the inequalities, such that: ; ; ;

, the core restricted to coalition (123) is a triangle bounded by the points in

the payoff space (65,55,15), (65,25,45), and (35,55,45). Figure 2-1, generated in Mathe-

matica 1 [29], exhibits this triangle pictorially.

This solution concept has remained popular ever since its introduction in 1953 by Gillies.

It is the simplest and most persuasive of all cooperative solution concepts. Essentially, it

consists of the set of imputations which leave no coalition in a position to improve the

1. Mathematica is a trademark of Wolfram Research, Inc.

v N( ) xi

i 1=

i n=

∑=

xi v i ( )≥ i N∈

xi v S( )≥∑ i S∈ S N⊂

v 1( ) v 2( ) v 3( ) 0= = =

x 1( ) 0≥ x 2( ) 0≥ x 3( ) 0≥

x 12( ) v 12( )≥

x 23( ) v 23( )≥

x 123( ) v 123( )≥

x 1( ) x 2( ) x 3( )+ + v 123( )=

v 12( ) 90= v 13( ) 80= v 23( ) 70=

v 123( ) 135=

18

,

whenever and are disjoint coalitions of players.

Games in which at least one possible coalition can increase the total payoff of its mem-

bers are called essential, and those in which there is no coalition that improves the total

payoff are called inessential. Mathematically, an essential game is one in which at least

one of the superadditive inequalities (2-2) is strict.

Assuming superadditivity and collective rationality, the grand coalition will form at the

end of the game. However, how will the players divide the joint payoff? The division of

the joint payoff , represented by the payoff vector is not evi-

dent. A payoff vector will not be a reasonable candidate for a solution unless it satisfies

(Group rationality) (2-3)

(for each ) (Individual rationality) (2-4)

If satisfies (2-3) and (2-4), we say that is an imputation. Equation (2-3) states that

any reasonable payoff vector must give all the players an amount that equals the amount

that a grand coalition would attain. Equation (2-4) says that player must receive a payoff

al least as large as what he can get by himself: .

2.4. Core, Stable Set, and Bargaining Set

The notions of group and individual rationality introduced in equations (2-3) and (2-4)

suggest that there may be a solution of the game including all possible coalitions of play-

ers as well. If we add coalition rationality to an imputation, i.e. rationality for every subset

of players, we have defined a new solution concept: the core.

v S T∪( ) v S( ) v T( )+≥

S T

v N( ) x xA xB … xn, , ,( )=

v N( ) xi

i 1=

i n=

∑=

xi v i ( )≥ i N∈

x x

i

v i ( )

17

for all , and (2-1)

.

These conditions state that each player belongs to one and only one of the m nonempty

coalitions within the coalition structure, and also specifies that none of the players in any

coalition m is connected to other players not in the coalition, finally, the mutually exclu-

sive union of all coalitions m forms the grand coalition.

2.3. Characteristic Function and Imputation

Von Neumann and Morgenstern (1944) [27] introduced the term characteristic function,

denoted , for the first time. More formally, we can define that:

Definition: For each subset S of N, the characteristic function of a game gives the big-

gest amount that the members of S can be sure of receiving if they act together and

form a coalition, without any help from other players not in S.

A restriction on this definition is that the value of the game to the empty coalition is zero,

that is, . A further requirement that is generally made is called superadditivity.

Superadditivity can be expressed as follows:

for all S, such that (2-2)

This means that the total payoff for the grand coalition is collectively rational, because the

total payoff to the players is always as much as what they would get individually. This

suggests the following definition.

Definition: A game in characteristic function form consists of a set

of players, together with a function , defined for all subsets of , such that

Si Sj∩ ∅= i j≠

Sj

∈∪ N=

v

v

v S( )

v ∅( ) 0=

v S T∪( ) v S( ) v T( )+≥ T N⊆ S T∩ ∅=

P P1 … PN, , =

v P

16

‘linear utility’, ‘transferable utility’, ‘comparable utility’, or indeed any utility whatso-

ever [pp. 14-15].”

The implication of this argument is that we can calculate any solution of a game in units

of money, provided there is no distortion of the solution that would have been obtained

had we used the utilities.

Although players in a game are autonomous decision makers, they may have an interest in

making binding agreements in order to have a bigger payoff at the end of the game. This

agreement or partnership is the basic ingredient of the mathematical model of a coopera-

tive game, and it is called a coalition.

