The Pennsylvania State University The Graduate School Department of Industrial and Manufacturing Engineering ON DISTRIBUTED OPTIMIZATION METHODS FOR SOLVING OPTIMAL POWER FLOW PROBLEM OVER ELECTRICITY GRIDS A Thesis in Industrial and Manufacturing Engineering by Jinwei Zhang c 2016 Jinwei Zhang Submitted in Partial Fulfillment of the Requirements for the Degree of Master of Science August 2016
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The Pennsylvania State University
The Graduate School
Department of Industrial and Manufacturing Engineering
programming, and particle swarm optimization [3, 4, 5, 6, 7, 8, 9]. The nonconvexity of
OPF problem is partially due to the nonlinearity of physical system governing real(active)
power, reactive power, and voltage magnitude [10]. Many of these methods are based on
the Karush-Kuhn-Tucker (KKT) necessary conditions; therefore they can only guarantee a
local solution [9].
In order to compute the global optimal solution, different conic and convex relaxations
[11, 12] have been proposed to convexify the OPF problem. Among multiple convexification
methods, Lavaei et al. [13] considered the OPF problem with a quadratic generation cost
function, and studied the Lagrangian dual of OPF, which can be formulated as a semidefinite
program (SDP). They studied sufficient conditions that guarantee the duality gap between
OPF problem and its dual is zero, which is the main advantage of the convexification of
OPF through its Lagrangian dual, since a global optimal solution of OPF can be found
(in polynomial time). Moreover, a second-order cone programming (SOCP) relaxation
2
has been proposed to convexify and solve OPF problem even more efficiently under certain
conditions [14]. The SOCP relaxation will be applied in this thesis as the main modification
to the OPF problem to approximate it with a convex model.
Suppose the generation and distribution system is given as a graph G = (N , E) of nodes
N = 1, ..., N connected with branches in E . Without loss of generality, assume that
(i, j) ∈ E implies i < j; and define
N (i) = j ∈ N : (i, j) ∈ E or (j, i) ∈ E (1.1)
as the set of neighboring nodes to i ∈ N . We assume that each node i has a composite
convex function,
Fi(x) = ξi(x) + fi(x) (1.2)
with a possibly non-smooth convex part ξi and smooth convex part fi, respectively; and we
are interested in minimizing an overall system objective subject to power flow constraints,
i.e.,minx∈X
∑i∈N
Fi(x), (1.3)
where X ⊂ Rn denotes the feasible set of decision variables x associated with power flow
and node-specific constraints. For details of how variables x are defined and X is modeled
based on OPF problem, see Section 2.1. Traditionally, OPF problem has been solved in a
centralized fashion by communicating all the objective functions, such as generation cost
functions, to a central node, and solving the overall problem at this node. However, such an
approach can be prohibitively very expensive both from communication and computation
perspectives. In such case, all the local data needs to be transmitted to the central node
which may also violate privacy constraints of the grid nodes. Furthermore, it requires that
the central node have large enough memory to be able to accommodate all the data [15].
Considering these disadvantages of centralized method,it is motivated to seek for consensus
among all the nodes on an optimal decision using local decisions communicated by the
neighboring nodes, shown as following:
minxi∈Xi,i∈N
∑i∈N
Fi(xi) : xi = xj ,∀(i, j) ∈ E
, (1.4)
3
where the goal is to collectively solve OPF problem in order to optimize a global objective
function of the system, and Xi is the feasible set split to each node from X . We assume
that only local information exchange is allowed, i.e., there is no central node where the data
can be consolidated, and only neighboring nodes can communicate. We show that the OPF
problem can be formulated as (1.4); hence, it can be regarded as a typical case of decentral-
ized multi-agent consensus optimization problem. (1.4) is a generic model, which can be
used, not only for power system but also for various applications in signal processing [16],
and machine learning [17]. Decentralized multi-agent consensus optimization is motivated
by the emergence of large scale networks such as mobile ad hoc networks [18] and wire-
less sensor network [19], characterized by the lack of centralized access to information and
time-varying connectivity. Therefore, optimization algorithms in such networks should be
completely distributed, relying only on local information.
