A Mixed-Integer Linear Programming Approach for Multi-Stage ...

IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 27, NO. 2, MAY 2012

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A Mixed-Integer Linear Programming Approach for Multi-Stage Security-Constrained Transmission Expansion Planning Hui Zhang, Student Member, IEEE, Vijay Vittal, Fellow, IEEE, Gerald Thomas Heydt, Life Fellow, IEEE, and Jaime Quintero, Member, IEEE

Abstract—The transmission expansion planning (TEP) problem in modern power systems is a large-scale, mixed-integer, non-linear and non-convex problem. Although remarkable advances have been made in optimization techniques, finding an optimal solution to a problem of this nature can still be extremely challenging. Based on the linearized power flow model, this paper presents a mixed-integer linear programming (MILP) approach that con1 security constraints siders losses, generator costs and the for the multi-stage TEP problem. The losses and generator cost are modeled as piecewise linear functions of the line flows and the generator outputs, respectively. The IEEE 24-bus system is used to compare the lossy and the lossless model. The results show that the lossy model provides savings in total cost in the long run. The selection of the best number of piecewise linear sections L is also shown. Then a complete planning framework is presented and a multi-stage TEP is performed on the IEEE 118-bus test system. Simulation results show that the proposed approach is accurate and efficient, and has the potential to be applied to large-scale power system planning problems. Index Terms—Generator cost modeling, loss modeling, mixedinteger linear programming, 1 contingency modeling, piecewise linearization, transmission engineering, transmission expansion planning.

Conductance of the line . scanning matrix. Slope of the th piecewise linear section for loss and generator cost modeling, respectively. scanning matrix. Number of piecewise linear sections used to approximate losses. Number of piecewise linear sections used to approximate generation cost. Total load connected at bus . Loss at the bus in state and in time period . Disjunctive factor for line , a positive number. Total number of generators in the system. nl

Total number of lines, including potential lines. Upper bound of the piecewise linear sections. Active power flow on line period .

NOMENCLATURE

Maximum active power capacity of the line . Active power loss on line period .

Admittance of the line . Investment cost of the line Cost of the generator Cost

in state in time

in state

in time

Maximum active power capacity of the generator .

in million $ (M$).

at time period in M$.

Total cost of investment and operation in M$.

Active power output of generator period .

Discount factor: 0.05.

15 min ramp for generator .

in state in

15 min ramp-up rate for generator . Manuscript received September 12, 2011; revised November 04, 2011; accepted November 18, 2011. Date of publication December 26, 2011; date of current version April 18, 2012. This work was supported in part by the U.S. Department of Energy funded project denominated “Regional Transmission Expansion Planning in the Western Interconnection” under contract DOE-FOA0000068. This is a project under the American Recovery and Reinvestment Act. Paper no. TPWRS-00862-2011. The authors are with the Department of Electrical Engineering, Arizona State University, Tempe, AZ 85287 USA (e-mail: [email protected]; [email protected]; [email protected]; [email protected]). J. Quintero is on leave of absence from the Universidad Autónoma de Occidente, Cali, Colombia. Digital Object Identifier 10.1109/TPWRS.2011.2178000

15 min ramp-down rate for generator . TPL

Total planning horizon in year.

TOP

Total operating horizon in year.

0885-8950/$26.00 © 2011 IEEE

Binary decision variable for line investment: 1 for build, 0 for not build. Maximum bus angle difference: Angle difference at the line time period .

radians.

in state and in

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Set of loads. Set of generators. Set of lines. I. INTRODUCTION