Mathematically, a coalition is a subset of the set of players N and we can denote it by S.

To form a coalition S, it is required that agreements take place involving all players in the

future coalition S. Whenever all players approve joining in a new entity called coalition,

we can say that the new coalition is formed. Joining a coalition S also implies that there is

no possible agreement between any member of S and any member not in S (set N-S). In

short, the essential feature of a coalition is its foundational agreement that binds and

reconstitutes the individuals as a coordinated entity. We denote a specific coalition by a

concatenation of its members, for example, coalition AB will refers to players A and B

acting as one decision making unit. The grand coalition of all n players will be referred as

coalition N; in a game of n players there are possible coalitions. The empty coalition

is a coalition made up of no members (the null set ).

A coalition structure is a means of describing how the players divide themselves into

mutually exclusive coalitions. Any exhaustive partition of the players can be described by

a set of the m coalitions that are formed. The set S is a partition of

N that satisfies three conditions:

,

2N

∅

S S1 S2 … Sm, , , =

Sj ∅ j,≠ 1 … m, ,=

15

Cooperative games allow players to split the gains from cooperation by making side-pay-

ments: transfers among themselves that change the prescribed payoffs. Cooperative game

theory incorporates commitments and side-payments in the solution concept, which can

be very sophisticated, unlike noncooperative game theory. The distinction between the

two types of games is therefore a modeling one. In the next sections of this Chapter, we

will provide an overview of n-person cooperative games only, following two excellent ref-

erences: Kahan and Rapoport [24], and Chapter 8 from Colman [25]. For noncooperative

games, the reader is referred to [26], where the rules of a noncooperative game are

explained in detail for a power transmission pricing scenario.

2.2. N-Person Cooperative Games

Any game terminates in an end-state called an outcome. The quantitative representation of

a player’s outcome at the end of a game is called a payoff. When a game terminates, each

player i receives a payoff, that we can denote as . The collection of payoffs to all play-

ers may be expressed as the row vector of each player’s payoff, in

alphabetical order. Such a vector is termed a payoff vector. We will consider the first

assumption posited by Aumann (1967) regarding the relationship of player’s utilities to

payoffs expressed in units of a real, desirable, infinitely divisible medium of exchange

(i.e., “money”):

“The utility of money to each player is fixed for any given game, and all players prefer

more money to less money [24].”

This assumption was justified by Aumann (1967), and is articulated in [24] as follows:

“To represent preferences between actual sums of money,...,utilities are not needed, as the

dollar amount is a perfectly good measure for this purpose. Therefore, we may calculate

[any solution] and the intuitive validity of the result is not based on any consideration of

xi

x xA xB … xn, , ,( )=

14

Chapter 2 Cooperative Game Theory: An Overview

2.1. Introduction

“Game theory is a branch of mathematics that is concerned with the actions of individuals

who are conscious that their actions affect each other [23].”

In everyday life all of us face decision making, and most of the times, we have to make

rational decisions using limited information. One typical example of practical game the-

ory is the famous prisoner’s dilemma. In this example, two prisoners are being interro-

gated separately. If both confess, they are sentenced to eight years in prison; if both

cooperate, each is sentenced to one year. If just one confesses, he is released, but the other

prisoner is sentenced to ten years.

What difference would it make if the two prisoners could talk to each other before making

their decisions?

“A cooperative game is a game in which the players can make binding commitments, as

opposed to a noncooperative game, in which they cannot [23].”

This definition draws the traditional distinction between the two theories of games, coop-

erative and noncooperative, but the difference is in fact in the modeling approach. Both

theories consider players: individuals who make decisions, actions: choices that the play-

ers can make, and the rules of the game. However, they differ in the kinds of solution con-

cepts employed. Cooperative game theory seeks equity and fairness; noncooperative game

theory has solution concepts based on players maximizing their own utility functions.

13

tency. A test example is provided, although the user should be able to define her own

problem following the instructions of the user’s guide. DistOpt is the result of a joint U.C.

Berkeley-Università degli Studi di Cassino research project initiated by Professor Felix F.

Wu at U.C. Berkeley, EECS Department.

