Transmission expansion plan: ordinal and metaheuristic multiobjective optimization

Transcription

1 Paper accepted for presentation at the 2011 IEEE Trondheim PowerTech 1 Transmission expansion plan: ordinal and metaheuristic multiobjective optimization J. D. Molina, GSM, IEEE and H. Rudnick, Fellow Member, IEEE Abstract-- Electric energy transmission is essential for the operation of competitive energy markets. Transmission expansion planning has been defined as a complex combinatorial optimization problem. This work puts forward a description of the solution techniques and alternatives to implement the transmission system s expansion. A model that considers Multi-objective Optimization MOO criteria is proposed under the concepts of Tabu Search TS Ordinal Optimization OO and Pareto optimality. The model proposed generates expansion plans under the Pareto optimality approach. It shows acceptable solutions under robustness and algorithmic speed criteria. The results obtained in the test systems show that the model developed is effective to find the solution for the combinatorial problem. Multi-objective optimization defines a set of feasible solutions that establishes expansion plans scenarios. Index Terms Transmission network expansion planning, combinatorial optimization, multiobjective optimization, metaheuristics, Tabu search, ordinal optimization. T I. INTRODUCTION he grid expansion is a complex problem. The definition of an optimal grid involves a set of possible combinations that makes its expansion a problem that is difficult to solve. In addition, complexity increases when considering the energy that is injected in different points of the grid and considering the uncertainty associated to time and demand. Basically, the aim is to minimize operating costs and the investment costs that are determined by the place and number of lines required by the system in addition to the moment in which such expansion is required. There are different ways to consider transmission in an electric system. Research has centered in mathematical and heuristic programming methodologies to define the optimal grid, both static and dynamic. For the case of the grid investment and expansion models, the idea is to minimize the investment and/or operating costs of the system subject to economic, reliability and sustainability constraints. The different techniques try to obtain feasible and robust solutions with acceptable processing times. In general, the evolution of the techniques has been based on the development of more robust models that support large size transmission models and that reduce the algorithm running times. It is important to bear This work was supported by Pontificia Universidad Católica de Chile, MECESUP(2), Fondecyt, and Transelec. H. Rudnick and J.D. Molina are with the Department of Electrical Engineering, Pontificia Universidad Católica de Chile, Casilla 306, Correo 22, Santiago, Chile ( hrudnick@ing.puc.cl). in mind that the transmission investment and expansion problems are non-linear with a non-convex nature. This type of problem belongs to the NP-Complete family of problems that have exponential running times as the system and the realism of the model grows. Below there is an introduction of the most recent applications for Transmission Expansion Planning -TEP-, focuses on meta-heuristic optimization [1], the proposer TEP model and conclusion are presented. II. TEP AND HEURISTIC OPTIMIZATION TECHNIQUES Meta-heuristic techniques are a hybrid version of classical and heuristic optimization. Basically, they try to take advantage of the two methods, namely, a feasible solution at a low computing cost. Recent advances in this type of technique have shown that it s possible to achieve better solutions in large size systems (as transmission systems actually are). These techniques achieve better scenario evaluations; better expansion plans options and better transmission models. According to literature, the most recent applications have been in the multiple period planning with evolutionary algorithms, Genetic Algorithms GA, swarm particles and simulated annealing. Other techniques have been the Greedy adapted random search, evolutionary strategies, planning with reliability criteria using simulated annealing and TS, Niche genetic algorithm, genetic Chu-Beasley, constructive heuristic algorithms with classical optimization techniques, fuzzy integer programming and the dynamic planning primaldual interior-point methods. Finally, emphasis is made on hybrid applications with the OO and TS [2], Genetic Algorithm considering social benefit functions, GA (NSGA II) with fuzzy decision processes and meta-heuristic hybrid algorithms [1]. III. TEP MODEL The model proposed is based on multi-objective optimization with the TS approach. TS is a meta-heuristic technique based on the adaptive memory (short and long-term adaptive memory) and solution strategies (intensification and diversification). Basically, it is an iterative local search process. The current solution is modified by a better solution located in the solutions space (neighborhood). In addition, it uses the Tabu list criterion to prevent falling into a local optimum. The TS multi-objective optimization considers a set of solutions by means of metrics, reduction to a single objective function, and weighting of objectives, among others. Basically, a boundary of non-dominated solutions is established. The solutions explored by MOO are defined according to the concepts of TS, path re-linking, and they are /11/$