W

ITH the advent of deregulated power markets and the need for the massive integration of renewable energy resources, the reliability and the economy of power systems have come into focus in recent years. The reliable and efficient operation of a power system largely depends on whether the transmission grid is well-planned or not. A rationally planned power system will not only alleviate the real time operation pressure caused by reliability issues, but will also contribute positively to the overall efficiency of the power system. The primary goal of transmission expansion planning (TEP) in power systems is to determine an optimal strategy to expand the existing transmission network to meet the demand of possible load growth and the proposed generators, while maintaining reliability and security performance of the power system. The commonly used objective function for TEP is to maximize the social welfare [2]. In general, the TEP problem is, in essence, a large-scale, mixed-integer, non-linear and non-convex optimization problem which is quite difficult to solve. In practice, the TEP is also a rather complicated process which requires a great amount of work: from the information collection to the scenario selection, from the choice of the pre-defined candidate pool to the final transmission plan, the planning process involves a series of studies to determine where, when and how many transmission facilities are needed. The classical mathematical representation for TEP problems is based on a mixed-integer nonlinear programming (MINLP) model [9]–[11]. The MINLP problems, especially those that have non-convex nonlinear constraints, are however extremely difficult, if not impossible to solve. Most of the existing MINLP solvers are only designed for solving convex MINLP problems [12]. For non-convex cases, the solvers either give a heuristic solution [14], or give no solution at all. Few global optimization solvers like Couenne may solve some special non-convex MINLP problems exactly, but the size of those problems is usually quite limited and the computational time can be extremely long. Considering the highly non-linear and non-convex nature of power systems, various heuristic methods such as genetic algorithms, greedy algorithms and simulated annealing [9]–[11] have been introduced. The advantage of the heuristic methods is that they normally can provide a more feasible solution with less computational effort. However, the drawback is also obvious: the solutions given by the heuristic methods typically do not include a mathematical indicator (e.g., duality gap), and hence provides few clues regarding the quality of the feasible solution. Due to the inherent complex nature of the TEP problem, the linearized DC power flow model is usually used [2]–[8]. This is based on two assumptions: first, the planning is at the transmission level, so the linearized power flow model provides a good approximation. Second, the optimal plan obtained by the TEP model will be validated later using the full AC power flow model and stability analysis.

Compared to the MINLP solvers, the existing mixed-integer linear programming (MILP) solvers are quite mature. With the development of cut plane generation and sophisticated branching strategies, the solvers are now capable of solving large-scale MILP problems with millions of variables within a reasonable time frame [13]. On the other hand, by properly selecting the variables to be optimized and using the disjunctive method discussed in [8], it is possible to reformulate the original TEP problem as an MILP problem without introducing additional approximations. In terms of security constraints, the NERC planning critecontinrion states that power systems must survive an gency [15]. For the linearized model, this criterion simply means that there should be no thermal limit violation with the outage of a single transmission or generation facility. The modeling of security constraints can be found in [2] and [3], where an MILP based disjunctive method is proposed for transmission line switching studies. The active power losses are usually neglected in the linearized power flow model. However, the losses may shift the generation economic dispatch solution and therefore influence the optimal transmission plan. An MILP based loss modeling approach is reported in [7], where the proposed model invokes a piecewise linear approximation to the quadratic loss term. For static planning, additional lines are planned only for a single year; while for multi-stage planning, lines can be built during several years in the planning horizon. That is: besides the problem of where to build the line, the planner also cares when to build the line. The significance of multi-stage planning is twofold. On one hand, some lines are crucial for future high load level, but not for the present load level. Building of these lines can be postponed to avoid unnecessary operational cost. On the other hand, utilities could face budget constraints, and can only afford to build certain lines in one year. It is to be noted that the available commercial software that specifically designed for solving TEP problems in power systems are limited. PROMOD IV is a production cost package which can be used to perform various market simulations and economic dispatch analysis [16]. PLEXOS 6 is another commercial software package designed for power market modeling, resource planning, portfolio optimization as well as renewable integration analysis [17]. In this paper, an MILP model that considers power losses security constraints for multi-stage TEP problems is and presented. The problem formulation is deterministic. The proposed model is formulated using AMPL [18] and solved using the linear solver Gurobi 4.5.1 [19]. The contributions of this paper include: 1) A security-constrained loss modeling approach is proposed in this paper and applied to solve a multi-stage planning problem. In the proposed model, the losses are represented as piecewise linear functions of the line flows instead of the bus angle differences. 2) A piecewise linear generator cost model is used in the mathematical programming model designed for planning studies. 3) A complete planning framework is proposed including the optimization and the security check sub-problems.