1.4. Thesis Outline

The dissertation is organized according to the following scheme:

Chapter 2 introduces the basic concepts of cooperative game theory from a general

mathematical standpoint.

Chapter 3 introduces the new deregulated environment in Power Systems and pre-

sents a new algorithm for decentralized coalition formation among transmission

expansion players.

Chapter 4 presents a new backward induction algorithm to allocate costs among the

players, once the coalition formation process has ended.

Chapter 5 presents a new decentralized software environment called DistOpt, and its

application to Power Systems problems.

Chapter 6 summarizes the major contributions of this dissertation and suggests direc-

tions for future research.

12

• Defining the rules that a coalition of players has to meet to become an “autonomous

coalition” in the expansion game

• Defining the role of an expansion coordinator, an arbitrator entity whose attributes are

exclusively regulatory and safety enforcing, and not related to the economics of the

game

• Using a cooperative game theory concept, the Bilateral Shapley Value (BSV), for the

players to make offers to potential expansion partners

• Introducing a new framework, the multi-agent system, in which we can allocate costs

once the negotiation process has ended. To do that we have developed a new algorithm:

we call it the backward induction algorithm. This algorithm takes into account the pre-

vious history of coalition formation and rewards valuable players according to their

contribution to cost reduction. It uses BSVs to allocate costs in a recursive fashion, i.e.,

it starts from the final coalition that was formed, splits the final cost among its two

founding members using BSVs, and proceeds to allocate the calculated costs recur-

sively until the first stage of the negotiation is reached. This scheme calculates both the

total cost and its division among the players.

A flexible software environment, suitable for the assessment of different proposals and the

modeling of new methodologies of cooperation and coordination in a decentralized con-

text was needed. We have developed a new software system, named DistOpt [21] that is

based on an existing package developed at UC Berkeley, called Ptolemy [22]. DistOpt is a

distributed software environment for solving large scale optimization problems developed

at the Department of Electrical Engineering and Computer Sciences, University of Cali-

fornia, Berkeley. DistOpt is an extremely flexible simulation software suitable to solve

large scale optimization problems. The modularity of its approach allows the user to

change the values of many parameters simultaneously with the execution of the program,

and rerun the simulation after that. Also, the number of subproblems and the convergence

parameters can be interactively selected, provided that some acceptable limits are calcu-

lated by the program. Tcl/Tk is used to interface with DistOpt whenever the user decides

to change a certain value or parameter, or when DistOpt points out a mistake or inconsis-

11

well known cooperative game theory concept. Gately’s approach was a centralized one,

where there was a central planner taking the allocation decisions. Gately’s main contribu-

tion was the creation of a new concept, the “propensity to disrupt”. Roughly speaking, the

propensity to disrupt of a player is the ratio of how much the other players would lose if

that player refused to cooperate to how much she would lose if it refused to cooperate. He

used cooperative game theory concepts: core, imputation and Shapley Value, to allocate

expansion costs in the most fair way possible.

1.3. Thesis Contributions

We believe that the power system structure that will emerge after the restructuring process

will be characterized by three principles: decentralization, cooperation and coordination.

To address these issues, we have developed a framework that introduces a new paradigm

of competition and cooperation in power systems: a methodology that is able to benefit

the grid as a whole, but letting the economic decisions to the players. In this dissertation,

we have tried to apply this principles to transmission planning, but the same paradigm can

be used to model other issues in power systems operations and planning1. The framework

that we have chosen is based on cooperative game theory.

We have modeled the power system as a multi-agent system, where the players are able to

take rational autonomous decisions that benefit the system as a whole. We have set the

rules of the cooperative game, and our model has determined how the coalitions of players

are formed and the way the total cost can be allocated among them. In detail, this has been

our contribution:

• Proposing what is the minimum cost that every individual player has to face at the

beginning of the game when expanding on her own

1. Coordinated multilateral trades in power system operations is a perfect example [10].

10

Phase II: The final network is reinforced to take into account the effect of single contin-

gencies: the algorithm initially simulates the effect of the removal of circuit elements

selected from a list of severe contingencies.

Very recently, Yoshimoto el al. [19] presented an approach for solving transmission

expansion planning based on neuro-computing hybridized with a genetic algorithm. They

used a 0-1 Hopfield Neural Network to solve the combinatorial transmission expansion

planning problem, and a genetic algorithm that improves the solution accuracy obtained

from the Neural Network model.

