EUROPEAN TRANSACTIONS ON ELECTRICAL POWER Euro. Trans. Electr. Power (2010) Published online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/etep.494
A multi-objective optimal power flow using particle swarm optimization J. Hazra1*,y and A. K. Sinha2 2
1 IBM India Research Lab, Bangalore, India Department of Electrical Engineering, IIT Kharagpur, India
SUMMARY This paper presents a multi-objective optimal power flow technique using particle swarm optimization. Two conflicting objectives, generation cost, and environmental pollution are minimized simultaneously. A multiobjective particle swarm optimization method is used to solve this highly nonlinear and non-convex optimization problem. A diversity preserving technique is incorporated to generate evenly distributed Pareto optimal solutions. A fuzzy membership function is proposed to choose a compromise solution from the set of Pareto optimal solutions. The algorithm is tested on IEEE 30 and 118 bus systems and its effectiveness is illustrated. Copyright # 2010 John Wiley & Sons, Ltd. key words:
multi-objective optimization; pareto-optimality; particle swarm optimization; optimal power flow
1. INTRODUCTION In operation and planning of power systems, operators need to make decisions with respect to different objectives. Hence, several tools have been developed to assist the operators. Optimal Power Flow (OPF) is one of them which helps the operators in running the system optimally under specific constraints. A lot of research starting from early 1960s has been done in this field to minimize the total generation cost. After the Clean Air Act Amendments (Kyoto Protocol) in 1990, operating at minimum cost maintaining the security is no longer the sufficient criterion for dispatching electric power. Minimization of polluted gases is also becoming mandatory for the generation utilities in many countries. Hence, OPF problem becomes a multiobjective optimization problem. Main challenges in a multiobjective optimization are generation of good quality solutions, generation of uniformly distributed Pareto set, maximizing the diversity of the developed Pareto set, selection of best compromise solution from the Pareto set, computational efficiency, etc. Several methods have been developed to solve mutli-objective optimization problems. By way of example, the penalty function method [1], weighted sum method [2], "-constrained method [3], non-dominated sorting genetic algorithm (NSGA) based approach [4], etc. have been used for solving various multiobjective optimization problems. However, these methods have difficulties. For example, in penalty function method choosing the proper penalty factors is a difficult task [5]. Weighted sum approach combines all the objectives to a single objective by using weight factors. This formulation may loose the significance of the objective function and, moreover, there is no rational basis of determining the weight factors for noncommensurate objectives [4]. " constrained method avoids the use of weight factors for multiple objectives but the difficulty with this method is that it requires repeated run for each relaxed level of " to get the Pareto-optimal set [4]. NSGA method is very sensitive to fitness sharing factor [6]. Moreover, *Correspondence to: J. Hazra, IBM India Research Lab, Bangalore, India. y E-mail:
[email protected] Copyright # 2010 John Wiley & Sons, Ltd.
J. HAZRA AND A. K. SINHA
these methods have not addressed the problem of diversity preserving, maximizing the diversity, selection of best compromise solution, etc. This paper emphasizes on the development of a multi-objective optimal power flow technique using particle swarm optimization method which has overcome some of the above mentioned difficulties. Instead of combining all the objectives to a single objective, objectives are solved simultaneously and hence retain the significance of each objective. To maintain diversity among the Pareto-optimal solutions a diversity preserving technique is proposed. To obtain diversification special care is taken in the selection process. Special care is also taken care to prevent non-dominated solutions from being lost. A fuzzy satisfying method is also proposed to help in choosing a compromise solution from the set of Pareto-optimal solutions. A power flow method which considers realistic situations such as load characteristics, generator regulation, on-load tap adjustment, etc. is used for solving the OPF problem. Proposed method has been tested on IEEE 30 and 118 bus systems and its effectiveness is presented. The rest of this paper is organized as follows. Section 2 presents the OPF formulation and Section 3 presents the brief overview of the power flow method used. Section 4 describes the PSO technique used in this paper. Section 5 introduces the Fuzzy membership function for choosing compromise Paretooptimal solution. OPF solution strategy is presented in Section 6. Section 7 presents the simulation results whereas Section 8 concludes the proposed work.
