... Member, IEEE, W. D. Rosehart, Senior Member, IEEE, O. P. Malik, Life Fellow, IEEE, and ... mation of the power flow in all the branches connected to bus.
IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 24, NO. 3, AUGUST 2009
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Modified Chance-Constrained Optimization Applied to the Generation Expansion Problem M. Mazadi, Student Member, IEEE, W. D. Rosehart, Senior Member, IEEE, O. P. Malik, Life Fellow, IEEE, and J. A. Aguado, Member, IEEE
Abstract—A modified chance-constrained solution approach applied to the generation expansion problem is presented. Index Terms—Chance-constrained, generation expansion.
I. INTRODUCTION
P
LANNERS must predict load demand growth to decide on expansion of generation capacity (i.e., solve a generation expansion problem); however, uncertainty associated with the forecasted load demand can make an optimal solution difficult to achieve. Since the system loading level is stochastic due to load uncertainty, the expansion problem includes stochastic variables. Chance-constrained optimization (CCO) is a type of stochastic programming which handles the stochasticity of the problem by specifying a confidence level at which it is required that the stochastic constraints hold [1]. A heuristic approach for distributed generation capacity investment planning was presented in [2]and [3]. Generation and transmission expansion under risk was studied in [4]. In [5] and [6] a CCO approach is used to solve a stochastic unit commitment problem and a generation planning model with reliability constraints, respectively. In this letter, a modified chance-constrained approach is used to solve the generation expansion problem considered in [2]. The context of the problem considered in this letter is for regulated utilities, not electricity markets, since in the latter, power producers do not tend to have the same obligation to serve load. II. GENERATION EXPANSION AND OPF PROBLEM In a centrally planned power system, a generation expansion problem can be formulated as (1a) (1b) (1c) (1d) (1e) (1f) Manuscript received September 09, 2008; revised February 02, 2009. First published May 08, 2009; current version published July 22, 2009. Paper no. PESL-00106-2008. M. Mazadi, W. Rosehart, and O. Malik are with the Schulich School of Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada. J. Aguado is with the University of Malaga, Malaga, Spain. Digital Object Identifier 10.1109/TPWRS.2009.2021198
and are the investment and production cost of where new generation units; is the production cost of existing and are the active power of new and existing units, units; is the sumrespectively; is the load connected to bus i and mation of the power flow in all the branches connected to bus is the susceptance of the line between bus and bus is is the total buses in the system. The the voltage angle; and and for noncandidate buses for new generation variables are fixed to zero. In (1a), the first term is the investment cost of installing new generation units and the second and third terms are the generation production cost. Equation (1b) is the active power balance equation at each system bus; (1c) is the summation of active power of all the branches connected to bus ; and (1d) and (1e) are the limits on the size of the existing and new generation units, respectively. Constraint (1f) models the binary variables, , which is 0 if there is no new generation unit and 1 if a new unit is installed at bus . The problem considered in this letter is not the full expansion problem [3], since the objective of this letter is to focus on the CCO solution approach. III. CHANCE-CONSTRAINED OPTIMIZATION (CCO) In chance-constrained problems, a particular constraint must hold with a prescribed probability as follows: (2) where is a given nonzero probability, are the decision varimeans the probability of . ables, and the symbol The solution to the CCO problem can be found by converting the probabilistic constrained to an equivalent deterministic form , [1]. For example, consider the constraint: where the random variable has a known distribution function, . If a scalar can be found such that , then this constraint is equivalent to the determin, where . istic in (1b) is the peak forecasted demand, The loading level plus a reserve margin, of the future system at each bus. If the uncertainty in the bus demands are assumed to be a random variables, it is desired that the problem is feasible within a pre; hence, (1b) of the assigned boundary of probability expansion problem can be rewritten as
(3) which is a multivariate probabilistic constraint and cannot be directly converted to deterministic form. Following the approach
