Optimal Capacity Expansion in Electric Power Subtransmission Networks

Optimal Capacity Expansion in Electric Power Subtransmission Networks Felipe Tapia1; Vladimir Marianov2; and Luis Vargas3 Abstract: A procedure is proposed to determine where and when to increase the capacity of lines and transformers in a power subtransmission network. The expansion plan must minimize cost while supplying the demand for energy over a time horizon, keeping the quality and reliability standards of the network and minimizing the impact over the environment. The procedure is iterative and takes into account AC flows, reliability analysis, different scenarios, demand uncertainty, discrete investment costs, voltage constraints, capacity, and power factor. A significant contribution of this procedure is that, along with the expansion plan, it considers the optimization of the operation, specifically the movement of transformer taps and the location and sizing of reactive banks. This allows significantly reducing or delaying the investment, thus reducing its present value. The approach ensures convergence when computing the power flows and allows making an analysis of the effects of distributed generation and, if necessary, load curtailment. The model is tested on the power subtransmission network of the most important electric power distribution company in Chile, serving a city of 6 million people. DOI: 10.1061/共ASCE兲0733-9402共2009兲135:3共98兲 CE Database subject headings: Networks; Electric transmission; Chile; Optimization.

Introduction Electric power distribution networks normally operate in the voltage range of 23 kV and lower. Occasionally, however, it is required to transfer major power flows between various points of the network and subtransmission networks are used for this purpose. These networks operate at voltages of hundreds of kilovolts. Unlike a transmission system, subtransmission networks have usually a low meshed structure and shorter transmission lines. This article deals with subtransmission networks and the optimal expansion of their capacity over a time horizon of several years. Generation, transmission, and subtransmission expansion planning is not addressed jointly here. Instead, this article describes what Sauma and Oren 共2006兲 call a “reactive network planner.” This approach, although suboptimal as compared to joint planning, reflects the situation of a vertically separated market, which is the case of many liberalized markets. Specifically, the practical case analyzed here refers to the industry in Chile, where generation, transmission, and distribution are separate markets and their planning is dealt with by different decision makers. Capacity expansion of the network lines can be performed in different ways: 共1兲 by replacing the type of conductor of a given 1

Graduate Student, Dept. of Electrical Engineering, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, 782-0436 Santiago, Chile. E-mail: [email protected] 2 Professor, Dept. of Electrical Engineering, Pontificia Universidad Católica de Chile, Vicuña Mackenna 4860, Macul, 782-0436 Santiago, Chile 共corresponding author兲. E-mail: [email protected] 3 Associate Professor, Dept. of Electrical Engineering, Universidad de Chile, Beaucheff 850, 837-0448 Santiago, Chile. E-mail: lvargasd@ing. uchile.cl Note. This manuscript was submitted on June 12, 2008; approved on February 9, 2009; published online on August 14, 2009. Discussion period open until February 1, 2010; separate discussions must be submitted for individual papers. This paper is part of the Journal of Energy Engineering, Vol. 135, No. 3, September 1, 2009. ©ASCE, ISSN 0733-9402/ 2009/3-98–105/$25.00.

transmission line; 共2兲 by adding a second circuit in parallel to an existing line; or 共3兲 by adding a new transmission line, that is, by linking two busbars not previously connected directly by a line. Capacity expansion of the transformers, on the other hand, can be achieved by replacing a transformer or by adding a new transformer in parallel. All these investments are costly and have a high degree of irreversibility so they should consider strategic decisions of the company. There has been much discussion on the effects on human health of the electrical power transmission lines. This still being an unclosed issue, we choose to expand the capacity of the network by only increasing the capacity of existing lines and transformers, as opposed to adding new lines. We avoid in this way any further effects of the network over the environment, particularly for the case of subtransmission networks in urban settings. This is more important when the lines are not necessarily buried but some of them are aerial. As the demand for energy rises, the subtransmission network capacity must increase to accommodate larger flows of power. It is also necessary to connect reactive compensation equipment at various points of the network or adjust taps at some transformers to reduce voltage drops at load busses. In general, reactive compensation and capacity expansion have been treated independently. First, the necessary capacity to supply the active power demand is computed and then the reactive injections required for maintaining voltage and power factors within the operational and legal ranges are determined. However, this methodology does not detect the possibility to delay, or even avoid investing in line and transformer capacity, by modifying reactive flows. We jointly address the capacity expansion planning and reactive injections so to reduce the net present cost of the network. This planning approach involves determining where and when to increase the capacity of lines and transformers and where and when to locate reactive banks or adjust transformer taps, within a time horizon, to supply future demand while keeping network quality and reliability, as well as operational standards. All this should be made at a minimal investment cost and at a minimum environmental impact. The underlying optimization problem is highly complex