2 2 stored in the so-called Tabu list [3]. Below there is a description of the fundamentals of MOO and OO. A. Multiobjective optimization MOO is centered in minimizing/maximizing more than one objective function. The solution to this type of problem is more complex. The concept that is mostly applied in MOO is the Pareto dominance. The solution to the MOO problem is a solution set or solution vector that is not dominated by another vector and they are an equivalent of Pareto. This vector set is commonly known as the optimal Pareto set. In the MOO, it s possible to identify at least two criteria for its solution. In the first one, the multiple objective functions reduce themselves to a single objective function by the creation of a compounded objective function, namely, the weighted sum of each objective. The compounded objective function is optimized by means of this single objective function. The most known classical methods are: weighted aggregation, programming by goals and ε-constraint [3]. Finally the second criterion solves the multi-objective problem through the direct search of the optimal Pareto set. In this type of approach, the intelligent or meta-heuristic methods are concentrated based on the population [4] or on the TS [3]. min y = F(x) [ f 1 (x), f 2 (x),, f n (x),] T s.a. B. Ordinal Optimization g j (x) 0, j = 1, 2,..,M. Ordinal Optimization -OO- is based on two ideas. The first one puts an order in a simple manner by means of the value of the objective functions. The second one transforms the optimization into an easier problem [5]. That is, it centers itself in identifying the best solutions that provide a reliability that is higher than its results considering that these solutions form part of a high-probability optimal solution. In brief, OO consists on an optimization problem with a single objective function, which is defined as: min y = J( θ) (2) θ Θ {constraints set } Where is a feasible solution, is the search space and J() is the cost function. So, M feasible solutions are considered in search space, for each feasible solution, where each J() is evaluated. This determines a set of solutions that can be ordered in ascending order, that is, from the lowest cost function to the highest one of the M feasible solutions. This type of arrangement creates an increasing monotonous curve that is known as a feasible solutions ordered curve. According to the concept of ordered curve, subset G of search space is enough and proper proof to find an estimated optimum that is found in G and S. Fig. 1 is described in the solution s space and the identification of the optimal. The amount of samples that are enough will depend on the calculation method, which uses criteria such as the hyper-geometric distribution and polynomial functions [5][6]. (1) Fig. 1. Expected optimal solution space In our model, OO is used to accelerate the calculation processing and the number of samples is determined with the hyper-geometric distribution. Additionally, we considered investment restrictions and number of possible lines in each random expansion plan. This approach is known a constraint ordinal optimization to determine the number of samples to consider in the model [5]. Random expansion plans are defined and for each one of these plans a DC power flow is ran to identify a representative sample of the solution space that has a certain probability of containing the optimal solution. For example, in network combinatorial problems, the solutions space grows exponentially with the amount of possible arrangements. OO reduces the processing of each one of these possible solutions when processing a certain amount of combinations that contain an optimal solution with a certain level of probability. For example, the hyper geometric distribution function is determined by Eq. 3 [5]. g N g min( g, s) i s i P ( k) = (3) i= k N s We assumed that the number N of alternative or possible combinations to analyze the expansion transmission is The number of representative samples g is 50, such that feasible plans are considered to analyze in the optimization process. A search space s of bounded solutions equivalent to 10% of the number of alternatives is considered, s = 100. Now, if we want to establish the probability of finding at least one solution, k=1, acceptable or good solution of the bounded space, an acceptable solution will be found in the bounded search space with a probability of 99.6%. This implies a significant reduction of the computational effort. It should be noted that as the number of buses of a power system grows, the combination of possible lines grows exponentially, but the information of lines is not available. It is therefore necessary to evaluate the real and feasible lines that have all technical and economic information. A form of considering the representative space N, is to establish that for at least an acceptable solution, the probability is P = 1 - (1 - k p ) N, where k p represents the percentage of samples that contains acceptable solutions regarding the total space of solutions. Therefore, the optimal number of samples N* is determined by Eq. 4 [5, 6].