From a policy standpoint, Walton [20] has proposed the use of a “rated system path”

model as an approach to defining firm transmission rights as part of the transmission plan-

ning process. The rated system described in [20] is an adaptation of system planning and

design procedures in the Western System within the WSCC area. Walton suggested that

with his rated system path, “by examining the contribution each facility makes to the

simultaneous transfer capability of the system, a more appropriate division of firm rights

can be made.” Therefore, the transmission right of every party involved in a transaction of

power is based upon the same rules used to design system additions that increase transfer

capability, so that the physical effects of one party’s schedules on another party are not

ignored.

Finally, in a game theoretic approach to the expansion problem, Gately was the first

author concerned not only with the transmission expansion cost, but also with the distri-

bution of the gains from cooperation between two or more players of the so called “expan-

sion game.” His analysis concerned regional cooperation in planning investment in

electric power, in the particular case of four states in southern India. He proved that the

formation of coalitions of players in a transmission expansion investment scenario is a

valid approach to solve the expansion problem,. In [2], he was concerned with regional

cooperation in planning investments and cost allocation using the Shapley Value (SV), a

9

types. The result is a nonlinear staircase function with unequal steps. In addition, for each

discrete value of capacity, the optimal combination of line types is determined.

2) The method works directly with discrete line additions. the single-stage expansion

problem is formulated as a zero-one integer program which is solved by a modified

branch-and-bound method with optimal ordering of variables.

3) A screening algorithm is included which reduces the size of the problem by eliminating

those rights-of-way which are ineffective, through the use of sensitivity coefficients.

4) A DC load flow model is used, which results in fast computation and reasonable accu-

racy. Single contingencies can also be handled readily.

The real innovation in [16] stems from the use of sensitivity coefficients to compare the

effectiveness of different line additions and the use of screening algorithms to discard

rights-of-way that are not in the optimal solution.

Garver, by adding the most effective line at each step of his expansion algorithm, and Lee

and Hnyilicza, were using sensitivity analysis concepts to determine new line additions.

However, Monticelli et al. [18] developed a new interactive software tool for long term

transmission system expansion planning that is totally based on sensitivity analysis. They

created analysis tools where the user can add or remove any circuit of the network and

evaluate its impact in terms of flows, overloads, etc. The algorithm that they used for

static studies proceeds as follows:

Phase I: Automatic network reinforcement until all overloads have been eliminated and

all disconnected buses with non-zero injections have been connected. The ranking of new

additions is based on a “least-effort” criterion that takes into account the pattern of flow

distribution in the network. The program will list the overload reduction or increase in

each overloaded circuit. Finally, the user can rank the n best candidates for addition in

accordance with the “least-effort” criterion. A DC power-flow program is needed.

8

From a different standpoint, a cooperative game theory approach to allocate the expansion

costs among the users of the lines was first developed by Gately in his seminal paper [2].

And more recently, a novel technique that uses neural networks hybridized with genetic

algorithms [19] has shown promising results. In this Section, we will briefly introduce the

mentioned techniques to address the expansion problem in chronological order.

Garver [14] was the first author that systematically addressed the line expansion problem

from a mathematical programming point of view. The steps that he took to solve the prob-

lem are quoted from [14] as follows:

1) Formulate the power flow equations as a linear minimization problem.

2) Use linear programming to solve the minimization problem for the needed power

movements. This result is called a linear flow estimate.

3) Select a circuit addition based on the location of the largest overload in this flow esti-

mate.

4) Repeat the flow-estimation and circuit-selection steps until no overloads remain.

The most important new feature of his flow estimation technique was the fact that over-

loads did not appear on circuits, but on a new type of network link called an overload

path. Overload path existed between every bus in the network. They were used in the lin-

ear flow estimation, where the overload paths with flows became the possible rights-of-

way for new circuit additions.

Lee and Hnyilicza [16] proposed a different approach that used branch-and-bound integer

programming. Their method can be summarized as follows, as quoted from [16]:

1) The discrete nature of the cost function is taken into account and for each right-of-way,

an optimal capacity curve is derived which recognizes space constraints and different line

7

Figure 1-2 : 6 Bus System Solution.

