Evaluation of Emerging Metaheuristic Strategies on Opimal Transmission Pricing

José L. Rueda, Senior Member, IEEE István Erlich, Senior Member, IEEE Institute of Electrical Power Systems Institute of Electrical Power Systems University Duisburg-Essen University Duisburg-Essen Duisburg, Germany Duisburg, Germany [email protected] [email protected]

Abstract--This paper provides a comparative assessment of the In practice, there are several factors that can influence the capabilities of three metaheuristic algorithms for solving the adoption of a particular scheme of transmission pricing. Thus, problem of optimal transmission pricing, whose formulation is existing literature on definition of pricing mechanisms is vast. based on principle of equivalent bilateral exchanges. Among the Particularly, the principle of Equivalent Bilateral Exchange selected algorithms are Covariance Matrix Adaptation (EBE), which was originally proposed in [5], has proven to be Evolution Strategy (CMA-ES), Linearized Biogeography-based useful for pool system by providing suitable price signals Optimization (LBBO), and a novel swarm variant of the Mean- reflecting variability in the usage rates and charges across Variance Mapping Optimization (MVMO-SM). The IEEE 30 transmission network. In [6], an optimization problem was bus system is used to perform numerical comparisons on devised based on EBE in order enable exploration of multiple convergence speed, achieved optimum solutions, and computing solutions in deciding equivalent bilateral exchanges. In the effort. 2011 competition on testing evolutionary algorithms for real- Index Terms--Equivalent bilateral exchanges, evolutionary world optimization problems (CEC11), this problem was mechanism, metaheuristics, transmission pricing. solved by different metaheuristic algorithms, most of them constituting extended or hybridized variants of genetic I. INTRODUCTION algorithm and differential evolution [7]. Optimal allocation of transmission costs is among key Results of CEC11 showed the potentials of some problems on power system operation, especially in the new metaheuristic strategies as generic optimization engines for context, where higher uncertainties and complex agent solving a variety of problems by using the same parameter interactions will be reflected in more complicated problem setting. Moreover, considerable performance differences were formulation having a high dimensional, non-linear, non- evidenced by statistical assessment of results obtained by convex, discontinuous, and multimodal landscape. These repetitive optimization with the contestant algorithms [7], features motivate the exploration of new solution strategies, thus, highlighting the need of further algorithmic for which heuristic optimization framework is an attractive developments. Along this spirit, the work presented in this alternative due to conceptual simplicity, easy adaptability, and paper concerns a performance comparison between three open architecture. Reported applications of heuristic emerging metaheuristic algorithms, namely, Covariance optimization algorithms to transmission pricing include Ant Matrix Adaptation Evolution Strategy (CMA-ES), Linearized Colony Optimization [1], and Tabu Search [2]. Biogeography-based Optimization (LBBO), and a newly developed swarm variant of the Mean-Variance Mapping Due to their stochastic nature, heuristic optimization Optimization (MVMO-SM), which are used for solving the algorithms do not strictly guarantee global optimality, EBE based formulation of optimal transmission pricing (EBE- especially for high-dimensional and complex problems, but OTP). Numerical comparisons, obtained by application on a usually provide near-to-optimal or good enough solutions in benchmark power system, include convergence speed, reasonable time [3]. Thus, their effectiveness and robustness achieved optimum solutions and computing effort. can be questioned. With the aim of achieving fast and enhanced global search capability, high level schemes, known Following this introduction, the rest of the paper is as metaheuristic algorithms, have been conceived to organized as follows: Section II introduces the formulation of subordinate and exploit strategically different heuristic EBE-OTP. The theoretical background of the compared procedures, which could include especial schemes for adaptive algorithms is briefly outlined in Section III. Numerical results parameter change, local search, reinitialization, adaptive are provided in Section IV. Finally, conclusions are population sizing and population information exchange [4]. summarized in Section 5.