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due to its large size, nonconvexity and nonlinearity, and dynamic and mixed-integer nature. Furthermore, an important aspect in planning the subtransmission network expansion is uncertainty. Uncertainty can be addressed using either deterministic or stochastic planning processes. Deterministic planning assumes a known and fixed future demand while stochastic planning considers volatility in demand forecast. Another source of uncertainty comes from the fact that the network can have failures, which are traditionally addressed using the well-known “n − 1 contingency criterion,” i.e., the assumption that only one element can fail at a time. In this paper, we propose a dynamic capacity expansion planning model, which minimizes the present value of the investment cost, considering that operational variables 共voltages兲 and power factors should be kept within an operational range. The model considers joint capacity and reactive requirements for all the periods of a time horizon. This implies a simultaneous determination of investments in lines, transformers, and reactive banks, as well as transformers tap positions. The problem is solved by taking into account that the network capacity should be enough to supply the distribution network demand in all possible oneelement-failure situations or n − 1 contingency criterion, as well as for different demand and injection scenarios. Additionally, it is considered that capacities in transmission lines can change for each scenario, as is the case for winter and summer seasons, in which capacities are different due to the differences in environmental temperature. Expansions are carried out in discrete time intervals during the planning horizon and keeping the discrete nature of investments. The model takes into account both active and reactive power and uses nonlinear and nonconvex equations for power flow. These features, together with the fact that investments are discrete, lead to a mixed-integer nonlinear programming 共MINLP兲 model. To solve this model, it is split by scenarios, contingencies, and time periods, using an iterative procedure that converges into a good solution. As a result, a minimum cost investment plan is obtained that provides the schedule for investments in lines and reactive equipment, together with tap positions while keeping variables within the preset ranges. The benefits of this new approach are based upon the accurate representation of the possible network status and on the joint treatment of the capacity and reactive compensation problems. This, plus the addition of other operational variables, allows extending the set of expansion alternatives, which in turn, increases the possibility of reducing the investment costs associated with capacity increase. Particularly, it allows delaying decisions on expansion investments, thus reducing the net present cost. This paper is organized as follows: “Literature Review,” followed by the “Problem Definition” section which provides further details on the topic. The “Operation Flow Model” section describes the optimization subproblem. “Capacity Expansion Planning Algorithm” section presents the expansion model. In the “Application to Real Case” section the model is applied to the network of an actual electric power distribution company in Chile and shows the relevant results. Lastly, conclusions and future work are proposed.

Literature Review In the literature, the issue of expansion planning has been addressed from different perspectives: decomposition methods, combinatorial algorithms, and algorithms based on approxima-