3 3 N * ln 1 ln 1 ( P) ( ) k p For example, the optimal number of samples N to find at least one acceptable solution that belongs to the top rate of 0.1% solutions, with a probability of 90% and 99.99% is 2301 and 9205, respectively. C. Security Criteria In power system assessments, it is fundamental to establish system behavior after fault occurrences. Commonly the approach N-n is used to evaluate the reliability of the power system, where N corresponds to the total number of elements of the system, such as transmission lines, generators and transformers, and n corresponds to the elements that fail (usually used n = 1 or 2). In this article, the expansion plans are evaluated with the reliability approach N-1, criterion used in the Chilean energy market. This market considers a restricted N-1 criterion, N-1*, comparing the annualized cost of expansion investment with respect to the failure cost, the cost of non supplied energy. This implies that it is possible to consider expansion plans with load shedding greater than zero. Moreover, we assumed that the lines belong to the transmission trunk system and their fault probability is such, that it is necessary to evaluate the system response to simple contingencies of any element of the system. A simple contingency is the untimely failure of an element of the system, it may be a generating unit, a load, or a series element of the transmission system. In this article we only considered a simple contingency on transmission lines to identify the worst scenario. IV. CASE STUDIES Below we introduce the methodology of the model proposed. Basically, the amount of possible combinations of the transmission system is determined. For this number of combinations, the minimum amount of random samples is determined so as to find at least one solution that with a probability p is considered within the range of acceptable or top solutions. The technical feasibility is tested for each one of these samples, a DC power flow is run and the plan s operating and investment cost is obtained. In this space of feasible solutions, an initial solution is determined to apply the Pareto dominance concept and a list of possible candidates, a Tabu list, Pareto list and a elite solutions are created, which eventually constraint the solution to expansion plans under the Pareto criteria. The weighted aggregation method, which assumes an equal weight for the two objective functions, was applied. Pareto solutions of the system determine possible solutions. In turn, we do a Tabu search process considering these solutions generated for each neighborhood and evaluate the reliability criterion. In each neighborhood the optimal solution is identified with a minmax criterion. We identify the worstcase scenarios after failure, the maximum load shedding, and with the Euclidean distance criterion we choose the minimum between the two objectives, load shedding and investment (4) annuity. Finally, the iteration in function of the optimum s search is determined by the amount of ordinal defined samples. A. Modeling Parameters The methodology was implemented in MATLAB 7.3. We used an Intel Core 2 Duet 1.50Hz computer with 2 GB of RAM. A six busbar system and the IEEE 24- RTS system were used to test the proposed methodology. The technical data of both systems is in Mathpower 4.0b3 [7]. We assumed that the capacity of the transmission lines is 50% of the initial value (normal state, short term and emergency). The values considered in the ordinal optimization and multi-objective Tabu search are defined in Table I. We defined Possible Corridors (PC), Feasible Corridors (FC), Reduced Space (RS) and Ordinal Samples (OS), number of Neighbors (NN), number of Elite-Pareto solutions (EPS) and the stop criterion (SC). In turn, the Energy Not Supplied Cost (EC) is in $/MWh and the factor is the rate between the investment of the transmission and the load shedding. For the annuity of the investments we considered 25 years, 3% for the administrative, operation and maintenance cost, a demand growth rate of 5% and a discount rate of 10%. We considered a budget restriction (BR), in millions of Euros M. The monetary unit is Euros. TABLE I MODELING DATA Bus PC FC RS OS NN EPS SC EC BR In Table II the characteristics of the demand curve are described. For each block of the demand, the capacity this expressed in function of the peak demand. TABLE II DEMAND CURVE Block Demand percentage (%) Hours (h) B. Six busbar system The 6-busbar system data is the one established in Mathpower s case6 [7]. Table III describes the base system s data. It establishes the lines that are present with the n column indicating the line status and the respective investment cost. For example, a random sample of 1000 possible expansion plans was determined. The 6-busbar system has 15 possible combinations, defining 2 15 possible states. It is assumed that the maximum additional amount of facilities by corridor is 3. Fig. 2 shows the investment cost of each plan in ascending order and the figure describes the bell-shape behavior [8]. For each set of random plans, the Pareto plans are determined. In our case, the operating annual cost and annuity objective functions are determined (Fig. 3). Five Elite-Pareto plans are obtained, and they are described in Table III. Cost of corridors are obtained from [9].

