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POWER SYSTEM EXPANSION PLANNING UNDER UNCERTAINTY

B.G.Gorenstin J.P.Costa Cepel

M.V.F.Pereira N.M.Campod6nico PSRI

Rio de Janeiro, Brazil

Keywords expansion planning, uncertainties, stochastic optimization, decision analysis, multi-objective optimization.

Abstract - This paper describes a methodology for expansion planning of interconnected power systems under several uncertainty factors such as demand growth, fuel cost, delay in project completion, financial constraints etc. The solution approach is based on stochastic optlm1zation techniques, decision analysis, and multiobjective tradeoff analysis. The work is being carried out through the Latin American Organization for Energy (OLADE), which represents 27 member countries in South America, Central America and the Caribbean, and Intcramerican Development Bank (IDB). Case studies with the system of Costa Rica are presented and discussed.

1. INTRODUCTION

One of the basic objectives of power system planning is to determine an investment schedule for the construction of generation plants and interconnection links which minimizes the sum of investment cost and operation cost. The latter includes fuel costs for the system thermal units, interchanges with neighboring systems, and penalties for failure in load supply.

The calculation of operation costs has to take into account the uncertainties in future operating conditions such as load variation and random equipment outages [2]. Probabilistic analysis tools such as production costing [3] and reliability evaluation [ 4] are now widely used by utilities in their planning and operation studies.

The implementation of these probabilistic tools has been an important step toward the incorporation of uncertainty into planning activih:":s. However, several other sources of uncertainty, which mta~ have an important impact on future supply conditil;fa:>, are still represented as deterministic parameters in '\;Ost planning studies [5] :

• load growth rates; • fuel costs; • construction time; • interest rates and financial constraints; • economic growth; • environmental constraints, etc.

A recent World Bank study [6] indicated that some of these sources of uncertainty have been more relevant to

the decision making process than the other probabilistic aspects already represented such as load variation and equipment failures.

In particular, it was felt that the use of deterministic load forecasts in optimal expansion planning models such as WASP or EGEAS favors the construction of large hydro plants. These plants are chosen because of their usually low $/MWh costs, due to economies of scale. However, the planning models cannot take into account one of their drawbacks, which is a long construction time. For example, if the program determines that the optimal dMe for a plant to enter operation is 2004, and the construction time is 8 years, the starting date for constmction is obtained by subtraction (1996, in this case). Unfortunately, this starting date is only optimal if the actual future load is close to the predicted load, which becomes less and less likely as the construction time increases. In other words, plants with long construction times are less flexible if there uncertainty in future loads, and may be less attractive than more expensive plants which can adapt more easily to changing situations.

The need to represent the benefits of flexibility under load uncertainty, as well as uncertainties in fuel costs, investment costs and financial constraints, led IDB and OLADE to investigate a systematic and consistent treatment of the various sources of uncertainty in power system planning.

A pioneering approach in the representation of load uncertainty and construction lead time was developed in EPRI's OVER/UNDER model [20] . In this model, the system expansion is simulated along several "branches" of future load scenarios. The capacity additions are sequentially determined based on a reserve margin criterion and a user-defined target generation mix. The expected operation cost at each stage is calculated by a simplified probabilistic production costing model. The model has some limitations, such as the representation of hydro dispatch in production costing (which requires an optimization over several stages), and the possibility of producing non-optimal expansion sequences due to the simplified capacity addition criterion. It was also felt that the use of expected values to measure the imp:ict of expansion strategies was important , but did not fully capture the "exposure" of the strategy against unfavorablc but plausible scenarios. Some of the uncertainties related to hydro systems were addressed in an expansion model developed for the Pacific Northwest [21] .

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Our proposed approach incorporates several . of OVERJUNDER's uncertainty-handling features into an optimization-based framework similar to those of WASP and EGEAS. It also represents the speeific features of hydrothermal system operation, as in Ref. [21]. The methodology is based on three classes of techniques: . decomposition and stochastic optimization provide the basic framework, and allow an implicit representation of all alternative investment strategies [7]; decision analysis is used to represent the dynamic aspects of decision making as uncertainties are resolved over time [8]; and "hedging" objectives from tradeojf analysis help select flexible and resilient expansion strategies [9, 10]. Case studies with the Costa Rica system are presented and discussed.