Given the importance of decentralized multi-agent consensus optimization, a number
of different distributed algorithms have been proposed to solve it. Particularly, Aybat et
al. [15] proposed a distributed first-order augmented Lagrangian (DFAL) algorithm to solve
(1.4). Based on the tests and results reported, the algorithm DFAL performs very well in
practice. However, its implementation on a network of distributed agents requires a more
complex network protocol. Specifically, checking the subgradient stopping criteria for inner
iterations requires evaluating a logical conjunction over G, which may cause trouble for large
networks. To overcome this disadvantage, Aybat et al. [20] proposed a proximal gradient
alternating direction method of multipliers (PG-ADMM) and its stochastic gradient variant
(SPG-ADMM) to solve composite convex problems. They implemented PG-ADMM and
SPG-ADMM on two different, but equivalent, consensus formulations, which gives rise
to two different node-baes distributed algorithms: DPGA-I and DPGA-II and their
stochastic gradient variants. Using only local communication, these node-based distributed
algorithms require less communication burden and memory storage compared to edge-based
distributed algorithms. Moreover, the proposed algorithms consist only a single loop, i.e.,
there are no outer and inner iteration loop; therefore, they are easy and practical to be
implemented over distributed networks. The contribution of this thesis is to implement
DFAL and DPGA-II algorithms to efficiently solve convex relaxation of the OPF problem
4
in a distributed manner. The IEEE benchmark networks with 3, 9, 14, and 30 buses are used
as test samples, and we compare DFAL and DPGA-II in terms of optimal solution accuracy
and convergence rate. Both DFAL and DPGA-II algorithms applied to OPF problem are
fully distributed, the buses are not required to know some global parameters depending on
the entire network topology.
The rest of this thesis is organized as follows. The OPF problem is formulated in Chapter
2. The multi-agent consensus problems and some algorithms to solve it are discussed in
Chapter 3. The practical implementation of DFAL and DPGA-II to solve SOCP relaxation
of OPF problem is proposed in Chapter 4, and numerical results are illustrated in this
chapter. Concluding remarks are given in Chapter 5. Finally, the data format used in
the implementation is summarized in Appendix. The following notations will be used
throughout this thesis.
• i: Imaginary unit.
• R: Set of real numbers.
• Hn×n
: Set of n× n Hermitian matrices.
• Re· and Im·: Real and imaginary parts of a complex matrix.
• ∗: Conjugate transpose operator.
• >: Transpose operator.
• : Matrix inequality sign in the positive semidefinite sense (i.e., given two symmetric matrices
A and B, A B implies A−B is a positive semidefinite matrix, meaning that its eigenvalues
are all nonnegative).
• Tr: The matrix trace operator.
• | · |: The absolute value operator.
Given complex values a1 and a2, the inequality a1 ≥ a2 means Rea1 ≥ Rea2 and
Ima1 ≥ Ima2
5
Chapter 2
Optimal Power Flow Problem
Power flow is also known as “load flow”. This is the name given to a network solution that
consists of complex currents, complex voltages, real power, and reactive power at every bus
in the system. Since we assume that the parameters of system elements, such as lines and
transformers, are constant, the power system in consideration is a linear network. However,
in the power flow problem, the relationship between voltage and current at each bus is
nonlinear, and the same holds for the relationship between the real and reactive power
consumption at a bus or the generated real power and scheduled voltage magnitude at a
generator bus. Thus even computing a feasible power flow for given amount of power injected
at each generator bus involves the solution of nonlinear equations. Therefore, OPF problems
are nonconvex, and in general large-scale optimization problems which may contain both
continuous and discrete control variables. Many different OPF formulations have been
developed to address specific instances of the problem under various assumptions, each
having different objective functions, controls, and constraints. Regardless of the different
names given to different cases, any power system optimization problem that includes a set
of power flow equations in the constraints may be classified as an OPF problem.
OPF is an important part of power system operation and planning. It describes the
optimal electrical response of the transmission system to a particular set of loads and power
injections. In traditional power systems, OPF is mainly used for planning purpose together
with a forecast engine, e.g., to determine the system state in the day-ahead market with the
given current system information. In the smart grid paradigm, due to highly intermittent
nature of the renewables, the later the prediction is made, the more reliable it would be.
Therefore, if OPF can be solved very efficiently in real time, some of the unpredictability
in the system will be mitigated.