2. PROBLEM FORMULATION Let fi ð~ xÞ, i ¼ 1; . . .; m be m objective functions defined over n dimensional search space. A multiobjective optimization problem can then be formulated as: minimize
fi ð~ xÞ ¼ ff1 ð~ xÞ; f2 ð~ xÞ; . . .; fm ð~ xÞg
(1)
subjected to the constraints. This will give a set of Pareto-optimal solutions. A decision vector, x (a set of control parameters) is said to be Pareto optimal, if there is no other decision vector, y dominating x with respect to the set of objective functions. The decision vector x is said to strictly dominate another vector y (denoted by ~ x~ y) if: fi ð~ xÞ fi ð~ yÞ
8i ¼ 1; . . .; m
(2)
xÞ < fi ð~ yÞ for at least one i: fi ð~
(3)
and
Though primary objective of OPF is to minimize the generation cost, with the increase in environmental awareness and after the Clean Air Act Amendments (Kyoto Protocol) in 1990, operating at minimum cost is no longer the sufficient criterion for dispatching electric power. Minimization of polluted gases such as sulfur dioxide (SO2), carbon dioxide (CO2), nitrogen oxide (NOx) etc. is also becoming mandatory for the generation utilities. Therefore, in this paper OPF is formulated as two objectives optimization problem as follows.
2.1. Minimization of cost of generation The generator cost function for any turbo generator is obtained from data points taken during ‘‘heat run’’ tests. Using these data points generator cost curves are approximately fitted by quadratic functions. However, for large turbo generators this approximation does not hold good. For large turbo generators, just before opening a valve, a turbine operates at maximum efficiency at that loading. However, as soon as a valve opens turbine operates at less efficiency because of the throttling of the steam passing through the throttled control valve. To model the effects of valve-points, a recurring rectified sinusoid contribution was proposed to add with the quadratic input-output equation as Copyright # 2010 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power (2010) DOI: 10.1002/etep
A MULTI-OBJECTIVE OPTIMAL POWER FLOW
follows [7]: f1 ¼
NG X
Fgi
i¼1
¼
NG X
(4) ðpi þ qi Pgi þ
ri P2gi Þ
þ jei sinðfi ðPgi Pi;min ÞÞj
i¼1
where f1 is the total fuel cost; Fgi is the generation cost of generator i; NG is the number of participating generators; Pgi is the generation of ith generator; Pi;min is the minimum generation of ith generator; pi ; qi ; ri are cost coefficients of generator i; and ei ; fi are coefficients of generator i reflecting valve point loading effect. 2.2. Minimization of polluted gas emission Like generator cost curve, emission curve is obtained using curve fitting from practical emission data of thermal plant. The amount of emission from a fossil-based thermal generator unit depends on the amount of power generated by the unit. Total emission generated is usually approximated as sum of a quadratic function and an exponential function of the active power output of the generators as follows [4,8]: f2 ¼
NG X
102 ðai þ bi Pgi þ g i P2gi Þ þ ji expðli Pgi Þ
(5)
i¼1
where, a, b, g, j, and l are emission coefficients. Subjected to the constraints of load and generation balance and operating limits of each equipment.
3. POWER FLOW MODEL 3.1. Load characteristics In general majority of the loads are frequency and voltage dependent. To take care of voltage and frequency dependence, both real and reactive power loads are modeled as follows [9]: Pdi ¼ Pdio ð1 þ ki Df Þðai þ bi jVi j þ ci jVi j2 Þ
(6)
Qdi ¼ Qdoi ð1 þ ki0 Df Þða0i þ b0i jVi j þ c0i jVi j2 Þ
(7)
where Pdi ; Qdi are actual active and reactive load at bus i; Pdio ; Qdoi are active and reactive load at bus i for nominal voltage and nominal frequency; Df is the system frequency deviation from normal frequency; jVi j is the bus voltage magnitude at bus i; ki ; ai ; bi ; ci are coefficients of active load at bus i with ai þ bi þ ci ¼ 1; and ki0 ; a0i ; b0i ; c0i are coefficients of reactive load at bus i with a0i þ b0i þ c0i ¼ 1. All quantities are in per unit. 3.2. Generator characteristics Real power output of any generator with free governor operation is adjusted by the governor droop characteristics. This can be expressed as: ngi ngi X X Pmaxik Pgi ¼ Pgik ¼ Psetik Df (8) Rik k¼1 k¼1 Pminik Pgik Pmaxik
(9)