2. PRELIMINARY DEFINITIONS AND NOTATION

2.1 Traditional Formulation

As discussed in the Introduction, the traditional objective of expansion planing is to determine an investment schedule which meet load demand and minimizes the present worth of investment and operation costs. The determination of . this optimal expansion plan can be formulated as the following optimization problem:

z= T

Min L Pt [ c Xt + d Ytl t=l

subject to

(1.1)

At Xt ;::_bi (l.2) t

I. E x't + FtYt;::: ht (1.3) 't=I

for t = 1, ... , T

where: xt vector of investment variables for stage t; for

example, xti = 1 indicates that plant i will be constructed in stage t; conversely, xti = 0 indicates that the plant will not be constructed in that stage

c vector of investment costs Yt vector of operation variables in stage t (generation

in each plant, load curtailment etc.) d vector of operation costs p discount factor for stage t

Eqs. (1.2) represent constraints on investment decisions (financial constraints, upper and lower bounds on starting dates etc.). Eqs. (1.3) represent operating constraints (generation limits, bounds on storage, load supply equation etc.). Note that the operating constraints depend on the plants x

1 constructed up to stage t.

In order to simplify the notation, the above formulation ( 1) and the following formulations (2) and (3) do not explicitly represent the "lead time", i.e. the time between the investment decision and the plant completion (start of

operation). In the actual model, the decision variables represent the lead time for construction.

2.2 Uncertainties in the Planning Process

In formulation (1), we assume that the problem parameters, {c, d, A, E, F, b, h}, are constant values, i.e. that they are known with certainty. In this case, the optimal solution x* of problem (1) is indeed the most adequate expansion plan. However, as discussed in the Introduction, there is a great uncertainty with respect to their "true" values, leading us to question the adequacy of x*. In other words given that the future "reality" will be different than the one assured in the solution of (1), how can we say that x* is optimal?

2.3 Alternative Solution Approaches

Some techniques have been proposed to address this question, such as the deterministic equivalent approach and sensitivity analysis.

2.3.1 Deterministic Equivalent

In the deterministic equivalent approach, the expansion problem (1) is solved for the forecasted values of the parameters, and the optimal investment decision associated to the first stage of the optimal solution is implemented (for example, the one corresponding to the current year). In the next stage, the forecasts are updated, based on the most recent information available, and the optimal expansion plan is recalculated for all the horizon. Once more, the decisions associated to the current stage are implemented, and the process is restarted. This approach is attractive in intuitive terms, because it recognizes the fact that the plan will be effectively readjusted as new data becomes available, and tries to accommodate the effect of uncertainty by constantly readjusting forecasts.

Unfortunately, the deterministic equivalent approach does not necessarily lead to the most adequate expansion strategy. The basic reason is that an investment decision for the current stage is only optimal under the assumption that the future conditions will occur as predicted. Otherwise, this decision may be inadequate and, even, in some cases the worst possible. For example, the construction of a large hydro plant today could be the optimal solution of (1) given a forecast of high load growth, which ensures that large amounts of available energy will be absorbed in the future. However, if the actual future loads are substantially lower than expected, the first-stage decision would be inadequate.

2.3.2 Sensitivity Analysis

In .this approach, we have N scenarios { c, d, A, E, F, b, h} 1, i = 1, ... , N. Next, we calculate the optimal expansion plan (1) for each scenario, obtaining a set of solutions

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{ x*} i, i = 1, .. . , N. Based on this set, several analysis are carried out. For example, if a given plant appears as part of the optimal solution in all scenarios, one concludes that the plant is "robust" and is therefore part of the best expansion strategy.

The basic limitation of the scenario approach is the difficulty of constructing one expansion plan which is adequate on the average for all scenarios, starting from optimal expansion plants which were tailor-made for each scenario. The only exceptions are, as mentioned previously, the robust plants which, because they are part of the optimal solution of each scenario, are also part of the global optimal solution.