6
2.1 Formulation of OPF problem
2.1.1 General Structure
Generally, OPF requires solving a system of nonlinear equations and inequalities, describ-
ing optimal and secure operation of a power system:
minx,u
f(x,u)
subject to g(x,u) = 0 (2.1)
h(x,u) ≤ 0,
where
i. f(x,u) is an appropriate cost to be minimized;
ii. x is the vector of state variables;
iii. u is the vector of control variables, which are usually the independent variables in an
OPF;
iv. g(x,u) is the set of equality constraints resulting from power flow equations;
v. h(x,u) is the set of inequality constraints resulting from the physical limits imposed
on the vector arguments x and u.
Depending on the objective functions and constraints, there are different mathematical
formulations for the OPF problem. They can be broadly classified as follows:
i. Linear problem in which objectives and constraints are given in linear forms with
continuous variables.
ii. Nonlinear problem where either objectives or constraints or both are nonlinear with
continuous control variables.
iii. Mixed-integer linear problem where control variables are both discrete and continuous
within the linear setting.
7
2.1.2 Objective function
The most common OPF objective is to minimize the total generation cost, sometimes also
considering system losses as well. Generation cost functions are often approximated with
quadratic cost curves [21, 22], or with piecewise linear functions [23]. Besides minimiza-
tion of generation cost, other commonly adopted objectives include maximization of power
quality (often by minimizing voltage deviation), and minimization of capital costs during
system planning. In nearly all cases, the objective is a function of real and reactive power
generated in the system. Other objectives considered in the literature and practice also
include power transfer capability, number of controls rescheduled or shifted, cost or VAR
investment, optimal voltage profile, load shedding, environment impact, system loadability,
etc. [24].
2.1.3 Variables
State variables x in OPF problems represent the electrical state of the system. These
continuous state variables include bus voltage magnitude, bus voltage angle, real and reac-
tive power injections at each bus, as well as MVar loads, line parameters. Control variables
u typically include a subset of the state variables as well as variables representing control
device settings, such as transformer tap ratios, p-shifter angels, values of switchable shunt
capacitors and inductors. Control variables may be continuous or discrete. The choice of
state variables is dictated by the form of power flow equations used, while control variables
differ widely among OPF formulations based on the nature of particular problem. Table
2.1 summarizes typical variables previous used in the literature [24].
2.1.4 Constraints
Typical equality and inequality constraints used in OPF are summarized in Table 2.2 [24].
Equality constraints g(x,u) include the power flow conservation constraints. Alternating
Current (AC) power flow has been adopted for use in real-life transmission systems. OPF
formulations incorporating the AC power flow equations are nonlinear; thus, nonconvex.
For the sake of simplification of the model in practice, the flow of real power in the system
may be “decoupled” from the flow of reactive power, which leads to decomposing OPF
8
Variable TypesState variablesBus voltage magnitude ContinuousBus voltage phase angel ContinuousBus voltage real & imaginary parts ContinuousNetwork power flow ContinuousBranch currents ContinuousSlack bus power ContinuousGenerator reactive power outputControl VariablesReal/reactive power generation ContinuousRegulated bus voltage magnitude ContinuousTransformer tap settings DiscreteTransformer phase shifters Continuous, DiscreteSwitched shunt reactive devices BinaryLoad to shed Continuous, DiscreteMW interchange transactions ContinuousHVDC link MW controls ContinuousFACTS controls Continuous, DiscreteGenerator voltage control settings ContinuousStandby start-up units BinaryLine switching Binary
Table 2.1: OPF typical variables
problem into separate subproblems for the real and reactive power flows [25]. Many OPF
algorithms take advantage of this decomposition because it provides significant algorithm
simplification while introducing only a “small” amount of error under certain condition.
However, the decoupling approach to OPF is not typically accurate when complex control
devices are present in the system [26]. Sometimes although the current type is AC, Direct
Current (DC) power flow equations are used to approximate the nonlinear balance equations
for AC power flow. However, this simplification, i.e., using DC to approximate AC power
flow, both neglects network losses and prevents accurate cost accounting for reactive power.
Neglecting network losses introduces unacceptable levels of error in large power system
models. Several methods are available to enhance the DC power flow equations to provide
an estimate of system losses [24].
The inequality constraints h(x,u) include minimum and maximum limits on control and
state variables, such as bus voltage and line current magnitudes. Many transient security
constraints may also be incorporated analytically.