1.2. Previous Work

Traditionally, the transmission expansion planning problem has been studied using two

different techniques:

• techniques based on mathematical programming, in particular linear programming

[14], dynamic programming1 [15], and branch-and-bound techniques [16]

• techniques based on sensitivity analysis [17, 18]

1. Since the number of possible network configurations for a typical power system is very large, dynamicprogramming is quite impractical.

Bus 6 Bus 4

Bus 2

Bus 1Bus 5

Bus 3

180

80

40

3 UNITS

120

240

240

4 UNITS

30

30

30

60

4 UNITS

60

60

120

120

240

240

32

188

4

357

51

53187

62

6

(existing and future) in the system is given in Table 1-1, where the capacity of each line is

determined based on thermal limitations and stability requirements.

The solution obtained by Garver in [14] without generation rescheduling (generator out-

puts 1, 3, and 6 are 50, 165 and 545 MW respectively) is shown in Figure 1-2. The arrows

indicate the value and direction of the active power flow across the line(s).The optimal

solution has a cost of 200 monetary units, and circuit additions are: circuits,

circuit, and circuits as shown in Figure 1-2.

Table 1-1. 6 Bus System Circuit Data

Should we allow generation rescheduling, i.e. the real power generation ranging from 0 to

the maximum generation available (150, 360, and 600 MW respectively), the optimal

solution had a cost of 130 monetary units, and circuit additions are: circuits, and

circuits.

Bus:from/to

Cost(units)

Suscept.(1/ )

Capacity(MW)

1/2 40 2.50 100

1/4 60 1.67 80

1/5 20 5.00 100

2/3 20 5.00 100

2/4 40 2.50 100

2/6 30 3.33 100

3/5 20 5.00 100

4/6 30 3.33 100

5/6 61 1.64 78

n26 4=

n35 1= n46 2=

Ω

n26 3=

n35 2=

5

Figure 1-1 : 6 Bus Example.

We assume that transmission line cost is 1 monetary unit/mile. Its initial configuration has

5 nodes: 1, 2, 3, 4, and 5 and 6 branches: 1-2, 1-4, 1-5, 2-3, 2-4, and 3-5, as shown in

Figure 1-1 and Table 1-1. When the system expands, there is a new bus: bus 6 and 4 new

possible rights-of-way: 2-6, 3-5, 4-6 and 5-6. The connection between any two buses is

allowed with a limit of 4 parallel paths in each right-of-way. The data of the different lines

Bus 6 Bus 4

Bus 2

Bus 1Bus 5

Bus 3

240

180

80

40

3 UNITS

120

240

240

4 UNITS

30

30

30

60

4 UNITS

60

60

120

120

80

100

100

100

100

100

240

LEGEND

Existing line

Possible line addition

4

(1-2)

where A is the node-branch incidence matrix, T is the branch power flows vector, and P is

the net power injections vector.

(b) Limits in branch power flow:

(1-3)

where is the branch power flow limit vector.

In a DC load flow model, each element of the branch power flow vector T in constraint (1-

2) can be described as follows:

(1-4)

where is the variable representing the total number of parallel links of line j, is the

reactance of a link of branch j and and are the voltages angles of the terminal buses

of branch j. Then, constraint (1-2) becomes:

(1-5)

where is the susceptance matrix whose elements are: for the off-diag-

onal terms, and for the diagonal terms, is the total reactance of

branch (k, l), are the branches connected to bus k, and is the vector of nodal

voltage angles. Finally, constraint (1-3) becomes:

(1-6)

where is a diagonal matrix whose elements are: .

Let us illustrate the problem formulation by introducing the classical 6 bus system

described in [14], as shown in Figure 1-1. We will use this simple example in the follow-

ing sections to simulate coalition formation schemes and to allocate total expansion costs.

A T⋅ P=

T Tmax Zj( )≤

Tmax

Tj

Zj

xj---- θj θl–( )=

Zj xj

θj θl

B Zj( ) Θ⋅ P=

B Zj( ) Bkl1

xkl------–=

Bkk Bkl

l Ω∈∑= xkl

l Ωk∈ Θ

BLAtΘ Tmax Zj( )≤

BL Zj xj⁄

3

This dissertation presents a cooperative game theory model of planning within a multi-

agent system environment, where the agents cooperate with each other to achieve the opti-

mal common expansion goal. Here, the agents have to fulfill certain number of tasks, i.e.,

adding new lines, and they want to cooperate by means of forming coalitions to reduce

overall costs. Each agent is rational, in the sense of being a utility maximizer, and she is

“an independently motivated agent, not willing to settle for a plan generated by a central

planner [5].”