ZHANG et al.: MIXED-INTEGER LINEAR PROGRAMMING APPROACH

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The remainder of the paper is organized as follows: Section II presents the linearized mathematical expressions for securityconstrained loss modeling and the generator cost modeling. In Section III, the complete mathematical formulation of the proposed MILP model is presented. Simulation results of the proposed model are provided and analyzed in Section IV by using the IEEE 24-bus and the IEEE 118-bus systems. Finally, concluding remarks are provided in Section V. II. LINEARIZED LOSS AND GENERATOR COST MODELING In this section, a security-constrained loss modeling approach is formulated, followed by the generator cost modeling method. All the equations derived in this section are only for one time period, the subscript is therefore omitted unless otherwise needed. A. Steady State Security Constraints The steady state security constraints are also known as the criterion. The criterion states that the transmission network should be robust enough to handle any single element loss in the system. For the linearized power flow model this implies that no thermal limit violation and load curtailment occurs or genwith the outage of a single transformer/branch facility. eration Similar to the approach proposed in [3], two contingency and are introduced to model the scanning matrices and contingency, respectively. The two matrices are both constant binary matrices .. .

..

..

.

Fig. 1. Piecewise linear approximation of P .

B. Linearized Loss Modeling In TEP studies, a DC lossless model is usually used based on the assumption that the line losses are limited and can be ignored without loss of generality. However, this assumption can be problematic for long term planning studies, where line losses may play a role over a long time horizon (i.e., 20 years) and result in a different network expansion plan. Suppose all transformer ratios are set to their nominal values, the line loss for AC power flow studies can then be expressed as (2) where the subscripts fr and to represent the physical quantities at from end and the to end, respectively. Setting the voltage magnitudes to 1 per unit, (2) can be simplified as (3)

.

(1)

is an by binary matrix and is an by In (1), binary matrix. In the two matrices, a 0 means the status of the corresponding element in the system is out, while a 1 means the includes all 1s, which is to status is normal. The first column represent the base case with all lines in service. Each column of the two matrices is called a state. Generally, for a system with lines and generators, the total number of states for a complete and contingency scanning is . contingency usually reA power system subjected to a quires generation re-dispatch. However, in real power systems, only certain generators can be re-dispatched. These generators usually have a higher ramp rate and serve non base load. On the contrary, the base load generators, though cheaper in cost, usually have a lower ramp rate and therefore do not participate. These base load units can be used to provide long-term system reserve, but are usually excluded as candidates for real time re-dispatch. This fact is considered in this paper: In the proposed model, each generator has a binary flag for re-dispatch. Only the will participate in the generation generators with the re-dispatch following a contingency.

Notice that in (3), the DC model active power loss expression is a quadratic term. In order to formulate an MILP model, the quadratic term in (3) needs to be linearized. One approach is to as proposed in [7]. The perform piecewise linearization of main drawback of this method is that the angle difference may not be zero and cannot be accurately bounded when the from node and the to node are not directly connected. In this method [8] should be used to ensure that the case, the big corresponding constraints are unbinding. In this paper, the proposed security-constrained loss model instead approach is to perform piecewise linearization on of . Notice that in the DC power flow model, the line flow equation is (4) Substituting (4) into (3), the relationship between can be obtained as

and

(5) is shown in Fig. 1, where The piecewise linearization of is approximated by linear sections.

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The mathematical formulation of the proposed security-constrained loss modeling approach is given in (6)–(10):

(6) (7) (8) (9)

Fig. 2. Piecewise linear approximation of the generator total cost.

(10) for existing lines for new lines where the linear combinations of the two non-negative variables and are used to represent the line flow and the absolute value as shown in (6) and (7). The upper and lower are defined in (8). Particularly, the limits of each interval are determined by a binary decision variupper limits of scanning matrix . For lines that are in able and the service, is 1, the upper limits are bounded by ; othto be 0. erwise, is 0, the right hand side of (8) forces and the line loss are 0 In this case, both the line flow as well. The slope of each linear section can be calculated using (10), where the slope increases with the growth of . Constraints with smaller will be (7)–(10) implicitly ensure that chosen to fill first. For instance, consider a line has a rating of 100 MW and the number of piecewise linear sections is 5. Suppose the solution requires the line flow to be 50 MW, should be non-zero then only the first three intervals MW, MW and the values must be MW. This is because the slope is proand portional to , any other combination will inevitably increase the value of the right hand side of (9) and will eventually result in a higher objective value. The main advantage of this loss modeling approach is that it takes advantage of the causal relationship between the line and the line flow. Unlike the angle difference , the line flow is bound to be 0 if the line between node and node does not exist, i.e., not selected or has a contingency. Therefore a “ ” can be applied to (9), rather than using the inexact disjunctive method in [7]. By using this proposed model, the additional variables need to be added are significantly reduced so that the efficiency of the algorithm is greatly improved. C. Linearized Generator Cost Modeling The goal for long term planning is to minimize the total cost of investment and operation. From the perspective of cost allocation, the transmission expansion investment cost is dependent on the line cost, while the system operational cost is mainly determined by the generator operating cost. In order to estimate the total operating cost accurately, a good generator cost model is important. In the existing literature, the total cost curve of a generator is usually assumed to be linear for simplicity purposes