tions. In Romero and Monticelli 共1994兲, Benders decomposition is used for dividing the problem into two subproblems: investment and operation. In Binato et al. 共2001兲, Tabu Search is used and in Da Silva et al. 共2001兲 GRASP is applied. A combined transportation model with linear equations of DC 共direct current兲 power flows is implemented in Villasana et al. 共1985兲. In Al-Hamouz and Al-Faraj 共2003兲, nonlinear programming is used for determining expansion planning, considering investments in new lines, active power losses in new and existing lines, and losses due to the crown effect. In De Oliveira et al. 共2005兲, an expansion planning static model is used in transmission systems, in which a modified version of the D.C. power flow equations is considered, which includes active power losses in transmission lines. Additionally, it uses sigmoid functions to approximate the integer variables and thus solves a linear optimization problem. On the other hand, Jin et al. 共2007兲 described a discrete method for particle swarm optimization, which is applied to solve the static problem considering DC power flow. The objective function includes investments in new lines and transmission losses. In Sepúlveda et al. 共2007兲, an iterative method based on response surface models is developed. These models have been proven to have a high degree of accuracy in modeling complex physical phenomena. The method splits the problem into two parts, a simulation phase and an optimization phase. The response in the simulation phase is used as a parameter in the optimization phase and vice versa. In the simulation phase, AC or D.C. flows, uncertainty in demand, economic dispatch with or without transmission losses, hydrothermal coordination, and transmission price studies can be included while in the optimization phase the response obtained in the simulation phase is used and different optimization algorithms are applied. In Mantovani and Garcia 共1996兲, the problem of dimensioning reactive power banks is solved using a binary search, linearizing the power flow equations and solving a mixed-integer linear programming problem at each node of the search tree. In Pilgrim et al. 共2002兲, an integer genetic algorithm is used to determine the reactive compensation planning that minimizes power losses and voltage deviation. Optimal tap positioning is a MINLP problem. In Adibi et al. 共2003兲, optimal tap positions are found using a model based on a modified interior point method. Penalization functions are combined with barrier functions. Finally, a classical technique to treat uncertainty is the use of different scenarios of future demand 共David and Wen 2001兲.

Problem Definition The net present cost of the expansion of a subtransmission network over a planning horizon must be minimized. This cost includes investments in new lines, transformers, and reactive banks. The network has to supply the necessary power to the distribution network in all the periods of the planning horizon, and it must keep load busbar voltages and power factors within the operational ranges. The subtransmission network capacity must be sufficient both under normal operation and different contingencies of the n − 1 criterion, which takes place when one network element goes off the system due to unpredictable situations. In operations under a contingency, power flows are redistributed and may produce oversaturation of lines or transformers which, under normal operations, are within the operational ranges. On the other hand, there are various scenarios that correspond to different demand and JOURNAL OF ENERGY ENGINEERING © ASCE / SEPTEMBER 2009 / 99

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injection situations. The most typical scenarios are those corresponding to winter and summer, in which load patterns experience changes, the environmental temperature modifies transmission lines capacities, and injections are affected due to the difference in rainfalls, which can also modify busbar types 共PQ, PV, or SL兲. A grid system is assumed, considering expansion of existing lines and transformer either by replacing the conductor or transformer by a larger capacity one or by building a new parallel circuit. No investments are made in new line layouts so the environmental impact of the expansion is kept to a minimum. Capacity increases are discrete. A major operational problem is that of adding reactive compensation equipment at various points of the network. Reactive compensation is added to maintain the power factors and busbar voltages within the legal and operational limits. The solution to this problem is relevant from the point of view of the network capacity since good handling of compensation reduces reactive power flows, reducing in turn the expansion capacity requirements of the network. Reactive power flows can be modified not only by adding or expanding reactive banks but also through tap movements in the transformers, which allow modifying the transformation ratio in magnitude and, in some cases, in angle. Optimal tap positioning is also an operational problem, which is typically solved independently from strategic decisions. We solve reactive compensation jointly with capacity expansion. To include reactive compensation in the problem, planning must be made using AC power flow. Although AC modeling is more complex, it gives precise values of power flows. On the contrary, D.C. modeling can produce significant errors in power flows due to the fact that in subtransmission networks, transmission lines have a high resistance to reactance ratio 共Overbye et al. 2004兲. In an AC model, the power flows are determined by solving a system of nonlinear equations, whose domain is analyzed in Hiskens and Davy 共2001兲. This system poses two difficulties for its solution. On one hand, it can have multiple solutions due to its nonlinearity. Furthermore, all the solutions are not necessarily feasible operation points. On the other hand, the functions are nonconvex. The nonconvex nature of the functions complicates the convergence of solution algorithms. Four types of operational constraints must be considered, which can have a physical, regulatory, or system stability origin: voltages per busbar, capacities by line and transformer, maximum variation in tap positions in adjustable transformers, and minimum power factor at each busbar. Even in transformers in which only transformation magnitude can be adjusted, the impact of tap positions can be very significant over the reactive flow. In this case, it is assumed that tap positions can be modified by period and scenario. For different contingencies, taps should remain fixed since the contingence occurrence cannot be foreseen.