1.1. The Transmission Expansion Planning Problem

The objective of the mathematical model of a transmission expansion planning problem is

minimizing the capital and operating costs associated with the system expansion over the

planning horizon. The constraints associated with this model are the physical and eco-

nomical constraints that are important when attempting to expand a utility system at a

minimum cost and yet meet all economic and demand restrictions that are placed upon the

system. We can formulate the transmission expansion planning problem in the following

terms:

(1-1)

where is the cost per unit power at node i for the whole planning period, is the real

power injected into the network by generators at node i, is the construction investment

cost per parallel link of line j, is the variable representing the total number of parallel

links of line j, is the initial number of parallel links of line j, is the set of genera-

tor nodes, and is the set of possible lines.

Assuming a DC load flow model for simplicity, the problem is subject to the following

constraints:

(a) Power nodal balance at each bus, Kirchhoff’s laws:

min CiPGiKj Zj Zj

0–( )

j A∈∑+

i N∈∑

Ci PGi

Kj

Zj

Zj0

Ng

An

2

to solve the day-to-day problems that a utility company has to address. The basis for

cooperation and coalition formation among entities that share common interests is there-

fore purely economical, but the results benefit the entire network, since the resulting sys-

tem is more reliable when the added lines are collectively agreed upon.

Game theory (GT) is sometimes described as multiperson decision theory or the analysis

of conflict. One of the applications of GT has been the modeling of sustaining cooperation

in apparently noncooperative environments through repeated interactions. As a modeling

tool, cooperative GT has been successfully applied in the power systems area to share the

gains of regional cooperation in centralized planning investments [2], to allocate wheeling

transaction costs [3], or to allocate cost savings in an energy brokerage system [4].

Research on Distributed Artificial Intelligence (DAI) has focused on how coalitions are

formed and on negotiation algorithms amongst players of economic games. Until now, the

range of applications of DAI has been restricted to other fields not related to power sys-

tems planning: stock market trading [5], cooperative databases [6], and algorithmic theory

[8]. Cooperative game theory concepts have been used, but they were suited to decentral-

ized multitask environments [5, 6, 7, 8, 9]. In the Power Systems area, Wu et al. [10] have

developed a decentralized algorithm to optimize multilateral trades among generators and

customers, although only looking at the operational side. Similarly, for transmission plan-

ning, Bushnell and Stoft [11], Chao and Peck [12], and Hogan [13] have described invest-

ment incentives and market mechanisms in a deregulated market, respectively.

In the light of these encouraging results, we propose a combined GT/DAI approach to

transmission planning that addresses and solves the pending issues of transmission expan-

sion in a deregulated electricity industry:

• Determining how coalitions are formed

• Implementing a negotiation algorithm

• Allocating total expansion costs to every single agent of the transmission game

1

Chapter 1 Introduction

Power systems transmission planning addresses the problem of determining the optimal

number of lines that should be added to an existing network to supply the forecasted load

as economically as possible, subject to operating constraints. “The objective is the mini-

mum cost expansion plan given the base network configuration, the generation facilities,

and the forecasted demands for a target year [1].”

Traditionally, expansion plans were carried out by the utilities alone, without considering

external factors: this was a monopoly where companies decided their own expansion

plans without competing utilities that offered their services. Since the initial PURPA regu-

lations starting in 1992, the electric utility industry is facing deregulation to allow trans-

mission open access to suppliers and customers. In the near future, transmission

expansion planning will involve decisions taken by some of the actors in the expansion

scenario (suppliers, customers, and/or transmission line owners) that can and will affect

decisions taken by other players. This intertwined decision process occurs when a new

transmission line is built, and it is shared by several “players” of the expansion game. The

decision whether to build the line or not, and the allocation of costs to the players who

will use the line is still an open issue in a decentralized environment.