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given in [5], the equivalent deterministic form of (3) is as follows: (4) where the demand uncertainty is assumed to be a normal random and variance . Different types of variable with mean input uncertainty can be considered; however, different expressions for the deterministic equivalent would be required [7]. The solution process for the CCO expansion problem comprises of two subproblems solved iteratively using the following steps. 1) The expansion problem, (1) with (1b) replaced by (4), is to determine the set of new solved for a fixed scalar . generation units given that particular value of 2) In order to check the probability in (3), an operation OPF problem, (1) with the investment term removed from the objective and (1f) removed from the constraints, given the determined set of new generation, is solved in a Monte Carlo simulation for varying demand for a number of . Using the number of feasible cases over the trials, total number of trials, the probability of feasibility is found. holds, the 3) If the termination condition is updated and the final solution is found; otherwise, process is repeated. in (4), is changed. Two different By updating the are used in this letter. The first one methods for updating updating method presented in [5], which is based on is the the interpolation of values in the multivariate and univariate probability spaces. Using this method, the system loading level is changed in one direction in all the system buses. In this letter, an alternative method based on the Monte Carlo simulation solution is proposed to reach the target probability . In this approach, an initial loading level is chosen and the expansion problem is solved for that particular loading level. The loading level is then updated in different directions for each system bus. The process is as follows. found in step 3 above, when a) Based on the , the number of infeasible cases that would need to be feasible to make is . b) The loading levels corresponding to the infeasible cases are sorted based on the loading stress (i.e., for this letter, norm of system loads). this is assumed to be the c) The th loading level from the sorted infeasible cases is selected and the expansion problem is solved for this loading level. , the loading levels of the feaIn the case where sible cases are sorted descendingly based on the loading stress. The th loading level from the sorted feasible cases is selected and the expansion problem is solved for this loading level. It should be highlighted that no additional simulations are required in this approach since the Monte Carlo simulation solutions are already determined to test for convergence. IV. NUMERICAL RESULTS Numerical results are presented based on the IEEE 30-bus test system. The resultant mixed integer program is solved using the
optimization solver CPLEX in AMPL [8]. For the simulations considered in this letter, the required feasibility was with . Monte Carlo simulation consisting of 1000 samples was used for the demand in the operation OPF problem. These values were chosen to demonstrate the effectiveness of the method. In practice, a value of close to 1 would likely be used. were applied to a sequence of Both methods to update nine simulations which considered the mean of the system load growth incremented by 10% from 110% to 190% of the base loading levels found in the standard 30-bus data file plus a 5% reserve margin. For each case, the uncertainty in the load growth was modeled using a normal distribution with three standard (where is the normalized deviation given by mean load growth). For all simulations considered, the two methods to update converged to the same answer; however, as expected, there was a noticeable trend in the number of iterations required to reach convergence. One iteration is defined as one solution of the expansion problem followed by the Monte Carlo simulation to determine the probability of feasibility of that solution. The method based on [5] required between four and eight iterations to converge with an average of 5.3 over the nine simulations. The proposed method in this letter required between two to three iterations to converge with an average of 2.7 over the nine simulations. This trend was found to be robust over a variety of initial starting points. No simulation was recorded where the original method took less iterations than the proposed approach. It is also noted that the method based on [5] needs to establish the interpolation points at the beginning, requiring two additional Monte Carlo simulations. V. CONCLUSION In this letter, a chance-constrained approach is applied to the generation expansion problem. A method is proposed to update the loading level, , in the expansion problem. Using this method of updating , the solution is obtained in less iterations. REFERENCES [1] A. Charnes and W. W. Cooper, “Deterministic equivalents for optimizing and satisficing under chance constraints,” Oper. Res., vol. 11, no. 1, pp. 18–39, Jan.–Feb. 1963. [2] W. El-Khattam, K. Bhattacharya, Y. Hegazy, and M. M. A. Salama, “Optimal investment planning for distributed generation in a competitive electricity market,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 1674–1684, Aug. 2004. [3] J. L. C. Meza, M. B. Yildirim, and A. S. M. Masud, “A model for the multiperiod multiobjective power generation expansion problem,” IEEE Trans. Power Syst., vol. 22, no. 2, pp. 871–878, May 2007. [4] J. A. Lopez, K. Ponnambalam, and V. H. Quintana, “Generation and transmission expansion under risk using stochastic programming,” IEEE Trans. Power Syst., vol. 22, no. 3, pp. 1369–1378, Aug. 2007. [5] U. A. Ozturk, M. Mazumdar, and B. A. Norman, “A solution to the stochastic unit commitment problem using chance constrained programming,” IEEE Trans. Power Syst., vol. 19, no. 3, pp. 589–1598, Aug. 2004. [6] G. J. Anders, “Generation planning model with reliability constraints,” IEEE Trans. Power App. Syst., vol. PAS-100, no. 12, pp. 4901–4908, 1981. [7] A. Prékopa, Stochastic Programming. Dordrecht, The Netherlands: Kluwer, 1995. [8] R. Fourer, D. M. Gay, and B. W. Kernighan, A Modeling Language for Mathematical Programming. Belmont, CA: Duxbury, 2003.