3. OUTLINE OF THE PROPOSED APPROACH

The proposed approach is based on two observations:

• the concept of a "plan" as an exlJansion schedule is inadequate; it is necessary to have expansion strategies which take into account the "tree" of possible future scenarios and the dynamics of the decision making process as the uncertainties get resolved over time;

• it is necessary to reformulate the traditional objective function of planning studies at least in two aspects: (a) the use of expected values, which do not capture the "exposure" of a plan to unfavorable, but plausible scenarios; (b) the use of only one scalar measure (usually cost), which is not adequate to represent conflicting objectives such as power production and environmental impacts.

3.1 Development of an Expansion Strategy

As discussed previously, there is not a unique investment schedule, which can be determined for all stages at the beginning of the first stage, but an expansion strategy, in which the decision in a given stage depends on the observed parameter values (demand, cost, etc.) observed in the previous stages.

The formulation of the expansion problem with updating of investment decisions will be illustrated for a two-stage problem, in which there is one scenario for the first stage, and two scenarios for the second stage. For notational simplicity, we consider that the discount factor ~ is equal to one, and that the investment and operation cost vector c and d are the same for all stages and scenarios. The stochastic expansion problem is represented as:

subject to (2)

Af21 "<::. b21

AzX22 2 b22 E1x1 +FJ11 "2:.h, E1x1 + E21X21 + F1.>'21 2 '121 E1x1 + E22Xi2 + F12Y 22 2 '122

In problem (2), the first-stage decision x1 is taken before the second-stage scenarios are known. This knowledge will be available in the beginning of that stage, so that the second-stage investment decisions Xii and Xi2 can be "taylor-made" for each scenario.

This formulation allows the representation of several important planning aspects, such as a "tree" of demand forecasts which allows the representation of the benefits of flexibility (for example, plants with smaller construction time, which can be adapted to different scenarios).

3.2 Minimax Objective Function

The use of the average costs as the decision criterion in stochastic problems is adequate for "high frequency" phenomena, in which a representative sample of all values is expected to occur along the planning period. One typical example is the operation cost of a thermal-based system. Along 10 years of operation, one expects the most likely combinations of generation outages (which would correspond to scenarios) to occur.

However, the same is not true for "low frequency" phenomena such as the load level in expanding systems. Due to the great uncertainty in the load growth rates, we can expect a great variability of investment and operation costs for those scenarios. As only one of these scenarios is actually going to take place, we can question the meaning of investment decisions which are optimal "on the average" for all scenarios.

One possible way of representing this aspect is to calculate the loss, or regret, associated to each scenario. The regret is the difference between the actual cost and the cost that would have been incurred if there was prior knowledge that a given scenario would take place. The minimax criterion, as defined in [14, 19], is to minimize the maximum regret associated to each scenario.

The expansion problem (2) is reformulated as:

z= Min a. subject to (3)

a. "<::. cx1 + cx21 +dy,+ dy21 - z, a. 2 cx1 + CXi2 +dy, + dy22 - Z2 A 1x1 2 b,

AzX21 2 b21

AzXn 2 b22 E 1x1 + FJ!, 2h, £ 1x1 + E2,x21 + F2iY21 2 h21

Etx• + E22Xi2 + F12Y 22 2 '122

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where the first two constraints represent the regrets associated to scenarios I and 2. Because ex. is greater than or equal to each regret, it is in particular greater than or equal to the maximum regret. Because the objective is to minimize ex., it will be equal to the maximum regret.

The value z1 is the optimal cost for the deterministic problem associated to scenario I:

z = I Min cx1 + cx21 + dy1 + dy21 subject to (4)

A1x1 "?.bi

AzX21 "?. b21

E1x1 +F.)I, "?.hi E 1x 1+ E21X21 + F1Jl21 "?. h21

In tum, z2 is the optimal deterministic cost for scenario 2, formulated in a similar way as (4).

4. SOLUTION ALGORITHM

The solution approach is based on the decomposition of the expansion problem into two subproblems (investment and performance analysis) which can be efficiently solved via mathematical programming decompositiOn techniques [7], and is illustrated in Figure I.

The investment subproblem represents implicitly all investment strategies for the multi-stage "uncertainty tree". It is modeled as a multi-stage mixed integer (or linear) programming problem, solved by a LP or Branch and Bound algorithm.