9
Equality constraintsFull AC power flowDecoupled AC power flowDC power flowNet active power exportSteady-state securityOther balance constraintsInequality constraintsActive/reactive power generation limitsDemand constraintsBus voltage limitsBranch flow limitsControl limitsTransmission interface limitsActive/reactive power reserve limitsSpinning reserve requirementsActive/reactive power flow in a corridorTransient securityTransient stabilityTransient contingenciesEnvironment constraints
Table 2.2: OPF typical constraints
2.1.5 Standard OPF Formulation
The bus injection model being used throughout this thesis is the standard model for
power flow analysis and optimization. It is built on nodal variables such as voltage, current
and power injection, while the branch flow model focuses on currents and powers on the
branches. The buses in a power system network are generally divided into three categories:
generation bus, load bus, and slack bus. Following quantities are specified at each bus: (1)
|V |: magnitude of the complex voltage V ; (2) θ: phase angel of the complex voltage; (3) P :
active (real) power; (4) Q: reactive power. Typically the bus voltage magnitude and phase
angel are represented by one complex variable V instead of two real variables separately.
Consider a power system network with the set of buses N := 1, 2, . . . , N, the set of
generator buses I ⊂ N , and the set of flow lines E ⊂ N ×N . Define:
• PDi + iQDi : Complex power of the load connected to bus i ∈ N .
• PGi + iQGi : Complex power output of the generator connected to bus i ∈ N .
• Vi: Complex voltage at bus i ∈ N .
10
• yij : Admittance of the transmission line (i, j) ∈ E , yij = gij − ibij .
Define V, PG, QG, PD and QD as the vectors Vii∈N , PGii∈N , QGii∈N , PDii∈N ,
and QDii∈N , respectively. Suppose fi(PGi) is a convex function representing the cost
associated with generator i ∈ I, and our objective is to minimize the total generation cost.
This problem can be formulated as follows:
minPG,V
f(PG) =∑i∈I
fi(PGi)
subject to PGi + iQGi = PDi + iQDi + Vi∑
j∈N (i)
y∗ij
(Vi − Vj)∗, ∀i ∈ N (2.2a)
Pmini≤ PGi ≤ P
maxi
, ∀i ∈ I (2.2b)
Qmini≤ QGi ≤ Q
maxi
, ∀i ∈ I (2.2c)
V mini≤ |Vi| ≤ V
maxi
, ∀i ∈ N (2.2d)
|Vi − Vj | ≤ ∆V maxij
, ∀(i, j) ∈ E (2.2e)
In this formulation PG and QG are the controllable parameters of the power network, and
PD and QD are fixed. Therefore, once PG and QG are fixed, this setting will set the value
of voltage V through out the network G according to power flow equation (2.2a). Given
the known vectors PD and QD, OPF minimizes the objective function over the unknown
parameters V, PG,and QG subject to the power flow equations at all buses as well as the
physical constraints. We assume that PGi = QGi = 0, if i 6∈ I, and the limits Pmini
, Pmaxi
,
Qmini
, Qmaxi
, V mini
, V maxi
, ∆V maxij
are given.
2.2 Hardness of OPF problem
There are three main difficulties that make OPF problem hard to solve.
(1) Active Constraints: Given a feasible point, the inequality constraints satisfied at
this point with strict inequalities are called inactive, while those constraints satisfied with
equality are referred to as binding or active constraints. Finding the active inequality
constraints has a combinatorial aspect, and it is a difficult part of solving the OPF problems.
11
No direct methods exist for solving OPF without using intermediate optimization method
to identify the active sets at an optimal solution [2].
(2)Nonconvexity : Even though some OPF problems can be formulated with linear objec-
tive function and constraints, e.g., DC-OPF problem. In most cases, the formulation has
a nonlinear form due to power flow equations representing the physical constraints on the
electric grid and explaining the highly nonlinear interplay among real power, reactive power
and complex voltage as shown in equation (2.2a). Hence, described by nonlinear equations,
the nonconvex feasible region may even be disconnected. Consequently the KKT conditions
are not sufficient for a global optimum in general. In particular, for AC power systems, the
OPF problem is inherently nonconvex giving rise to many local optima. As a result, existing
solution methods used extensively in practice rely on iterative optimization methods, which
can only return local optimal solutions at best. To summarize, the nonconvexity of OPF
problem in general prevents solving OPF in polynomial time, which makes OPF NP-hard
to solve.