978-1-4799-6415-4/14/$31.00 ©2014 IEEE II. PRICING SCHEME where Xmn is the reactance of line connecting buses m and n,

The transmission pricing mechanism defined in [6] is whereas Xmi , Xni , Xmj , and Xnj are entries into the adopted in this paper. Broadly speaking, it assigns some of the reactance matrix X. total charges due transmission system usage to bilateral customers, whereas the rest is distributed through pool The set of optimization variables is defined by a vector customers. The first assignment is due to the fact that bilateral containing all GDij . Each GD accounts for an exchange transactions are usually known a-priori, so the usage rate between a generator and a load in the system, so the size of the associated to them can be determined, for instance, by vector is Ng⋅Nb. Therefore, the scale of the optimization evaluating power transfer distribution factors (PTDs) [8]. By problem increases with system size. Besides, the problem contrast, optimization is needed to properly decide equivalent possesses a multimodal search space, so a powerful bilateral exchanges such that the charges on pool customers optimization solver is needed to find the most feasible are close to those had bilateral transaction been absent. solution. The lower bound for each GDij is set to zero, A. Optimization problem statement whereas the upper bound is determined by using (6). It is assumed that an input file, containing bus and line max{GD} =−− min{ P BT , P BT } (6) specifications, fixed cost to be recovered and bilateral ij gi ij dj ij transactions data, is previously defined. Then, EBE is applied Once the best values of GDs are found, the usage rates for on the given information and the transmission charges are pool generations and pool demands can be easily calculated by evaluated at each bus, such that the transmission usage rates using simple expressions, which are functions of GDs and can i j be found in [6]. They are not reproduced here due to space for a generation at bus i (TU g ) and a demand at bus j (TU d ) constraints. are obtained. Next, equivalent bilateral transactions between pool generations and pool demands are obtained by III. METAHEURISTIC ALGORITHMS minimizing usage rate deviations due to bilateral transactions. This Section provides a brief review on background of the Mathematically, this optimization problem can be formulated employed algorithms. N and N are used to denote number as: p var of particles and number of optimization variables, Minimize respectively. The superscript g denotes iteration (generation) 22number. ⎛⎞⎛⎞FGD() FGD () OF=−+−⎜⎟⎜⎟ij TUij ij TU (1) A. CM A-ES ∑∑⎜⎟⎜⎟PP−−**gd PP ij⎝⎠⎝⎠gi gi dj dj The framework of this type of evolutionary strategy bases considering that on the assumption that the elements (optimization variables) xi of the optimization vector x follow a multivariate normal FCk FGD()= GD γk (2) distribution with mean vector m ∈ℜn , and covariance matrix ijkk∑∑ ij ij ⋅γ + ⋅γ nn× ∑∑GDij ij ∑∑ BT ij ij jk C∈ℜ . Additionally, a step size σ is used to perturb C [9]. ij ij The underlying iterative procedure is summarized as follows: and subject to • Step 1: Set N , the initial covariance matrix C=I, draw =−* ∀∈ p ∑GDij P gi P gi i Ng (3) the elements of m from a random uniform distribution i within the problem space, set initial σ to 30% of the =−* ∀∈ ∑GDij P dj P dj j Nd (4) constrained region in search space and the evolution j paths pc and pσ to zero. Set the number of best parents where Ng and Nb are the number of generator and load buses, μ to Np/2. Additional strategy parameters to be k ω μ μ respectively. FC stands for fixed cost of a line k that needs initialized are i (i=1… ), eff, cσ, dσ, cc, c1, and cμ. to be recovered, BT denotes bilateral transaction between Default values for these parameters can be found in ij [9]. generator at bus i and demand at bus j, Pgi is total generation • Step 2: Sample a new population of N particles at bus i, P* is the sum of generations due to all bilateral p gi according to: * gggg+1 transactions at bus i, Pdj is the total demand at bus j, Pdj is the =+σ = xmk N (0, C ) k 1K Np (7) sum of demands due to all bilateral transactions at bus j, and • GD is the equivalent bilateral transaction that need to be Step 3: Evaluate all particles x, rank them in ij ascending order according to their fitness, and update γk evaluated. ij denotes the sensitivity of a line k connecting m by (8), which computes the weighted intermediate buses m and n for a transaction between buses i and j, and is recombination of the best μ parents. given by (5). μμ mxgg++11=ω ω=ωω ω XXXX−− − ∑∑ii, where i 1, 12 > >K >μ >0 (8) γ=k mi ni mj nj ii==11 ij (5) Xmn Each ωi is decreased linearly throughout iterations. • Step 4: Update pc by using (9). gg+1 gg+1 gg + mm− xx=+μ−() xx (15) ppgg1 =−()12ccc +cc() − μ (9) ij,, ij i k,, j ij cccσg cceff • Step 4: The new habitats are evaluated and the worst are μeff is inversely proportional to the sum of squared ωi. replaced with the elites of previous generation. Step 5: Update C according to • Step 5: Perform local, boundary, global grid, or Latin μ + TTHypercube search if there is no improvement of fitness in CCppyyg11111=1--()cc g + c g+() g+ + c ω g+ () g+ (10) 1 μ 1c c μ ∑ ii i two consecutive generations according to the predefined i=1 thresholds in Step 1. gggg++11=+ σ yxmii()/ (11) • Step 6: Re-initialization is performed, that is, Np new • Step 6: Update σ according to (12) and (13). individuals are randomly sampled within the search gg+1 boundaries. Also, two new individuals are created by mm− −1/ 2 ppgg+1 =−()12ccc +cc() − μ() C g (12) randomly sampling around the search boundaries. Thus, σσσσg σσeff there is a temporary population size is 2⋅Np+2, from which the N best individuals are selected for the next generation. ⎛⎞⎛⎞pg +1 p gg+1 cσ σ σ=σexp⎜⎟⎜⎟ − 1 (13) • Step 7: If there is no improvement of fitness after all search ⎜⎟d ⎜⎟E N ()0I, ⎝⎠σ ⎝⎠ strategies performed in Step 5, the entire population is restarted. E N ()0I, is expected length of a random normal vector. • • Step 8: Stop if the termination criterion is met; else go to Step: 7: Stop if the termination criterion (e.g. max. Step 2. number of function evaluations) is met; else go to Step 2 to generate new Np particles by using updated C. MVMO-SM m and C. Mean-variance mapping optimization (MVMO) is a B. LBBO recently introduced evolutionary algorithm, which has some basic conceptual similarities to other heuristic approaches, but The mathematical model of this algorithm is inspired by it constitutes a fundamentally new evolutionary mechanism the study of geographic distribution of biologic organisms with two salient features. Firstly, MVMO performs by x [10]. Each candidate solution i is termed as habitat, and has a considering normalized range of the search space for all habitat suitability index –HIS, which corresponds with the optimization variables within [0,1]. This ensures that new fitness of a solution. Hence, the potential of each x is i values generated for optimization variables in offspring measured by two parameters, namely, emigration and creation stage are always within their bounds. The μ* immigration rates ( i and λi), which are directly and inversely optimization variables are de-normalized before every fitness μ* ∈ proportional to its fitness, respectively. Both i and λi evaluation. Secondly, MVMO exploits the statistical attributes [0,1]. The algorithmic procedure is described as follows: of search dynamics by using a special mapping function for mutation operation on the basis of the mean and variance of • Step 1: Set Np, the mutation probability Pmutate, the amount of the n-best solutions attained so far and saved in a continually- best individuals be to keep from one generation to the next updated archive [11]. and to use local search, and the thresholds for deciding use of local search, boundary search, global grid search, or Latin The original MVMO represents a single particle approach, Hypercube search. Guidelines for choosing these settings which has shown a great potential for solving different can be found in [10]. Next, generate an initial random optimization problems. This paper presents a new variant of population of N particles within the search boundaries. MVMO, termed as MVMO-SM, which adopts a swarm p intelligence scheme and incorporates a multi-parent crossover • Step 2: For each habitat (particle), determine the HIS index strategy to increase the search diversity while striving for a μ* μ* μ* balance between exploration and exploitation. The overall (fitness), i and λi. Consider that λi = 1- i , where i is procedure is described as follows: computed by using (14). The best habitats be are selected as elites. • * Step 1: Define Np, the initial and final values ( fs_ini and max{HIS} − HIS μ=* * i (14) f ) for scaling factor f , solution archive size, dynamic max{}HIS− min {} HIS s_final s shape factor Δd , the initial and final proportion of good g +1 • Step 3: To create a new solution x , λi is used to i particles ( g* and g* ), and the initial and final number probabilistically decide whether to immigrate or not. If it is p_ini p_final g * * decided to immigrate, a xk emigrating solution is chosen by of dimensions ( mini and mfinal ) to be selected for mutation μ* operation. Next, generate an initial random population of N using a i -based roulette wheel tournament selection. p particles within the search boundaries and normalize the Dimensions of xg are selected with a probability P to i mutate sampled optimization variables by considering the range of be linearly combined with the corresponding dimensions of search within [0, 1]. g xk as shown in (15). • Step 2: De-normalize each particle from [0, 1] range to their The shape factors s and s of the variable x are assigned r1 r2 r original [min, max] boundaries and evaluate its fitness. by using a sequential scheme which accounts for mean and • * Step 3: Fill/update the solution archive associated to each variance of xr , quadratic decrement of fs from fs_ini to particle. The archive stores the n-best child solutions * achieved so far in a descending order of fitness. The archive fs_final, and Δd in order to exploit the asymmetry of h [11]. size is fixed for the entire process. For each particle, an update of its archive takes place only if the new solution is • Setp 7: Stop if the termination criterion is met; else go to better than those in the archive. Step 2. • Step 4: The first ranked solutions (i.e. local bests) of all IV. CASE STUDY solution archives are classified into two groups: A set of GP Numerical experiments were performed on a computer “good particles”, and the set of remaining Np-GP “bad with an Intel® Core™ 2 i7 -3820 central