Operational Flow Model Lines and transformers are modeled using their equivalent ␲-circuit. Particularly, transformers are treated as if they were transmission lines but having adjustable tap positions, i.e., transformation ratio. Double-circuit lines are modeled as an equivalent single line. To find feasible power flow values in each period, scenario, and contingency, instead of using a system of equations, a novel

procedure is proposed: use a nonlinear optimization model, with dummy variables representing the deviation from the solution of the equation system. The values of these variables in the solution have physical meanings related to distributed generation or load curtailment and to sizing and location of reactive banks. The nonlinear optimization problem is formulated as follows: Z min CD =

共CPi Pdi + CQiQdi兲 兺 i苸I

共1兲

subject to Pij = −

Qij = −

V2i ViV j y ij cos共␪i − ␪ j − ␥ij兲 + 2 cos ␥ij aija ji aij

∀ 共i, j兲 苸 A 共2兲

冉

y ij ViV j y ij bij sin共␪i − ␪ j − ␥ij兲 − V2i 2 sin ␥ij + aija ji 2 aij

冊

苸A

∀ 共i, j兲 共3兲

兺j Pij = Pinji − Pdemi + Pdi

∀i苸I

兺j Qij = Qinji + Qdemi + Qbani + Qdi 兺j Qij ⱕ 兺j Pij · tan共cos−1 fp−i 兲 P2ij + Q2ij ⱕ Cap2ij V−i ⱕ Vi ⱕ V+i a−ij ⱕ aij ⱕ a+ij Pdi,

Qdi ⱖ 0

∀i苸I

∀i苸I

共4兲

共5兲

共6兲

∀ 共i, j兲 苸 A

共7兲

∀i苸I

共8兲

∀ 共i, j兲 苸 A

共9兲

∀i苸I

共10兲

where the variables are Pdi , Qdi = dummy variables, active and reactive power shortage at busbar i; Pij , Qij = active and reactive power flows through line 共i , j兲; aij = tap position 共off-nominal ratio兲 at transformer 共i , j兲, tap is modified at busbar i; Vi = magnitude of the voltage at busbar i; ␪i = angle of the voltage at busbar i; Pinji = active power injection at busbar i; and Qinji = reactive power injection at busbar i. Note that the variables Vi, ␪I, Pinji, and Qinji are, in some cases, known beforehand and consequently, act as parameters. In fact, the network has slack 共SL兲, load 共PQ兲, and generation 共PV兲 busbars. For slack busbars, Vi and ␪I are parameters 共known a priori兲 while Pinji and Qinji are variables; for load busbars, the situation is exactly reversed while for generation busbars, Pinji and Vi are parameters while Qinji and ␪I are variables. The remaining parameters are: I , i , j Busbar set 共nodes兲 in network; A transmission lines set 共arches兲 in network; CD cost related to total power curtailment in network; CPi cost of active power curtailment 共or additional injection兲 in busbar i; CQi cost of reactive power curtailment 共or additional injection兲 in busbar i; y ij ⬔ ␥ij admittance of line 共i , j兲; bij line susceptance 共i , j兲; Pdemi active power demand at busbar i; Qdemi reactive power demand at busbar i; Qbani reactive power of an existing reactive bank at busbar i; Capij nominal line capacity of line 共i , j兲; V+i , V−i maximum and minimum operational voltage at busbar i; a+ij , a−ij maximum and minimum tap positions at transformer 共i , j兲; and fp−i minimum power factor at busbar i.