(3)Network Complexity : Traditional mathematical optimization methods have been used
to effectively solve conventional OPF problems where the constraints are represented in
steady-state without considering system contingencies that can occur temporarily. An-
other difficulty in this respect is to accurately model and deal with the binary and integer
variables representing the node levels switch controls in the power systems in addition to
conventional OPF problems. Moreover, due to emergence of a deregulated electricity mar-
ket and consideration of dynamic system properties, the traditional concepts and practices
of power systems are overruled by an economic market management. Therefore, the dif-
ficulty of solving OPF problems increases significantly with increasing network size and
complexity [13].
12
2.3 Techniques to solve OPF
2.3.1 Traditional Optimization Methods
The majority of traditional techniques discussed in the literature use one of the following
methods: gradient descent, Newton’s method, simplex method, sequential linear program-
ming (SLP), sequential quadratic programming (SQP), and the interior point methods
(IPM). The simplex method is suitable for LP-based OPF, and can be directly applied to
DC-OPF formulation[27, 28]. SLP is an extension of LP introduced by Griffth and Stew-
art [29] that allows optimizing problems with nonlinear characteristics via a series of linear
approximations. In certain cases, the original NLP formulation can be reduced to an LP
using linear approximations of the objective function and constraints around an initial esti-
mate of the optimal solution [23, 30, 31, 32]. SQP is a solution technique for NLP problems,
and similar to SLP, it solves the original problem by solving a series of QP problems, of
which solutions converge to the optimal solution of the original problems [33]. In most SQP
implementation for OPF problem, conventional power flow equations are linearized at each
iteration, which can increase the computational efficiency at the cost of an increase in the
number iterations. A significant amount of work has been proposed in the literature on the
implementation of SQP for solving OPF problems [25, 34, 35, 36]. The following methods
focus on directly solving NLP rather than solving LP or QP approximation problems.
(1) Gradient Methods: Gradient methods were among the first attempts around the
end of the 1960s to solve practical OPF problems. Gradient methods can be divided
into three mainstreams of research: Reduced Gradient method (RG), Conjugate Gradient
Method (CG), and Generalized Reduced Gradient Method (GRG). Gradient methods use the
first-order derivative vector ∇f(xk) of the objective function at the current iterate xk to de-
termine an improving direction. Gradient methods are easy to implement, and guaranteed
to converge for well-behaved functions. However, gradient methods are slow, i.e., require
more iterations compared to higher-order methods. Moreover, because they do not evaluate
the second-order derivative, they are only guaranteed to converge a stationary point (which
may not be a true local optimal solution). Global optimality can only be proven for convex
problems, which excludes most OPF formulations [24].
13
RG was first applied to solve OPF problems by Dommel and Tinney [37]. It used penalty
techniques to enforce box constraints on state variables and the functional constraints.
The CG method is an improvement of the RG method, and is one of the most well-known
iterative methods for solving NLP problems with sparse systems of linear equations. Instead
of using the negative reduced gradient as the direction of descent, the CG method chooses
the descent direction such that it is conjugate to previous search directions by adding
the current negative gradient vector to a scaled, linear combination of previous search
directions. There are several advantages of applying CG method for solving OPF problems;
particularly the improvement in search characteristics over the RG method [25]. GRG
method is an extension of RG method which enables direct treatment of inequality and
nonlinear constraints. Rather than using penalty functions, GRG method modifies the
constraints by introducing slack variables to all inequality constraints; and all constraints are
linearized about the current operating point. Thus the original problem is transferred into a
series of subproblems with linear constraints that can be solved by RG or CG method [38].
However, since linearization introduces error in the constraints, an additional step is required
to modify the variables at the end of each iteration to recover feasibility. In OPF, this
feasibility recovery is performed by solving a conventional power flow [39].
(2) Newton’s Method : Newton’s method is a second-order method for unconstrained
optimization based on Taylor series expansion. The search direction at this point is set
to dk = −H(xk)−1∇f(xk) , where H(xk) denotes the Hessian of f at point xk. Then the
method computes a step size αk > 0 in direction dk satisfying certain step-size selection
rules such as inexact line search with backtracking. However, the method is not guaranteed
to converge to a local minimum as the Hessian matrix H may not be positive semidefinite
in a sufficiently large vicinity of the minimum point. Newton’s method requires the use of
Lagrangian function when applied to OPF problems. Inequality constraints originated from
the power system physical limits should either be treated as equality constraints or omitted,
depending on whether they are binding at the optimal solution or not. Since the active
inequalities are not known prior to the solution, identifying active inequality constraints is
a major challenge for Newton-based solutions for OPF problem [40] as mentioned in Section
2.2. After Sasson et al. [41] have presented an early version of Newton-based OPF method,
14
and Sun et al. [42] have presented a more efficient algorithm employing the Lagrangian,
researchers have made significant contributions focusing on identifying and enforcing the
active inequality constraints [43, 44, 45, 46, 47].