Fig. 3. Total generation and losses versus the linear sections L.

[6], [7], [10], [11]. This means the incremental cost of the generator is a fixed value. However, a more realistic representation of the generator total energy cost curve is a convex quadratic curve as shown in Fig. 3. The quadratic cost curve can be expressed as (11) where is the output of the generator is the total generation cost, and is the coefficient of the th-order polynomial cost. Similar to the strategy in Section III-B, the generator total cost curve is represented by a series of linear sections as shown in Fig. 2. The mathematical formulation of the linearized generator cost model is shown in (12)–(15): (12) (13) (14) (15)

where is the th linear section for the output of generator is the slope of the th linear section for generator , is the maximum capacity of generator . Using this and method, the quadratic cost curve can be linearized by a series of


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linear sections, and the resultant linearized model is more accurate than simply assuming a flat generator cost curve. III. MATHEMATICAL FORMULATION OF THE TEP MODEL In this section, a complete multi-stage security-constrained TEP model will be presented. A. Objective Function For traditional TEP problems, the objective is to minimize the line investment cost. However, from the economic perspective, considering only the investment cost will probably not capture the economic benefits of a TEP project accurately. This is because in comparison to the one time investment cost, the continuous operating cost for a long term TEP project can play a substantial role in the total cost since the expected life of the transmission facilities are usually quite long. Furthermore, transmission planning will enhance the competitiveness of the power market where more players can participate. These market players, many are profit driven, consider the operating cost of their generation units as a major issue. As a result, it is important to coordinate the investment cost and the operating cost properly when formulating the objective function. The ideal objective for the TEP problems is to maximize the social welfare, where the social welfare, sometimes regarded as the market surplus, is defined as the sum of the producer surplus, customer surplus and the congestion cost. Assuming a perfect inelastic demand curve, an equivalent objective of maximizing the social welfare is to minimize the sum of the investment cost and the operating cost. The proposed objective function shown by (16) is to minimize the total discounted line investment cost and the total discounted operating cost over the entire planning horizon

explicitly), however, should be distinguished from minimizing the cost of losses, because the former does not necessarily minimize the losses. If one intends to minimize the losses, then only the term related to the cost of losses should be used in the objective function. Some people may argue that it is always good to have minimal losses, and the people who have this thought tend to penalize the term related to the losses by a large number to get a solution with fewer losses. Without entering into how to properly choose the penalty factor, the authors do not recommend this approach. The authors believe that for every dollar spent, either for the investment or for the operation (to fulfill the load or losses), should be treated in a nondiscriminatory manner. If both investment cost and the operating cost are properly discounted as shown in (16), then it makes no sense to bias the cost of losses since doing so may distort the overall economic value of the TEP project in the long run. B. Constraints The complete constraints set of the proposed linear TEP model is listed as

(17) (18) (19) (20) (21) (22) (23) (24) (25)

(16) (26) where the first term represents the total planning costs and the second term represents the total generator operating costs. The summation of these two costs is denoted as Cost. The term is the cost of building line is a binary decision variable which is used to determine whether to build a line or not; and guarantees that is a onetime investment, the is set to be zero. TPL is the total initial decision variable planning horizon in year and TOP is the total operating horizon is the corresponding energy cost of in year. The term the generator. In the objective function, the discount factor is denoted by , where is set to be 5% throughout this paper. Notice that the cost of losses is not explicitly modeled in (16). This is because the losses are a byproduct of system operation. equals plus , which In the DC power flow model, means that additional MWs need to be generated to balance the load due to the presence of losses. Thus, the cost of losses will eventually be reflected in the generators operating cost. In other words, the losses are modeled implicitly. The inclusion of the cost of losses in the objective function (no matter implicitly or