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The objective function 共1兲 minimizes the total cost associated with nonsupplied power, both active and reactive. Assuming that the active power injection into the network is enough to cover the demand and losses, shortage of active power at a busbar is due to an insufficient line capacity while shortage in reactive power is due to practical constraints in busbar voltages. Cost can be interpreted as the cost of not supplying an amount Pdi, of power at busbar i, or as the cost associated with injecting the same amount of power at the busbar. If the optimal value of the objective is zero, it means that the total demand can be supplied by the network as it is, within operational constraints. The main purpose of the planning exercise is to achieve this optimal value in all scenarios, contingencies, and power demands. Constraints 共2兲 and 共3兲 determine the active and reactive power flows through each of the network lines. These equations also capture, through variables aij, the degree of freedom provided by transformers with adjustable taps 共transformation ratio兲. For lines, aij = a ji = 1. Constraints 共4兲 and 共5兲 require power balance, and dummy variables are activated only when there is power shortage. Note that for reactive power balance, a constant capacitor bank injection is assumed. Constraint 共6兲 keeps the power factor of each busbar above a minimum value and also relates active and reactive power flows. Constraints 共7兲, 共8兲, and 共9兲 are the network operational constraints: line and transformer capacities must not be exceeded in terms of apparent power, voltage must remain within limits at busbars, and tap mobility range has lower and upper bounds. Constraints 共10兲 are non-negativity constraints for dummy variables, which are activated only when there is power shortage. Optimization problem Z can be unfeasible only if there is excess injection. If operational constraints are not binding, the optimal solution is the usual power flow. Problem Z is nonlinear and nonconvex because of the integer nature of some of its variables and the presence of nonlinear constraints including neither convex nor concave functions. However, Constraints 共2兲 to 共5兲 define a domain that has good convergence properties, as discussed in Hiskens and Davy 共2001兲. Furthermore, the operational Constraints 共8兲 and 共9兲 guarantee the existence of a unique solution within the preset ranges. Unfortunately, there are no global optimization methods that are of practical use for this problem and although the optimal solution is not guaranteed, we tested an interior point method that proved to be effective on this problem. Dummy variables, as used in Constraints 共4兲 and 共5兲, represent the additional power required 共or the power not supplied兲 at each busbar to satisfy the demand, given the specified network parameters 共line and transformer capacities, minimum power factor, and operational limits of voltage and taps.兲 Consequently, if the value of one or more dummy variables in the solution is greater than zero, it means that the operation for the provided conditions is not feasible. Since we assume that the reference busbar 共or the aggregated injection points兲 has enough generation capacity, the presence of dummy variables Pdi that are greater than zero means that there are power flows that are constrained by lines or transformers whose capacity limits have been reached. In this case, the requirements of extra power at the busbars with positive-valued active power dummy variables can be satisfied by increasing the capacity of the saturated line or lines or transformers so the power flows reach the busbars with active power shortage. If there were no binding line or transformer capacities but the aggregated generation in the network was not enough, an option to make the operation feasible is to inject, at each busbar i at

which the active power dummy variable Pdi is greater than zero, Pdi megawatts 共MW兲 at a cost CPi per MW. This is to use a distributed generation approach. Alternatively and if its cost is lower, load curtailment can be used, which reduces the active power requirements. In this last case, Pdi MW of load should be disconnected at busbar i. Similarly, for each busbar i at which the dummy variable Qdi is greater than zero, a capacitor bank providing Qdi M value at risk 共VAr兲 can be added, at a unit cost of CQi.

Capacity Expansion Planning Algorithm Problem Z is at the core of the planning algorithm. By solving Z for each period, contingency, and scenario, infeasibility can be detected and located in the network by looking at the dummy variables. This infeasibility can be avoided by increasing capacities, injecting extra power, managing loads, and adding reactive banks, whatever leads to a minimal cost. Let Ztke the optimization problem Z defined for contingency k of scenario e in period t, i.e., a subproblem of the global case. Through solving subproblems Ztke, all types of contingencies and topological changes in the network can be considered, as well as changes in demand, in distribution, and network parameters 共e.g., capacities can vary by scenario兲. Each subproblem is solved in an independent basis, which facilitates the implementation of parallel techniques. The algorithm starts with an initial set of line capacities and, at each iteration, it determines which line and/or transformer requires a higher capacity. This is done by using the information supplied by the dual prices associated with capacity Constraint 共7兲. An investment is made at every iteration until a stop criterion is met. The increase in capacity is discrete and it depends on the chosen technology and the costs associated to each line. Let ␭tijke be the dual price associated with capacity Constraint 共7兲 for line 共i , j兲 in the subproblem Ztke. This dual price is an estimator of the impact of an investment in capacity of line 共i , j兲 on the total cost. To capture a joint 共contingencies and scenarios兲 and global effect 共all periods over the time horizon兲, the following joint dual price is defined:

⌳ij =

␭t

兺 ␣tke 共1 +ijker兲t k,e,t

where r = discount rate; and ␣tke = probability of occurrence of contingency k in scenario e and period t. ⌳ij represents the expected value of the impact of an investment in line 共i , j兲 in period t, on the shortage cost. Let Fij be the cost of increasing the capacity of line 共i , j兲. The new investment in capacity is defined by comparing this joint dual price with Fij and the next investment is made on the line with the highest ratio ⌳ij / Fij. We remark that the capacity and reactive compensation problem is solved for normal operation, all possible n − 1 states of the network, and different scenarios, at each iteration. After solving all these independent problems, the most convenient investment is made 共in terms of capacity and reactive compensation兲. This investment is checked at the next iteration, for assessing its overall effect. An investment in line 共i , j兲 is made at the first period at which the line is saturated ˆtij JOURNAL OF ENERGY ENGINEERING © ASCE / SEPTEMBER 2009 / 101

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1. •

• 2. 1

2

3

4

Total Cost

5

6

Iterations

7

8

Total Shortage Cost

9

10

Total Expansion Cost

Fig. 1. Ideal convergence

再兺

ˆtij = min t: t

k,e

␭tijke ⬎ 0

冎

Let EP苸 A ⫻ H the set defining the expansion plan, where H corresponds to the time horizon. Initially, this set is empty and one element is added at each iteration. Let CPT be the total investment cost of the expansion plan, and CDT the expected value of the total active power shortage cost, respectively. Then CPT =

兺

共i,j,t兲苸P

3.

t Fij t CDke CDT = ␣ ke 共1 + r兲t 共1 + r兲t k,e,t

兺

where CDtke corresponds to the power shortage cost in contingency k in scenario e and period t. In each iteration an investment is efficiently selected, adding an element to EP and increasing the expansion plan cost. In ideal conditions, the total shortage cost decreases at each iteration as a result of the increased capacities. Fig. 1 shows the ideal convergence of the algorithm, where the total cost curve CT= CPT+ CDT corresponds to the shortage plus expansion costs. The stop criterion can be defined in several different ways: either constraining the shortage risk to be of at most a preset value or by iterating until there are no binding capacity constraints; or the shortage of active power is zero; or iterating until the total cost is minimized 共Iteration 8 in Fig. 1. Note that after Iteration 8, the total cost increases because satisfying all the demand is more expensive than leaving some shortage, i.e., applying load curtailment.兲 In the example of Fig. 1, the algorithm ends at Iteration 10 because ⌳ij = 0 for all the lines, which implies that there are no binding capacity constraints. Note also that in the last iteration the total shortage cost is not necessarily zero, in spite of the fact that there are no binding capacity constraints. This is because the reactive power requirements cannot be met only by increasing capacity; investment in capacitor banks is required. At each iteration, the choice of a new investment is made without taking into account the change in the admittance matrix 共which is a consequence of the new line parameters兲. However, this change is included in the next iterations. A consequence of the modified admittance is that the total shortage cost curve in Fig. 1 is not necessarily monotonically decreasing and an increase in shortage could happen at some iterations. This increase in shortage cost is exclusively due to the change of parameters in the network and it does not imply a bad investment decision. Finally, the capacity expansion planning algorithm can be stated as follows:

4. 5.

6.