Traditional planning tools that have been used in the utility industry have to be revised

and adapted to the new competitive environment. We envision the new planning environ-

ment as a group of loosely linked entities, the players of the expansion game, that cooper-

ate on the basis of mutual benefit. This means that current widespread planning tools,

optimal power flow and unit commitment are two examples, have to fit the new demands

that a decentralized and competitive environment presents, but without losing their power

ix

A mi madre, con cariño

viii

And last but not least, my family. My mother has been the only solid pillar I have had dur-

ing times of trouble and anxiety. My wonderful sister María del Carmen and her husband

José Antonio have always encouraged me to pursue higher ideals and do it with all my

heart. I do not want to forget my sister María Teresa and my lovely nephew and nieces:

Ignacio, Cristina, and Tineke.

Thanks to all.

vii

Acknowledgments

This dissertation would not be possible without the support and collaboration of many per-

sons in the EECS Department. First, I would like to thank Professor Felix F. Wu for his

continuous advice and mentorship. His valuable comments during this period have been

invaluable and worth for a lifetime. This Thesis owes to him a sense of ambition and inno-

vation that would be impossible to attain without his leadership.

I also thank Professors Shmuel Oren and Pravin P. Varaiya for their patience, support and

wise counseling before and after the Qualifying Examination. Their knowledge and

encouragement have been a key factor during my graduate student years. Thanks also to

the EEE Department of the University of Hong Kong, who provided me with a fantastic

working environment to pursue my goals. My special thanks to Professors Mario Russo

and Arturo Losi, Dr. Matthias Klusch and Dr. Onn Shehory, and all the LG201 students at

HKU.

I do not want to forget my great office mates, first at 341 Cory, and later at the new 273

location. Thanks to Linda Kamas, Heath Hoffmann, Jeff Kao, Tetiana Lo, Liam Murphy,

Zhigang Qin, S.Venkatesan, Angela Chuang, and Sandra Ellis. I am also really proud to be

a friend of Reinaldo García, José Antonio Ortega Osona, María del Mar Fernández Vega,

Carlos Mora, and Leonor Chico. I will never forget the evenings we remembered our

beloved Spain (and Recife too!).

Generous funding was provided by the Spanish Ministry of Education and Science, first

with a Fulbright Scholarship and later with a Doctores y Tecnólogos Scholarship. Without

their economic support, this Thesis would never be written. I also want to thank the EECS

Department, and Professor Wu in particular, for their funding during my last year of grad-

uate studies.

vi

List of Tables

Table 1-1 6 Bus System Circuit Data................................................................. 6Table 3-1 6 Bus System Circuit Data................................................................ 46Table 3-2 Coalition Expansion Costs................................................................ 56Table 4-1 Coalition Expansion Costs................................................................ 64

v

List of Figures

Figure 1-1 6 Bus Example ................................................................................... 5Figure 1-2 6 Bus System Solution........................................................................ 7Figure 2-1 The Core of a Three Person Game.................................................... 20Figure 2-2 The Nucleolus of a Three Person Game ........................................... 24Figure 3-1 Coordinating Plans for Agents.......................................................... 43Figure 3-2 6 Bus Problem................................................................................... 47Figure 3-3 Two Examples of Autonomous Coalitions ....................................... 50Figure 3-4 Overall Coalition Formation Framework ......................................... 55Figure 4-1 Cost Allocation Procedure ................................................................ 70Figure 4-2 Overall Coalition Formation and Cost Allocation Procedure........... 71Figure 4-3 BSV versus SV Allocations .............................................................. 72Figure 4-4 Core, Nucleolus, and Shapley Value Calculations ........................... 74Figure 5-1 DistOpt Palette .................................................................................. 85Figure 5-2 Measures Galaxy .......................................................................... 86Figure 5-3 APP_3 Galaxy.................................................................................. 87Figure 5-4 Final3 Universe............................................................................. 87Figure 5-5 IEEE-57 bus Case Subproblem 1 Objective Function...................... 89Figure 5-6 IEEE-57 bus Case Subproblem 2 Objective Function...................... 90Figure 5-7 IEEE-57 bus Case Subproblem 3 Objective Function...................... 90Figure 5-8 IEEE-57 bus Case Subproblem 1 Maximum Error .......................... 91Figure 5-9 IEEE-57 bus Case Subproblem 2 Maximum Error .......................... 91Figure 5-10 IEEE-57 bus Case Subproblem 3 Maximum Error .......................... 92

iv

Chapter 4. A Cost Allocation Method: Backward Induction Algorithm ............. 61