(27) (28) (29) (30) (31)

for existing lines for new lines. In this model, the constraint (17) guarantees the power balance at every bus. In (18) and (19), a disjunctive factor is introduced to eliminate the nonlinearities caused by the product

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of continuous variables and binary variables in line flow conis 1, straints. The constraint (18) indicates the following: if i.e., the line exists or the line is selected to be built, then the DC line flow equation is forced to hold; otherwise, if for any is 0, i.e., the line is subjected to a contingency or the reason ensures the line is not selected, then the disjunctive factor constraint is not binding. Similar logic also applies to (19): if is 1, then the line flow is limited by the rating of that the line; otherwise, the flow on the line is forced to be 0. On one should be sufficiently large so that the constraints are hand, not binding when lines do not exist. On the other hand, an that is too large will sometimes cause numerical difficulties. The can be calculated by the method in [4]. minimum value of The constraint (20) poses a steady state limit on the angle difference between two nodes. The constraint (21) provides the limit on generation dispatch. The power output of a certain generator must be within its capacity range if the generator is not subjected to a forced outage. The constraint (22) states that the generators must be re-dispatched taking into account their ramp rate. The constraints (23)–(31) represent the linearized loss model and generator cost model proposed in Section II. C. Reduction of the Number of States As mentioned in Section II-A, the total number of system states is , where and are the number of lines and generators, respectively. However, for large power system analysis is usually too explanning studies, a complete are very pensive, if not impossible to afford, because and large with the increase of system size. Notice that most of the constraints shown in this section have three subscripts, e.g., kct or gct. This means that every single constraint actually repreor constraints. In fact, a complete sents analysis is usually not necessary for large power system planning studies, because adding a few lines in one area will have little effect on those lines in areas that are far away. In this case, states, then if the algorithm still goes through all the the efficiency of the algorithm will be severely affected. In order to improve computing speed, the total number of states should be set equal to the number of contingencies plus one instead of . This function is implemented in the proposed algorithm by adding a binary index indicating whether the line analysis or not. or the generator will participate in the Every time before solving the optimization model, the program and contingencies are will first count how many there in the system, and then assign these two numbers to the line and generator states, respectively. Thus, when running the optimization model, certain constraints will only apply to those analysis is needed. lines or generators for which the IV. CASE STUDIES The proposed model has been successfully applied to two test cases. In this section, some simulation results of the proposed model are reported. The first test case is the IEEE 24-bus system [20], on which the performance of the proposed loss modeling approach are tested and analyzed. The second one is the IEEE 118-bus system [21], on which a two-stage network expansion planning will be performed, and a complete planning framework will be presented. The computer that is used to perform all the

TABLE I TEP RESULTS COMPARISON FOR THE IEEE 24-BUS SYSTEM

simulations has an Intel Core(TM) 2 Duo E8500 [email protected] GHz with 3.21 GB of RAM using Gurobi solver 4.5.1 under AMPL. A. IEEE 24-Bus System The IEEE 24-bus reliability test system (RTS) used in this paper has 24 buses, 35 existing branches, 32 generators connected at 10 buses, and 21 loads. All the system parameters including the line investment cost data can be found in [7]. The total load is 2850 MW. The lower bounds of all the generators are set to 0. The total operating horizon is assumed to be 20 years. The first task is to compare the TEP results using the lossy model and the lossless model for the cases that consider and security criterion. In this task, only one do not consider candidate line is allowed to be added for every existing corridor, i.e., the maximum number of lines in a corridor is 2. Since there are 35 existing branches and no parallel branches, the number analysis is of binary decision variables is 35. A complete performed on every line (35 existing lines and 35 potential lines) and all the generators. In this task, the number of the piecewise are set to be 5 and 20, respectively. linear sections and As observed from Table I, the lossy model results in completely different network expansion schemes as compared to crithe lossless model. For the case that considers the terion, more new lines are required to be built, which lead to a higher objective value. As discussed in the previous section, modeling losses tend to shift the cost from operation to line investment, and thus influence the optimal transmission plan. The lossless model may require building fewer lines initially and give a lower estimate of the planning cost, but in the long run, the solution will cost more because of the presence of losses in the real system. As observed from Table I, the lossy model requires building more lines as opposed to the lossless model. As a result, the lossy model gives higher investment costs in both cases. However, the results show that the annual operating costs of the resultant network based on the lossy model are lower than that is given by the lossless model for both cases. Since the line