(3) Interior Point Methods(IPM): The difficulty of enforcing inequality constraints was a
motivating factor for applying IPMs to solve OPF. IPMs are a family of projective scaling
algorithms for solving linear and nonlinear optimization problems. IPMs attempt to deter-
mine and follow a central path through the feasible region to the set of optimal solutions.
The key point of feasibility enforcement is achieved either by using barrier terms in the aug-
mented objective function or by directly manipulating the required KKT conditions [48].
When applied to OPF, IPMs have achieved several improvements over years. The popular
methods at this type include Primal-Dual Interior Point Methods (PDIPMs) [49, 50], Mehro-
Table 4.3: Copmarison of DFAL and DPGA-II in Generation Cost
Fig. 4.3: The convergence behavior for Line Loss with DFAL alg.
48
Fig. 4.4: The convergence behavior for Line Loss with DPGA-II alg.
49
Fig. 4.5: The convergence behavior for Generation Cost with DFAL alg.
50
Fig. 4.6: The convergence behavior for Generation Cost with DPGA-II alg.
51
Chapter 5
Conclusion
In this thesis, we studied distributed first-order augmented Lagrangian (DFAL) and dis-
tributed proximal gradient ADMM (DPGA-I and DPGA-II) methods for optimal power
flow (OPF) problem with composite convex node-specific objectives Fi = ξi + fi for each
bus. We assume that each bus can only access its local data, and is able to exchange infor-
mation with its neighboring nodes only. The advantage of solving OPF problem in a such a
distributed way is that buses are not required to know any global parameters depending on
the entire network topology, and by using local communication and consensus constraints,
one can avoid central planner leading to a more robust operation of the system as a whole.
Because if the computation is distributed to network, rather than having it all done at a
central node, there will not be a single point of failure or attack. Moreover, by eliminating
the central planner, the communication burden and memory storage are reduced, which
leads to higher computational efficiency, and at the same time, this mode of operation
also respects possible node-level privacy requirements in the network. DFAL and DPGA-II
algorithms are implemented on IEEE benchmark system with 3, 9, 14 and 30 bus cases.
Since OPF problem is in general nonlinear and nonconvex, the SOCP relaxation is adopted
in order to convexify the OPF problem. The numerical results show that both algorithms
can compute ε-optimal and ε-feasible solution, and for certain cases the SOCP relaxation
was able to recover the optimal solution to the original nonconvex OPF (this is verified by
checking the rank constraints).
52
Appendix
MATPOWER DATA FORMAT
For the sake of completeness, the details of the MATPOWER case format are given in
the tables below taken from MATPOWER 6.0b1 User’s Manual1. First, the baseMVA
field is a simple scalar value specifying the system MVA base used for converting power into
per unit quantities. Following field descriptions are given in (Table A.1-Table A.4)
Table A.1: Bus Data (mpc.bus)
name column description
BUS I 1 bus number (positive integer)BUS TYPE 2 bus type (1=PQ, 2=PV, 3=ref, 4=isolated)PD 3 real power demand (MW)QD 4 reactive power demand (MVAr)GS 5 shunt conductance (MV demanded at V = 1.0 p.u.)BS 6 shunt susceptance (MVAr injected at V = 1.0 p.u.)BUS AREA 7 area number (positive integer)VM 8 voltage magnitude (p.u.)VA 9 voltage angle (degrees)BASE KV 10 base voltage (kV)ZONE 11 loss zone (positive integer)VMAX 12 maximum voltage magnitude (p.u.)VMIN 13 minimum voltage magnitude (p.u.)
LAM P† 14 Lagrange multiplier on real power mismatch (u/MW)
LAM Q† 15 Lagrange multiplier on reactive power mismatch (u/MVAr)
MU VMAX† 16 Kuhn-Tucker multiplier on upper voltage limit (u/p.u.)
MU VMIN† 17 Kuhn-Tucker multiplier on lower voltage limit (u/p.u.)
† Included in OPF output, typically not included (or ignored) in input matrix. Here weassume the objective function has units u.