processing unit particles”. Local best-based parent assignment is adopted for (CPU), 3.60 GHz processing speed, and 8 GB RAM Windows each particle classified as good, whereas for each bad 7 pro, 64 bit OS. The implementation of all algorithms was parent particle xp, the parent xp is synthesized by using the done in Matlab® Version R2013a. The IEEE 30 bus benchmark system is used to test the performance of all following multi-parent criteria. metaheuristic optimization algorithms. Details of system data parent =+β− as well as lower and upper limits for optimization variables xxxxpkij() (16) can be found in [7]. Reference parameter settings provided in where xi, xj, and xk represent the first (global best), the last, [9]-[11] were used for the studied algorithms. A static penalty and a randomly selected intermediate particle in the group of scheme is defined for all algorithms in order to properly good particles, respectively. The factor β is a random number, consider the fulfillment degree of constraints as well as to which is drawn according to ensure fair comparison. The fitness f * is calculated as β= − ⋅α α= 0.5 0.25 , ii / max (17) follows: where i denotes fitness evaluation number, and r is a random Ncon n * =+ ρ []2 number with uniform distribution in [0,1]. ff∑ iimax 0, g (23) i=1 parent An element of xp is set to 1 or 0 if it is outside the range where f stands for objective function value, Ncon is the number of constraints, g denotes the i-th constraint, and ρ is the [0, 1]. i penalty coefficient for each constraint. • Step 5: Create a child vector xnew for each particle by Performance comparisons were done by running 30 * combining a subset of N-var m directly inherited independent repetitions for each optimization algorithm. Every search process was terminated upon completion of a dimensions from xparent and m* selected dimensions (via p pre-specified number of 15E+04 iterations. A comparative roulette wheel tournament selection) that undergo mutation summary concerning the obtained best (minimum), worst operation through mapping function based on the means and (maximum), median, mean, and standard deviation of fitness variances calculated from the particle’s solution archive. m* in the last iteration, is presented in Table 14. is progressively decreased as follows: TABLE I. COMPARATIVE PERFORMANCE STATISTICS † mm*=+− round( m rand() m ) (18) Algorithms final final Fitness MVMO-SM LBBO CMA-ES m† =−α−round() m() m m (19) Best 6.764877E+00 8.304106E+00 6.765758E+00 ini ini final Worst 1.252368E+01 1.409256E+03 7.013549E+00 • Step 6: The new value of each selected dimension x of Median 7.186816E+00 3.937011E+02 6.808229E+00 r Mean 7.605154E+00 4.567834E+02 6.824101E+00 new x is determined by Std. 1.173334E+00 3.635885E+02 5.421104E-02 * Average run =+−+⋅− 4.133541E+00 5.328682E+00 3.264886E+02 xhrx(1 hhxh 10r0 ) (20) time (min) * where xr is a randomly generated number with uniform Note that the best results are achieved by using MVMO- distribution between [0, 1], and the term h represents the SM and CMA-ES, the latter having the smallest median, mean transformation mapping function defined as follows: and standard deviation, which highlight its robustness in spite −⋅ − − ⋅ of randomness involved in initialization and some algorithmic =⋅−x sxs12 +− ⋅(1 ) (21) hxs(,12 , s ,) x x (1 e ) (1 x ) e steps. Results achieved with both MVMO-SM and CMA-ES hx, h1 and h0 are the outputs of the mapping function are competitive and even better than those obtained in the calculated for CEC11 by using a variant of differential evolution algorithm (ADAP-DE) that incorporates adaptive parameter control hhxx==(),(0),(1)* hhx == hhx == (22) xr01 strategies, a center based differential exponential crossover and hybridization with local search [12]. The obtained statistics of fitness were: best = 7.585491E+00, worst = 1.159561E+02, median = 4.730413E+01, mean = global grid, or Latin Hypercube sampling as in the case of 1.930413E+01, standard deviation = 5.690016E+00. The best LBBO. Fig. 2 shows the usage rates for pool generations and achieved with LBBO is also competitive, but the remaining demands, which were calculated based on the best GD values statistical parameters are far from that of MVMO-SM, CMA- provided by MVMO-SM. Note that higher rates are charged to ES, and ADP-DE. This issue could be due to its embedded loads at buses 26, 29, 30, which are located far from system strategies for local search and reinitialization, which require a generators, and their supply entails a heavy use of the significant amount of function evaluations to enhance the network. This observation is in agreement with the findings on search progress, thus reducing the amount of evaluations EBE application given in [5] available for improvement of remaining best particles. V. CONCLUSIONS In this work, a comparative performance assessment of three selected metaheuristic algorithms, when employed to solve the EBE based formulation of optimal transmission pricing, was presented. Considering reference results from CEC11, it was found out that CMA-ES and MVMO-SM algorithms have a great potential to solve this task, since they are able to quickly find near-to-optimum solutions. Moreover, it was found that the algorithmic procedure of MVMO-SM entails the lowest computational effort. 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