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TABLE II CANDIDATE LINE INFORMATION FOR THE IEEE 118-BUS SYSTEM

Fig. 4. Computation time and objective value versus the linear sections L.

investment is only a onetime cost, the cost of operating a network based on the lossless model will eventually overtake the one based on the lossy model. Define the term cost turnover as (32) where the numerator and the denominator represents the investment cost and the operating cost difference between the lossy and the lossless model, respectively. For any year beyond cost turnover, the lossy model will give a lower total cost as compared to the lossless model. For the problem considered, the cost turnover for each case is 11 years and 20.6 years, respectively. Having understood the differences brought about by including losses in the model, the next task is to evaluate the loss model itself with respect to the variations of certain model parameters, i.e., the number of the linear sections used to approximate the losses. Since the impact of performing the analysis was demonstrated in the previous task, in order to speed up the program, the criterion is not considered is set to be 20. Figs. 2 and 3 show in this task. In this task, the performance of the proposed loss model versus the number of linear sections . As observed from Fig. 3, the total losses reduce as the number of increases. This result can be easily explained by observing Fig. 1: the larger is, the closer the linear approximations approach the original quadratic curve, and a lower system loss is therefore expected. Since this is a minimization problem, a lower objective is also expected. The objective value and the computing time with respect to are shown in Fig. 4. A larger gives more accurate results; however, too large an will add unnecessary computing burden to the algorithm. This is especriterion is considered: If too large cially true when the an is selected, then the simulation time may become unacceptably long. As a result, a compromise has to be made between accuracy and computing time to obtain the right balance. To be specific: if similar results are obtained using a different , then should be chosen as small as possible. As one can ob, the changes in the serve from the above two plots, with loss and objective value are small; however, the simulation time grows significantly as the grows. Therefore one can conclude is a sufficient number of linear sections for acceptable algorithm performance in this case. A similar study shows that for this case is 10. the sufficient number for

B. IEEE 118-Bus System In this case study, a multi-stage planning study will be performed on the modified IEEE 118-bus system [21]. The system has 118 buses, 186 existing branches, 54 generators, and 91 loads. The line ratings have been reduced to create congestions in the initial network. Similar to the previous case, the same assumption for the line cost data has been made. Notice that in the previous case study, every corridor in which a line exists is considered as a valid corridor for building another line. However in actual power systems, due to physical and regulatory limits it is often impractical and unnecessary to consider building a line in every corridor. There will almost always be a candidate line set in which only a few valid candidate lines are listed. Consideration of these candidate lines is either based on previous operational experience, or due to the generation expansion plan that utilities have for the future. In this case study, seventeen lines have been selected as the candidate lines without loss of generality. Thus the number of the binary decision variables is 17. The complete candidate line set and the related information are provided in Table II. A complete planning framework including generating and the validating of the optimal plan is proposed for this study as described in Fig. 5. As shown in Fig. 5, the planning problem can be divided into two sub-problems. The first sub-problem is the optimization problem. After determining the candidate line set, the MILP based optimization model as proposed in this paper will be solved. If the optimization model yields no result, i.e., the model is infeasible or no line needs to be built, then a larger candidate line set should be considered. Otherwise, the initial network expansion plan given by this optimization model will be recorded and will serve as the input for the security check sub-problem, in which the given initial plan will be checked and validated. Since the DC model is used in the optimization, analysis will be performed as a check. here the full AC The transient stability test will also be performed for critical contingencies to ensure the resultant system will not incur transient stability issues. This paper focuses on the derivation of the mathematical formulation of the network planning model.

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TABLE III TEP RESULTS FOR THE IEEE 118-BUS SYSTEM

Fig. 6. Comparison of the LMPs at each bus. Fig. 5. Complete planning framework of the proposed approach.