GEN BUS 1 bus numberPG 2 real power output (MW)QG 3 reactive power output (MVAr)QMAX 4 maximum reactive power output (MVAr)QMIN 5 minimum reactive power output (MVAr)VG 6 voltage magnitude setpoint (p.u.)MBASE 7 total MVA base of machine, default to baseMVAGEN STATUE 8 machine status, > 0=in-service, < 0=out-of-servicePMAX 9 maximum real power output (MW)PMIN 10 minimum real power output (MW)PC1* 11 lower real power output of PQ capability curve (MW)PC2* 12 upper real power output of PQ capability curve (MW)QC1MIN* 13 minimum reactive output at PC1 (MVAr)QC1MAX* 14 maximum reactive output at PC1 (MVAr)QC2MIN* 15 minimum reactive output at PC2 (MVAr)QC2MAX* 16 maximum reactive output at PC2 (MVAr)RAMP AGC* 17 ramp rate for load following/AGC (MW/min)RAMP 10* 18 ramp rate for 10 minute reserves (MW)RAMP 30* 19 ramp rate for 30 minute reserves (MW)RAMP Q* 20 ramp rate for reactive power (2 sec timescale) (MVAr/min)APF* 21 area participation factor
MU PMAX† 22 Kuhn-Tucker multiplier on upper Pg limit (u/MW)
MU PMIN† 23 Kuhn-Tucker multiplier on upper Pg limit (u/MW)
MU QMAX† 24 Kuhn-Tucker multiplier on upper Qg limit (u/MVAr)
MU QMIN† 25 Kuhn-Tucker multiplier on upper Qg limit (u/MVAr)
* Not included in version 1 case format.† Included in OPF output, typically not included (or ignored) in input matrix. Here weassume the objective function has units u.
54
Table A.3: Branch Data (mpc.branch)
name column description
F BUS 1 “from” bus numberT BUS 2 “to” bus numberBR R 3 resistance (p.u.)BR X 4 reactance (p.u.)BR B 5 total line charging susceptance (p.u.)RATE A 6 MVA rating A (long term rating)RATE B 7 MVA rating B (short term rating)RATE C 8 MVA rating C (emergence rating)TAP 9 transformer off nominal turns ratio, (taps at “from” bus,
impedance at “to” bus, i.e., if r = x = 0, tap =|Vf ||Vt| )
QF† 15 reactive power injected at “from” bus end (MVAr)
PT† 16 real power injected at “to” bus end (MW)
QT† 17 reactive power injected at “to” bus end (MVAr)
MU SF‡ 18 Kuhn-Tucker multiplier on MVA limit at “from” bus (u/MVA)
MU ST‡ 19 Kuhn-Tucker multiplier on MVA limit at “to” bus (u/MVA)
MU ANGMIN‡ 20 Kuhn-Tucker multiplier lower angle difference limit (u/degree)
MU ANGMAX‡ 21 Kuhn-Tucker multiplier upper angle difference limit (u/degree)
* Not included in version 1 case format. The voltage angle difference is taken to be un-bounded below id ANFMIN< −360 and unbounded above if ANGMAX> 360. If bothparameter are zero, the voltage angle difference is unconstrained.† Included in power flow and OPF output, ignored on input.‡ Included in OPF output, typically not included (or ignored) in input matrix. Here weassume the objective function has units u.
55
Table A.4: Generator Cost Data† (mpc.gencost)
name column description
MODEL 1 cost model, 1=piecewise linear, 2=polynomialSTARTUP 2 startup cost in US dollars*SHUTDOWN 3 shutdow cost in US dollars*NCOST 4 number of cost coefficients for polynomial cost function,
or number of data points for piecewise linearCOST 5 parameters defining total cost function fp begin in this column,
units of f and p are $/hr and MW (or MVAr), respectively(MODEL=1) ⇒ p0, f0, p1, f1, . . . , pn, fn
where p0 < p1 < · · · < pn and the cost fp is defined bythe coordinates (p0, f0), (p1, f1), . . . , (pn, fn)of the end/break-points of the piecewise linear cost
(MODEL=2)⇒ cn, . . . , c1, c0n+ 1 coefficients of n-th order polynomial cost, starting with
highest order, where cost is f(p) = cnpn + · · ·+ c1p+ c0
† If gen has ng rows, then the first ng rows of gencost contain the costs for active powerproduced by the corresponding generations. If gencost has 2ng rows, then rows ng + 1through 2ng contain the reactive power costs in the dame format.* Not currently used by any MATPOWER functions.
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