4 4 TABLE III SIX SYSTEM DATA AND EXPANSION SOLUTIONS # i j r x MW n Cost M Expansion Elite Opt. The results show that the optimal solution of the model is among the Elite-Pareto solutions proposed. The optimal solution has an investment cost of 1.8 M and is the corridor between nodes 1 and 5. The solution proposes a backup between generator of bus 2 and load of bus 5. The 6 bus system is a highly meshed one. This implies that after faults, the system is robust and requires scarce reinforcements of the transmission system. Moreover, if we consider solutions with load shedding different from zero, the investment plans will cost less. The type of form of the samples described in Fig. 2 implies that the number of random samples to choose from is 45, to find an acceptable solution with 95% probability. Pareto solutions and solutions generated by each neighborhood present zero load shedding. There are only three cases with load shedding, with values between 20 to 25 MW. In turn, Fig. 3c shows the savings that can be achieved in the operation by the expansion plan proposed. These savings are determined between the annual operating cost with the expansion plan and without the expansion plan (cost of the congestion). The Pareto solutions proposed present savings close to 4.3x10-3 M. We show that the exploration of representative samples and its Pareto multi-objective analysis find feasible solutions at a low computational cost. For example, for 1000 samples it is necessary to run an optimal power flow for each demand block, each proposed neighbor and each fault condition. In this six bus system, a small case, the algorithm takes 5.28 minutes. This time can be less if one considers only the ordinal optimization approach. (a) (b) Fig. 2. Ordered performance curve for six bus case. Now, when considering the multi-objective approach we establish 11 Pareto solutions between the annual operation cost and the annuity of transmission investment. Fig. 3a shows how the proposed solutions do not significantly impact the annual operating costs. On the other hand, it must be emphasized that the proposed solutions have a dispersion of about 3 M. Fig. 3b shows that a high percentage of the Elite- (c) Fig. 3. Non-dominated solutions for six bus case. (a) Trade-off between annual operation cost and annuity. (b) Trade-off between maximum load shedding (worst scenario) and annuity. (c) Trade-off annual savings (congestion cost) and annuity.

5 5 C. IEEE 24 RTS The case study described in this section is based on the IEEE 24-bus Reliability Test System (RTS). The transmission network comprises 24 buses, 34 existing corridors, and seven expansion corridors. We consider that all lines in the same corridor are identical and that the maximum number of lines per corridor is three. Cost of corridors are obtained from [9]. In Fig. 4 the topology of the IEEE 24-RTS system is illustrated. For this case we assumed that the representative samples are 5 millions, of approximately 2.2x10 12 possible combinations. We carried out an analysis regarding the number of samples and we settled down that the number of samples modifies the behavior of the ordered curve. For example, in Fig. 5 we show that for 1000 samples, the ordered performance curve has a bell-shape behavior. On the other hand, if we consider an ordinal optimization approach with budgetary constraints, the behavior of the ordered curve is the steep-shape. This implies that fewer samples are required to determine an acceptable solution with 90% probability. Besides, we carried out an analysis with 3000 and 5000 samples and the behavior is similar to the obtained one for the 1000 samples. However, to explore this larger number increases the computational cost with little benefit to find better solutions, if they exist. Hence, the importance of considering a meta-heuristic search to strengthen the solutions found by the ordinal optimization. In this case, the ordinal optimization proposes two possible expansions, corridors 7-8 or 6-10, with a cost of 1.8 M each one. These expansions aim at reinforcing the generator in bus 7 and the load in bus 6 or injection in bus 2. Fig. 5. Ordered performance curve for 24 IEEE-RTS case. (a) (b) (c) Fig IEEE-RTS system. When considering the multi-objective search with Elite- Pareto solutions, we find approximate annual operation costs between 322 and 290 M and 11 Pareto solutions. In turn, we identify that the conditions of the worst scenario show load shedding between 150 and 250 MW and maximum annual savings of 32 M (Fig. 6). Fig. 6. Non-dominated solutions for 24 IEEE-RTS case. (a) Trade-off between annual operation cost and annuity. (b) Trade-off between maximum load shedding (worst scenario) and annuity. (c) Trade-off annual savings (congestion cost) and annuity. Finally, the optimal solution is the expansion of corridor 14-23, with an investment cost of 5.15 M. In Table V the Elite- Pareto solutions are shown. For example, plan 1 is similar to the one proposed in [10] and makes part of the solution proposed in [11] and the optimal solution is part of the solution proposed in [9] and [12].