The output of the investment subproblem is a trial expansion strategy, which determines the investment decisions contingent to each node of the "uncertainly tree".

The trial strategy is then evaluated in the performance analysis subproblems. These subproblems comprise production costing for each combination of hydrological alternatives, construction delay and fuel cost scenarios, and financial analysis for each branch of the "uncertainty tree".

The production costing module represents the chronological operation of the hydro plants, the thermal dispatch, and power exchanges with neighboring areas. This subproblem is solved by a minimum cost network flow with gains [16].

The financial module represents constraints on yearly expenditure: the yearly revenues from operation plus external loans should exceed the yearly disbursement of

Investment Subproblem

Scenario 1

0-0-------G

Scenario 4

Scenario 1 ... 4

Figu-e 1 - Oec()IT1)0sition Scheme

Scenario 1

Scenario 2

Scenario 3

Scenario 4

Operation subproblem Hydrological scenario 1

Operation subproblem Hydrological scenario n

Operation subproblem Hydrological scenario 1

Operation subproblem Hydrological scenario n

Financial Analysis 1----------~

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each project under construction, plus repayment of previous debts. There are ceilings on the amount of money that can be borrowed from each class of financial institution (e.g. IDB, private banks etc.)

One ever-present concern in stochastic opt1m1zation problems is the "curse of dimensionality", i.e. the combinatorial "exlllosion" of scenarios. In our model, the "uncertainty tree" represents load uncertainty. and is completely enumerated. This decision was based on discussions with planning engineers from several utilities, which considered that a more limited set of load scenarios judiciously chosen by the users was more effective than a multitude of scenarios generated by a stochastic model.

In turn, the hydrological scenarios are sampled from historical records or synthetic streamflows. A similar sampling scheme is used to represent the impact of fuel cost and construction delay uncertainties.

The integration of investment and analysis subproblems is made through the Benders decomposition technique [17] . In the Benders scheme, the feedback from the analysis subproblems is given by sensitivity vectors, which indicate the variation of the analysis results (operation costs or financial costs, in this case) with respect to incremental variations in the investment plan. A formal derivation of the Benders decomposition technique can be found in [18].

One attractive feature of the decomposition method is the availability, at each iteration, of lower and upper bounds on the optimal value of the objective function. Each upper bound is associated to a feasible solution, and the smallest value can be taken as the solution if the procedure is terminated short of optimality.

The extension to handle the mm1max criterion is straightforward. The optimal deterministic costs {zi} are obtained in preliminary runs of the expansion planning model using deterministic loads, expected fuel costs, and no construction delays.

5. CASE STUDY

5.1 System Description

The Costa Rica power system is predominantly hydro (about 90% of the installed capacity of 790 MW). The existing system consists of 6 hydroplants and 4 groups of thermal units.

For the planning period under consideration (15 years), the total number of candidate hydroplants is 12, with a total capacity of 944 MW. There are also 8 candidate groups of thermal units, including: 165 MW in geothermal units, 300 MW in coal plants, 140 MW in gas

turbines and 224 MW of low speed diesel units. The hydro system configuration is shown in Figure 2.

Pl•witll ... rH«VOir

• Run-of-the riverpllnl &p..ioapl ...

'\7' widr. rncrvoir

0 EJcr>-"ioaNM)f-chl river-pi•

_..._ Garita

_..._ Ventanas Garita

-o- Pirris

--0- Los Llanos

-0--0-Toro 1 Toro 2

Rio mac o

Cachi Pacuare

Angostu a

Siquirre i Af•"'' Corobici

Sandillal

-0- Ayil

-0- San Fernando

Figure 2 - Hydrosystem configuration

5.2 Uncertainties

The following uncertainties were represented:

5.2.1 Delay in construction time

Candidate Plant Delay (years) Probabilitv Siauirres (hydro) 02 - 03 - 05 0.4 - 0.4 - 0.2

Pirris (hydro) 02 - 03 - 05 0.4 - 0.4 - 0.2 Carbon (thermal) -01-00-01 0.4 - 0.4 - 0.2

Geo (thermal) 02 - 03 - 05 0.4 - 0.4 - 0.2

Table 1 - construction time uncertainty

obs: a negative delay time in table 1 means that the plant Carbon could start-up one year before the schedule, (probability: 40%).