The network is initialized: Network topology is defined for all the assessment periods, for all contingencies and scenarios. Recall that the topology can change during the planning horizon as a consequence of changes in the transmission system. These changes are reflected in the subtransmission network as new contact points with the transmission system 共busbars兲 and changes in the nature of existing busbars. Demand and load distribution forecast are included. Ztke is solved for all the contingencies, scenarios, and time periods. Tap movement and reactive compensation is considered. The next investment is made on the line with the highest ratio ⌳ij / Fij. The period ˆtij of the new investment is determined. EP= EP艛 兵共i , j , ˆtij兲其, where 共i , j兲 is the line selected in Step 3. The capacity of line 共i , j兲 is increased ∀t ⱖ ˆtij. The admittance matrix is updated using the new parameters resulting from the capacity expansion. If 兺共i,j兲苸A ⌳ij ⱕ ␧, where ␧ is a small enough value, then stop. Else, go to Step 2.

Application to a Real Case The algorithm was implemented in AMPL 共Fourer et al. 2002兲 and tested with two different optimization solvers: IPOPT 共Wachter and Biegler 2006兲 and KNITRO 共Waltz and Plantenga 2007兲. Both solvers use interior point methods to find a stationary point, that is, a point at which the first order optimality condition is met 共Luenberger 2005兲. The same results with both solvers were obtained but execution times were significantly better with KNITRO. In all the cases analyzed where operational constraints were not binding, the interior point methods implemented in KNITRO and IPOPT got the same solution as that obtained with a standard power flow method. Such comparisons were made against DigSilent software 共DIgSILENT 2004兲. The capacity planning model was applied on the subtransmission network of Chilectra, the most important electric power distribution company in Chile, serving the city of Santiago, with a population of 6 million people. The network, whose diagram is shown in Fig. 2, was completely analyzed, including busbars from 220 to 12 kV, all the transformers, and capacitor banks. The network has 366 busbars, 222 lines, and 73 capacitor banks. A 5-year planning horizon was used, and four possible scenarios per year and four critical contingencies of the system, scenarios, and equally probable contingencies were analyzed. All the system’s power flows were validated 共with relaxed operational constraints兲 using DigSilent results. Taps movement was considered only under normal operation. In the event of a contingency, the position of taps remains fixed. Furthermore, in contingency cases a 10% overcharge is accepted in transmission lines, which is modeled by increasing the actual line capacity by the same amount. The goal of the planning exercise is to supply all the demand in all contingencies, scenarios, and throughout all the assessment horizon, for which a high enough value was used as shortage cost CPi = 100 MUS$ / MW. For reactive power shortage, an average capacitor bank installation cost was used CQi = 20 MUS$ / MVAr. We were able to find a solution in which all shortage costs are

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Fig. 2. Unilinear diagram of Chilectra⬘s subtransmission network

made zero by investing in larger line capacities and some extra reactive banks. In the last iteration of the procedure, all the active power dummy variables were 0 while reactive banks were installed to make zero the corresponding reactive power dummy variables. Each line has its own expansion alternatives and costs: change of conductor, a new circuit in parallel, or a new line in parallel. Capacity expansion in transformers can be made by replacement or by connecting a new one in parallel. Given the number of scenarios, contingencies, and time periods, subproblem Z needs to be solved 100 times in each iteration 共one per each contingency, scenario, and period.兲 However, subproblems whose optimal value is zero do not need to be solved again since they do not have binding operating restrictions. Each of these subproblems has about 3,850 constraints and 3,123 variables, although these figures change slightly per contingency, scenario, and period. The model converged in 19 iterations, solved 629 optimization subproblems, and required a total execution time of 304 min. On average, each subproblem was solved in 29 s, executed on a 3 GHz processor. Fig. 3 shows the lines and transformers that change as a consequence of the expansion plan. Fig. 4 shows the convergence of costs associated with shortage variables 共dummy variables兲, where CTPd and CTQd correspond to the total active and reactive power release cost, respectively, and CDT= CTPd + CTQd is the total shortage cost. Fig. 4 shows how active power shortage decreases as capacity is increased at each iteration. In the last iteration, there is no