6 6 However, it is important to note that each model has different approaches. For example, [11] defines a constraint of 1/3 of the capacity of the lines, and with a loss minimization approach proposes an expansion plan with 7 lines. [10] considers a constraint of 50% of the lines and uses an incentive approach, proposing an expansion plan with 2 lines. [9] and [12] increase the magnitude of the values of load and generation, the first uses a reliability approach and the second a reliability and risk approach, proposing expansion plans with 4 and 8 lines, respectively. Obviously, if we increase the penalty factor or the failure cost the optimal solution is more robust. In this case, the expansion plan are 7-8, 14-16, 16-17, with an investment cost of M. This plan is equal to the first Elite-Pareto plan. TABLE V ELITE-PARETO EXPANSION AND OPTIMAL SOLUTION Plan Corridor Cost(M$) 1 7-8, 14-16, , 14-16, 16-17, , 2-8, 12-13, 14-16, , 15-24, 16-17, , 14-16, 14-23, 15-16, Opt (7-8, 14-16, 16-17) 5.16 (10.49) Fig. 7 shows the behavior of the objective function during the multi-objective meta-heuristic search. The function begins with the minimum value found in the constraint ordinal optimization and solutions are explored considering sensitization regarding the power flows and path re-linking with the Elite-Pareto solutions. It is observed from the beginning of the iterative process that the optimal solution is found, with a maximum load shedding of 150 MW and an approximate annuity of 0.5 M. The computer time was minutes, acceptable for the system size. The ordinal optimization identifies acceptable solutions in combinatorial problems like transmission expansion. The ordinal optimization approach is deterministic and it is possible to implement it in large systems at an acceptable computational cost. The meta-heuristic Tabu search is more flexible than other combinatorial approaches and finds acceptable solutions in large systems. The consideration of reliability security approaches increases the difficulty of the search. The path re-liking approach explores high quality subspaces and close to the Elite-Pareto solutions. This diversification process finds solutions at low computational cost. The resulting plans can be valued under additional criteria such as social benefits, sustainability and risks. VI. REFERENCES [1] J. D. Molina and H. Rudnick, "Transmission of Electric Energy: a Bibliographic Review," Latin America Transactions, IEEE (Revista IEEE America Latina), vol. 8, pp , [2] H. Mori and Y. Iimura, "Transmission Network Expansion Planning with a Hybrid Meta-heuristic Method of Parallel Tabu Search and Ordinal Optimization," presented at the International Conference on Intelligent Systems Applications to Power Systems, ISAP, Toki Messe, Japón., [3] K. Lee and M. El-Sharkawi, Theory and Applications to Power Systems: Modern Heuristic Optimization Techniques, [4] P. Maghouli, et al., "A Multi-Objective Framework for Transmission Expansion Planning in Deregulated Environments," Power Systems, IEEE Transactions on, vol. 24, pp , [5] Y.-C. Ho, et al., Ordinal Optimization: Soft Computing for Hard Problems: SPRINGER, [6] R. A. Jabr and B. C. Pal, "Ordinal optimisation approach for locating and sizing of distributed generation," Generation, Transmission & Distribution, IET, vol. 3, pp , [7] R. D. Zimmerman, et al., "MATPOWER's extensible optimal power flow architecture," in Power & Energy Society General Meeting, PES '09. IEEE, 2009, pp [8] M. Xie, et al., "Multiyear transmission expansion planning using ordinal optimization," Ieee Transactions on Power Systems, vol. 22, pp , Nov [9] L. P. Garces, et al., "A Bilevel Approach to Transmission Expansion Planning Within a Market Environment," Power Systems, IEEE Transactions on, vol. 24, pp , [10] J. Contreras, et al., "An incentive-based mechanism for transmission asset investment," Decision Support Systems, vol. 47, pp , [11] N. Alguacil, et al., "Transmission expansion planning: A mixedinteger LP approach," Ieee Transactions on Power Systems, vol. 18, pp , Aug [12] F. Risheng and D. J. Hill, "A new strategy for transmission expansion in competitive electricity markets," Power Systems, IEEE Transactions on, vol. 18, pp , VII. BIOGRAPHIES Fig. 7. Search objective function and worst scenario of load shedding. V. CONCLUSIONS The results obtained in the test systems show that the model developed is effective to solve the combinatorial problem. The multi-objective optimization under Pareto dominance defines a set of feasible solutions that establish expansion plans scenarios. Hugh Rudnick (F 00) is a professor of electrical engineering at Pontificia Universidad Catolica de Chile. He graduated from University of Chile, later obtaining his M.Sc. and Ph.D. from Victoria University of Manchester, UK. His research and teaching activities focus on the economic operation, planning and regulation of electric power systems. He has been a consultant with utilities and regulators in Latin America, the United Nations and the World Bank. Juan D. Molina (M 05, GSM 08) is a doctoral student at Pontifica Universidad Católica de Chile. He graduated from Universidad de Antioquia, later obtaining his M.S.c from the same university. His research focus on energy planning, planning and regulation of electric power systems and transmission expansion.