4.1. Introduction ...............................................................................................614.2. The Backward Induction Algorithm..........................................................624.3. Coalition Stability......................................................................................664.4. Simulation Results .....................................................................................684.5. Discussion..................................................................................................71

Chapter 5. DistOpt: A Distributed Optimization Software Environment .......... 76

5.1. Introduction ...............................................................................................765.2. The Auxiliary Problem Principle and Problem Transformation................775.3. Ptolemy Environment ................................................................................815.4. DistOpt Structure .......................................................................................835.5. DistOpt Simulation ....................................................................................885.6. An Optimal Power Flow Application ........................................................895.7. Conclusions ...............................................................................................92

Chapter 6. Conclusions and Future Work ............................................................. 94

6.1. Contributions .............................................................................................946.2. Future Research .........................................................................................946.3. Concluding Remarks .................................................................................95

BIBLIOGRAPHY .................................................................................................. 96

iii

Table of Contents

List of Figures ........................................................................................................... v

List of Tables ........................................................................................................... vi

Acknowledgments .................................................................................................. vii

Chapter 1. Introduction .............................................................................................. 1

1.1. The Transmission Expansion Planning Problem.........................................31.2. Previous Work .............................................................................................71.3. Thesis Contributions..................................................................................111.4. Thesis Outline............................................................................................13

Chapter 2. Cooperative Game Theory: An Overview ............................................ 14

2.1. Introduction ...............................................................................................142.2. N-Person Cooperative Games....................................................................152.3. Characteristic Function and Imputation ....................................................172.4. Core, Stable Set, and Bargaining Set.........................................................182.5. Excess Theories: Kernel and Nucleolus ....................................................222.6. Shapley value.............................................................................................252.7. Equal Excess Theory Model......................................................................272.8. Dynamic Coalition Formation Models: Transfer Schemes .......................282.9. Summary....................................................................................................31

Chapter 3. A Coalition Formation Algorithm in Transmission Planning ............ 32

3.1. New Policies in a Deregulated Environment.............................................323.2. Transmission Planning under the Open Access Paradigm ........................373.3. Multiagent Environments in Power Systems.............................................423.4. Simplified Network Expansion Model ......................................................453.5. Coalitions in Expansion Planning..............................................................483.6. Four Phases of the Coalition Formation Algorithm: .................................503.6.1 Self-Calculation Phase ............................................................................523.6.2. Communication Phase.............................................................................533.6.3. Bilateral Shapley Value Calculation Phase.............................................533.6.4. Bilateral Negotiation Phase.....................................................................543.7. Simulation Results .....................................................................................553.8. Discussion..................................................................................................59

1

Abstract

A Cooperative Game Theory Approach to Transmission Planning in Power Systems

by

Javier Contreras

Doctor of Philosophy in Electrical Engineering and Computer Sciences

University of California at Berkeley

Professor Felix F. Wu, Chair

The rapid restructuring of the electric power industry from a vertically integrated entity

into a decentralized industry has given arise to complex problems. In particular, the trans-

mission component of the electric power system requires new methodologies to fully cap-

ture this emerging competitive industry. Game theory models are used to model strategic

interactions in a competitive environment. This thesis presents a new decentralized frame-

work to study the transmission network expansion problem using cooperative game the-

ory. First, the players and the rules of the game are defined. Second, a coalition formation

scheme is developed. Finally, the optimized cost of expansion is allocated based on the

history of the coalition formation.

________________________

Chairman


Copyright 1997by

Javier ContrerasAll rights reserved

The dissertation of Javier Contreras is approved:

University of California at Berkeley

1997

DateChair

Date

Date


by

Javier Contreras

B.S. (Universidad de Zaragoza, Spain) 1989M.S. (University of Southern California) 1992

A dissertation submitted in partial satisfaction of the requirements for the degree of

Doctor of Philosophyin

Engineering-Electrical Engineeringand Computer Sciences

in the

GRADUATE DIVISION

of the

UNIVERSITY of CALIFORNIA, BERKELEY

Committee in charge:

Professor Felix F. Wu, Chair

Professor Pravin P. Varaiya

Professor Shmuel Oren

1997

Thesis_JC a Cooperative Game Theory Approach to Transmission Planning in Power Systems

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