The AC contingency check and the stability check are outside the scope of this paper and will not be included. The following simulations show the results for a two-stage security-constrained planning study on the IEEE 118-bus system. The time span of this planning problem is ten years. The planned lines are to be built at the beginning of Year 1 and Year 6 respectively. In total, 8 lines are required to be built in this ten-year planning horizon, and no more than 5 lines should be built in the first stage due to a budget constraint. Starting from Year 1, the objective is to minimize the total cost of investment and operation in this ten-year horizon. The estimated load growth is 20% in five years, which means at the beginning of the sixth year, the total load is 20% higher than the initial load. Furthermore, the increased load is assumed to be distributed in proporanalysis except for the tion to each load bus. A complete single outlet transformer (line 9–10, 71–73, 86–87, 110–111, and 68–116) is performed for each stage. The optimal network expansion scheme and the total estimated cost obtained by the model is shown in Table III. As observed in Table III, four new lines are needed for this two-stage security-constrained planning problem, and all of them need to be built in Year 1 for economic purposes. During the ten-year planning horizon, the total investment and the operating cost are about 4.38 billion dollars. The computing time is about 68.5 min. Notice that the result obtained above only gives a high level picture of how the system can be designed reliably and economically for the long run based on the best estimate at the present. The line flow and the generator

dispatch obtained in the planning studies, however, may not be economical for real time operation due to unforeseen load levels and many uncertainty factors that cannot be forecast. For the planned system, a lower and flatter locational marginal price (LMP) profile is usually expected, and this is observed for most cases. However, notice that the objective of power system planning is to maximize the social welfare, not to minimize the LMP. Therefore the planned system does not necessarily lower the LMP at every bus. The AC model based LMP profile of the initial network without additional lines, the planned network at the end of Year 1, and the planned network at the end of Year 6 are plotted and compared in Fig. 6 using Matpower 4.0, a MATLAB based power system simulation package [20]. One can observe that the LMPs for the planned system at the end of Year 1 are much flatter than the initial network. This is because initially there is more congestion in the system. At the end of Year 6, the initial network without the transmission additions fails to converge using an AC OPF due to the significant load growth, while for the planned system, a higher LMP profile is shown (see Fig. 6). V. CONCLUSION A MILP based multi-stage security-constrained transmission planning model is proposed in this paper. A piecewise linearization approach is employed to model the system losses as well as the generator cost, from which a more accurate and more realistic model is obtained. The security-constrained model expliccriterion using the two contingency scanitly models the ning binary matrices, so that the security requirement is


automatically ensured. A complete planning framework is proposed in this paper including the optimization sub-problem and the security check sup-problem. The first case study shows that the network expansion plan based on whether the loss model is used or not can be quite different. The cost turnover index is introduced in the paper to show that the lossy model provides savings in the total cost in the long run. The simulation results also show that with a proper selection of the piecewise linear sections and , the proposed algorithm can solve the planning problem efficiently. The multi-stage planning studies on an IEEE 118-bus system show that the proposed algorithm has the potential to be applied to large power system planning problems. ACKNOWLEDGMENT The authors would like to thank Dr. K. Hedman from Arizona State University for the helpful discussions and valuable suggestions he provided. The authors also would like to thank Mr. I. Aguayo and Mr. B. Nickell from WECC for their input and support throughout the research work. REFERENCES [1] G. Latorre, R. D. Cruz, J. M. Areiza, and A. Villegas, “Classification of publications and models on transmission expansion planning,” IEEE Trans. Power Syst., vol. 18, no. 2, pp. 938–946, May 2003. [2] R. P. O’Neill, E. A. Krall, K. W. Hedman, and S. S. Oren, “A model and approach for optimal power systems planning and investment,” Math. Program., to be published. [3] K. W. Hedman, R. P. O’Neill, E. B. Fisher, and S. S. Oren, “Optimal transmission switching with contingency analysis,” IEEE Trans. Power Syst., vol. 24, no. 3, pp. 1577–1586, Aug. 2009. [4] S. Binato, M. V. F. Pereira, and S. Granville, “A new benders decomposition approach to solve power transmission network design problems,” IEEE Trans. Power Syst., vol. 16, no. 2, pp. 235–240, May 2001. [5] L. S. Moulin, M. Poss, and C. Sagastizabal, “Transmission expansion planning with re-design,” Energy Syst., vol. 1, no. 2, pp. 113–139, Feb. 2010. [6] J. H. Roh, M. Shahidehpour, and Y. Fu, “Market-based coordination of transmission and generation capacity planning,” IEEE Trans. Power Syst., vol. 22, no. 4, pp. 1406–1419, Nov. 2007. [7] N. Alguacil, A. L. Motto, and A. J. Conejo, “Transmission expansion planning: A mixed-integer LP approach,” IEEE Trans. Power Syst., vol. 18, no. 3, pp. 1070–1077, Aug. 2003. [8] L. Bahiense, G. C. Oliveira, M. Pereira, and S. Granville, “A mixed integer disjunctive model for transmission network expansion,” IEEE Trans. Power Syst., vol. 16, no. 3, pp. 560–565, Aug. 2001. [9] A. H. Escobar, R. A. Gallego, and R. Romero, “Multi-stage and coordinated planning of the expansion of transmission systems,” IEEE Trans. Power Syst., vol. 19, no. 2, pp. 735–744, May 2004. [10] R. Romero, R. A. Gallego, and A. Monticelli, “Transmission system expansion planning by simulated annealing,” IEEE Trans. Power Syst., vol. 11, no. 1, pp. 364–369, Feb. 1996.