5.2.2 Fuel costs

The probability distribution of coal prices was set as follows: reference cost - 60% probability; high cost (20% increase) - 40% probability.

5.2.3 Load growth

Four load scenarios were represented, as shown in Table 2. The "tree" structure of the load scenarios is similar to that of the investment strategy, shown in Figure 3. The yearly load was decomposed into quarterly stages in the operation subproblems. Two load levels were represented in each quarterly stage: peak load: 60% of total load, olT peak load: 40%.

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Load Scenarios Year LL LH HL HH 1992 462 462 462 462 1993 483 483 492 492 1994 505 . 505 524 524 1995 527 527 558 558 1996 551 551 594 594 1997 576 576 633 633 1998 602 602 674 674 1999 623 635 711 725 2000 644 670 750 779 2001 667 706 792 837 2002 690 745 835 900 2003 715 786 881 968 2004 740 830 930 1040 2005 765 875 981 1118 2006 792 923 1035 1202

Table 2- Demand Scenarios (average MW)

5.2.4 Inflows

Inflow stochasticity is represented by two 15-year hydrological sequences (60 quarters each).

5.3 Problem Solution

5.3.1 Investment Decisions and Objective Function

The minimax criteria was used to obtain an ·energy expansion strategy for the Costa Rica electric system for the period 1992/2006.

Decisions are considered as a tree representing the different decisions accordingly the demand growth scenarios. There are 45 decision nodes for the 15 years of the planning period. The lowe.st decision scenario in Figure 3 is associated to the demand scenario LL; the highest, to HH.

Figure 3 - Tree of Investment Decisions

In this case study there are 60 constraints and 419 variables in the investment problem. Each operation subproblem is composed of 289 constraints and 648

variables. The operation subproblem is solved 144 times per iteration. of the decomposition scheme. The total number of constraints and decision variables for the whole problem is respectively 41,676 and 93,731.

The investment subproblem in this case was modeled as a linear programming problem (i.e. continuous values of the decision variables were allowed). The operation subproblem uses an aggregate representation of the hydro plants [19], and is solved by a minimum cost network flow with gains.

The optimal minimax expansion strategy was obtained in 29 iterations, for a tolerance of 1 %. The total execution time was 79 min in a 486/66MHz microcomputer.

The regrets associated to each scenario are presented in Figure 3.

LL LH HL

Figure 3 - Minimax regrets (millions of US$ dollars)

6. CONCLUSIONS

HH

A methodology for expansion planning of interconnected power systems that takes into account uncertainty factors such as demand growth, fuel cost, delay in project completion, financial constraints etc. was presented. The solution approach is based on decomposition and stochastic optimization techniques. Decision analysis was used to represent the dynamic aspects of decision making as uncertainties are resolved over time; and "hedging" objectives from tradeof! analysis help select flexible and resilient expansion strategies. The methodology was illustrated in case studies with the system of Costa Rica

7. ACKNOWLEDGMENTS

This work is being carried out through the Latin American Organization for Energy (OLADE), directed by Dr. Gabriel Sanchez and sponsored by the Interamerican Development Bank (IDB), through Dr. Jaime Millan. The guidance and support of Drs. Fernando Montoya, Rafael Campo and Manuel Tinoco (OLADE) and of the advisory panel, composed of Steban Scocknic (Endesa, Chile), Javier Gutierrez (ISA, Colombia), and Jose Manuel Mejia (consultant, Colombia), is gratefully acknowledged. This project is also supported by the World Bank,

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DOE/ Argonne National Laboratories, Bonneville Power Administration (BPA) and the International Atomic Energy Agency (IAEA), through Dr. Pablo Molina. Last, but not least, the authors wish to thank Eng. Talita Oliveira Porto for her contributions to the work.

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[14] AN.Halter, G.W.Dean, Decision Under Uncertainty, South-Western Publishing Co., 197 l .

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[18] J.F.Benders, "Partitioning procedures for solving mixed variables programmng problems", Numer. Math 4, 1962.

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