active power shortage but there is still some shortage of reactive power that can be compensated at a cost of $3.7 M, by distributing 186 MVAr throughout the network and along the assessment period. Fig. 5 shows the convergence of the total investment 共CT兲, composed of both the total shortage cost 共CDT兲, and the total investment in capacity 共CPT兲. In the last iteration the total shortage cost is $3.7 M, exclusively due to reactive requirements in the network, the total investments in capacity being $85.1 M. All in all, an $88.8 M investment is required to meet all the requirements in the capacity expansion plan. In Figs. 4 and 5, the increase in cost between Iterations 16 and 17 is due to an imbalance in reactive flow after the increase in capacity of a line. This imbalance is corrected adding an important amount of capacitor banks, whose cost is reflected in Iteration 17. In subsequent iterations, the situation is normalized.

Conclusions Although the proposed algorithm does not guarantee a global optimum, it is a heuristic procedure that efficiently seeks a minimum cost expansion plan considering all the operational constraints and the cost of investments in lines and reactive, in addition to the possible contingencies in the network, as well as different demand and injection scenarios. JOURNAL OF ENERGY ENGINEERING © ASCE / SEPTEMBER 2009 / 103

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Fig. 3. Highlighted lines and transformers belong to the expansion plan

The inclusion of dummy variables ensures a fast convergence to the problem of finding power flows for each contingency, scenario, and period. This multiple solution approach would not be possible using traditional power flow techniques as NewtonRhapson or Gauss-Seidel. Furthermore, using these methodologies, the algorithm could converge to an unfeasible solution from the point of view of practical operational constraints. The whole procedure has also good convergence features and finds a good solution for a medium-size network in a few itera-

tions. If the network is too large or if there is a large number of scenarios, contingencies, or periods, the execution time can become a limiting factor. However, the procedure has features that easily allow a parallel implementation and different processors can be used to solve each of the subproblems. Additionally, the dummy variables have a physical interpretation. They represent active and reactive power shortage at different busbars of the network and can be used to analyze and

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Fig. 5. Total cost of the expansion plan

104 / JOURNAL OF ENERGY ENGINEERING © ASCE / SEPTEMBER 2009

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quantify the supply risk under different contingencies and multiple scenarios. Note that it is possible to determine the risk associated with the expansion plan and to determine the VaR. Any situation in which these dummy variables are greater than zero is infeasible. To make the operation feasible, line and transformer capacities are increased or still missing active power shortage can be compensated through either load curtailment, additional power injection, or a combination of both, at the locations at which the dummy variables are nonzero. If the reactive power dummy variables are nonzero at some locations of the network, it means that reactive banks are needed at those locations, possibly with automatically adjustable injection. The algorithm has an investment selection criterion with a global and dynamic vision of the network, that is, the investment decision is made based on the joint analysis of all the time horizon and all the possible contingencies and scenarios. The procedure locates reactive banks and optimizes tap positions. By considering these operational decisions, some of the long term investment decisions are delayed or even cancelled, thus optimizing the present value of the total expansion cost. The optimization subproblem can be used as a basis for a simulation model, which could in turn, be used for analyzing probability distributions of nonsupplied power in each of the network’s busbars. This information can be used for negotiating energy prices that could be made dependent on the failure likelihood for risk analysis purposes or to determine the best distributed generation alternatives. Such a simulation model could include a large number of stochastic variables. The simulation model, together with the planning model, would allow making expansion plans that include risk criteria. From the viewpoint of network management, the model allows to accurately determine the utilization factor of each of the network’s lines and transformers. Slight modifications of the optimization subproblem allow capturing the dynamics of those elements providing flexibility to the system, as FACTS equipment. Thus, it could be also possible to maximize the utilization factor of currently underloaded lines.

Acknowledgments We gratefully acknowledge Chilectra for allowing us to use their subtransmission network for the real case analysis. We also acknowledge the support provided by FONDECYT 共Grant No. 1070741兲 and the Instituto Milenio “Complex Engineering System.” We thank the Centro de Modelamiento Matemático at the Universidad de Chile for providing us with computing support.

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