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[11] P. Maghouli, S. H. Hosseini, M. O. Buygi, and M. Shahidehpour, “A scenario-based multi-objective model for multi-stage transmission expansion planning,” IEEE Trans. Power Syst., vol. 26, no. 1, pp. 470–478, Feb. 2011. [12] M. R. Bussieck and S. Vigerske, MINLP Solver Software. [Online]. Available: http://www.math.hu-berlin.de/~stefan/minlpsoft.pdf. [13] J. T. Linderoth and A. Lodi, MILP Software. [Online]. Available: http:// homepages.cae.wisc.edu/~linderot/papers/Linderoth-Lodi-10.pdf. [14] R. A. Waltz and T. D. Plantenga, Knitro User’s Manual, Sep. 2010, version 7.0. [Online]. Available: http://seldon.it.northwestern.edu/sscc/ knitro/Knitro70_UserManual.pdf. [15] NERC System Performance Under Normal Conditions, NERC Standard TPL-001-0.1, Oct. 2008. [16] PROMOD IV. [Online]. Available: http://www.ventyx.com/analytics/ promod.asp. [17] PLEXOS. [Online]. Available: http://energyexemplar.com/products/plexosdesktopedition. [18] R. Fourer, D. M. Gay, and B. W. Kernighan, AMPL: A Modeling Language for Mathematical Programming, 2nd ed. Florence, KY: Duxbury, 2002. [19] Gurobi Optimization, Gurobi Optimizer Reference Manual. [Online]. Available: http://www.gurobi.com. [20] R. D. Zimmerman and C. Murillo-Sánchez, MATPOWER User’s Manual. [Online]. Available: http://www.pserc.cornell.edu/mat-power. [21] Electrical and Computer Engineering Department, Illinois Institute of Technology (IIT), IEEE 118-Bus System Data. [Online]. Available: http://motor.ece.iit.edu/Data/JEAS_IEEE118.doc. Hui Zhang (S’09) is from Nanjing, China. He received the B.S. degree from Hohai University, Nanjing, China, in 2008 and the M.S. degree from Arizona State University, Tempe, in 2010, both in electrical engineering, specialized in electric power systems. He is currently pursuing the Ph.D. degree at Arizona State University. He is a research associate at Arizona State University.

Vijay Vittal (S’78–F’97) received the Ph.D. degree from Iowa State University, Ames, in 1982. He is currently the Director of the Power Systems Engineering Research Center (PSERC) and is the Ira A. Fulton Chair Professor in the Department of Electrical Engineering at Arizona State University, Tempe. Dr. Vittal is a member of the National Academy of Engineering.

Gerald T. Heydt (S’62–M’64–SM’80–F’91–LF’08) is from Las Vegas, NV. He received the Ph.D. degree in electrical engineering from Purdue University, West Lafayette, IN, in 1970. Currently, he is the Site Director of PSERC at Arizona State University, Tempe, where he is a Regents’ Professor. Dr. Heydt is a member of the National Academy of Engineering.

Jaime Quintero (M’06) received the Ph.D. degree in electrical engineering from Washington State University, Pullman, in 2005. He is on leave as an Associate Professor from Universidad Autónoma de Occidente, Cali, Colombia. Currently, he is a postdoctoral researcher at Arizona State University, Tempe.