where ngi is the number of generators at bus i; Pgi is the total active power generation at bus i; Pgik is the active power generation of kth unit at bus i; Psetik is the set value of active power generation of kth unit Copyright # 2010 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power (2010) DOI: 10.1002/etep
J. HAZRA AND A. K. SINHA
at bus i; Pmaxik ; Pminik are maximum and minimum generations of kth generator at bus i; and Rik is the regulation constant of kth generator at bus i. All quantities are in per unit. 3.3. Real power flow equations Static real power flow equations considering follows [9]: 2 0 3 2 DP1 6 jV j 7 6 B11 B12 : : 6 1 7 6 7 6 6 DP20 7 6 7 6 6 B21 B22 : : 6 jV2 j 7 6 7 6 6 6 : 7¼6 6 7 6 : : : : 6 : 7 6 7 6 6 : : : : 7 6 6 6 DP 0 7 6 4 n5 4 Bn1 Bn2 : : jVn j
load and generation characteristics are derived as 3 þ b1 k1 Pdo1 72 Dd 3 1 7 k¼1 R1k jV1 j 76 7 ng2 7 7 6 P Pmax2k Dd 2 7 7 6 þ b2 k2 Pdo2 76 7 k¼1 R2k jV2 j 76 7 7 7 6 : : 76 7 7 7 6 : : 74 5 7 ngn 5 P DðDðf ÞÞ Pmaxnk þ bn kn Pdon k¼1 Rnk jVn j ng1 P
Pmax1k
(10)
3.4. Reactive power flow equations Static reactive power flow equations considering load and generation characteristics are derived as follows [9]: 2 0 3 DQ1 32 3 2 0 DV1 6 jV j 7 B þ 2c Q B : : B 11 do1 12 1n 1 6 1 7 6 76 7 6 0 7 76 7 0 6 DQ2 7 6 7 6 B B þ 2c Q : : B 7 6 6 21 22 do2 2n DV2 7 2 7 7 6 6 6 jV2 j 7 6 76 7 6 : 7 6 : : : : : 7 7 6 7 6 6 76 : 7 (11) 7¼6 6 7 7 6 : : : : : 6 : 7 6 76 7 7 6 6 76 7 7 6 0 76 7 6 DQ 0 7 6 B B : : B þ 2c Q n1 n2 nn don 7 7 6 6 : n 6 n7 5 5 4 4 7 6 4 jVn j 5 DVn
4. PARTICLE SWARM OPTIMIZATION (PSO) Particle Swarm Optimization (PSO) is a simple and efficient population based optimization method proposed by Kennedy and Eberhart [10]. In PSO, potential solutions are called particles and population of particles is called swarm. Each particle in a swarm flies in the search space toward the optimum or a quasi-optimum solution based on its own experience and experiences of nearby particles. Let us define search space, S in n dimension and the swarm consists of N particles. Each particle i, has its position defined by xit ¼ fxi1;t ; xi2;t ; . . .; xin;t g and a velocity defined by vit ¼ fvi1;t ; vi2;t ; . . .; xin;t g in variable space S at time t. Position and velocity of each particle changes with time (generation). Velocity and position of each particle in the next generation (time step) can be calculated as: vij;tþ1 ¼ w vij;t þ c1 randðÞ ðpij;t xij;t Þ þ c2 randðÞ ðpgj;t xij;t Þ xij;tþ1 ¼ xij;t þ vij;tþ1
8i ¼ 1; . . .; N
(12) (13)
where N is the number of particles in the swarm; n is the dimension of solution vector; w is the inertia weight; t is the generation number; c1 ; c2 are acceleration constant; randðÞis the uniform random Copyright # 2010 John Wiley & Sons, Ltd.
Euro. Trans. Electr. Power (2010) DOI: 10.1002/etep
A MULTI-OBJECTIVE OPTIMAL POWER FLOW
number in the range [0,1]; vij;t is the velocity of particle i at generation t; xij;t is the position of particle i at generation t; pgj;t is the global best at generation t; pij;t is the best position that particle i could find so far. In Equation (12), global best pgj;t conceptually connects all the particles in the population to one another, so that each particle is influenced by the very best performance of the entire population. It encourages the particles to move toward itself, and particles move in its direction. 4.1. Initialization PSO is initialized with a group of random particles (solutions) and then searches for optima by updating particles. In a PSO, each particle is a k-dimensional real-valued vector, where k is the number of the control variables. Each control variable of any particle i is initialized randomly within its range max ½xmin as follows: j ; xj þ randðÞ ½xmax xmin xij;t¼0 ¼ xmin j j j for j ¼ 1 to k
(14)
At initial step (t ¼ 0) as there is no idea about particles velocity, at t ¼ 0, velocities of the particles are initialized with their initial positions as follows: vij;t¼0 ¼ xij;t¼0 for j ¼ 1 to k
(15)
4.2. Limiting velocity and position The particle swarm algorithm proceeds by modifying the distance that each particle moves on each dimension per iteration. Changes in the velocity are stochastic and can expand into wider and wider cycles through the problem space, eventually approaching infinity. Commonly used method is to limit the velocity within its maximum value (Vmax ) as follows: if vij;t >Vmax ; vij;t ¼ Vmax
(16)
if vij;t xmax j j
(18)
if vij;t
