Multiobjective Optimization of Distribution Networks Using Genetic Algorithms Fatemeh Afsari Department of Computer Engineering, School of Engineering Shahid Bahonar University of Kerman, Kerman, Iran
[email protected] Abstract-This paper presents the application of multiobjective optimization, for finding out the optimal power distribution network. A multiobjective optimization model simultaneously minimizing the system expansion costs while achieving the best reliability (optimal sizing and location of future feeders and substations). Furthermore, the problem has some constraints. The proposed methodology has been tested for real distribution systems with dimensions that are significantly larger than the ones frequently found in the literature. We have used the Pareto-optimum model to find the suitable curve of non dominated solutions, composed of a high number of solutions. I. INTRODUCTION The optimal design of a power distribution system has been usually approached as the minimization of a single objective (mono-objective) function which represents the economic costs of the global system expansion. However, minimizing costs and getting the best reliability state cannot be solved by previous methods. Therefore, a multiobjective model is used. Genetic algorithms are global methods, which aim at complex objective functions (e.g., non differentiable or discontinuous). In addition, some constrains should be satisfied (e.g. radial structure, voltage and current limits). The multiobjective optimal design has been dealt with by few authors showing examples of application to distribution systems. In previous works [1, 2] several optimal multiobjective planning models were tested and validated intensively by computer experiments, optimizing simultaneously various objectives. However, only classic multiobjective optimization techniques were used [3, 4] to obtain a subset of satisfactory optimal non dominated solutions. This paper presents a new application of genetic algorithm to the multiobjective optimal design of distribution systems that allows optimizing n objectives simultaneously (based on Pareto optimality [3, 4]), using a new toolbox in MATLAB environment. The distribution systems used here present significantly larger dimensions and optimization complexity than most of the networks frequently found in the specialized papers. Also this paper uses a real alphabet that is more flexible than binary alphabet. However, this paper with genetic algorithm method obtains the optimal configuration of the distribution network with the best reliability and the lowest economic costs.
II. POWER DISTRIBUTION PROBLEM The problem that we want to solve is a simplified form which shows a series of sources (substations) and a series of drains or demand nodes. Each source has a maximum limit of the power supply. Moreover, the possible routes for the construction of electric line to transport the powers from the sources to the demand nodes are known. Each line possesses a cost that depends principally on its length (fixed costs) and the power value that transports (variable costs). The model contains an objective function that represents the costs corresponding to the lines and substations of the electric distribution system. Multiobjective optimization (also called multi criteria optimization, multi performance or vector optimization) can be defined as the problem of finding [6]; A vector of decision variables which satisfies constraints and optimizes a vector function whose elements represent the objective functions. These functions form a mathematical description of performance criteria which are usually in conflict with each other. Hence, the term “optimizes” means finding such a solution which would give the values of all the objective functions acceptable to the designer. Formally, we can state it as follows:
[
Find the vector x = x1∗ , x2∗ ,..., xn∗ m inequality constraints: g i (x ) ≥ 0 i = 1,2,..., m and the p equality constraints hi (x ) = 0 i = 1,2,..., p and optimize the vector function f (x ) = [ f1 (x ), f 2 (x ),..., f k (x )]T
]
T
which will satisfy the (1) (2) (3)
where, x = [x1 , x2 ,..., xn ] is the vector decision variable. T
The constraints given by (1) and (2) define the feasible region X and any point x in X defines a feasible solution.
The vector function f (x ) is a function which maps the set X in the set F which represents all possible values of the objective functions. The k components of the vector f (x ) represent the non commensurable criteria which must be considered. The constraints g i (x ) and hi (x ) represent the restriction imposed on the decision variables. The vector will be reserved to denote the optimal solutions (normally there will be more than one).
In this paper, the multiobjective design model is basically a nonlinear mixed-integer one for the optimal sizing and location of feeders and substations, which can be used for single stage or multi-stage planning (under a pseudo dynamic methodology [7, 8]). The vector of objective
functions to be minimized is f (x ) = [ f1 (x ), f 2 (x )]T , where the objective function of the global economic costs is f1 (x ) , and f 2 (x ) is a function related with the reliability of the distribution network. Then, the objective function f1 (x ) is: f1(x ) =
∑ (Ii Ri )(Yi )+ ∑ (CF )(Y ) nf
ns
i =1
k =1
k
k
(4)
where: n f : Number of routes (between nodes) associated with lines in the network. ns : Number of nodes associated with substations in the network. I i : Current carried through route, i. Ri : Resistance of route, i. CFk : cost of a substation to be built, in the node k. Yi : 1, if feeder associated with route i is built. Otherwise, it is equal to 0. Yk : 1, if substation associated with node k is built. Otherwise, it is equal to 0.
and the objective function f 2 (x ) is: f 2 (x ) =
nf
∑ (u )(X ) i
i =1
f i
(5)
where: n f : Number of “proposed” routes (proposed feeders) connecting the network nodes with the proposed substation. : Power flow, in kVA, carried through the route f, i
(X f )
which is calculated for a possible failure of an existing feeder (usually in operation) on the route i. In this case, reserve feeders are used to supply the power demands. (ui ) : constants obtained from other suitable reliability constants in the distribution network, including several reliability related parameters such as failure rates and repair rates for distribution feeders, as well as the length of the corresponding feeders on routes. The simultaneous minimization of the two objective functions is subject to technical constraints [7], which are: 1-The first Kirchhoff’s law (in the existed nodes of the distribution system). 2-The relative restrictions to the power transport capacity restrictions of the lines. 3-The relative restrictions to the power supply capacity restrictions of the substations.
The presented model corresponds to a mathematical formulation of non linear mixed-integer programming, which includes the true non linear costs of the lines and substations of the distribution system, to search the optimal solution from an economic viewpoint. III. GENETIC ALGORITHM IMPLEMENTATION The full description of genetic algorithms is given elsewhere [8] and for brevity is omitted here. In the present work, the representation of the solutions uses an alphabet with dimensions greater than two for the simplicity that contributes to the codification; it means that it can contain more information than a binary alphabet. Thus, the employed alphabet represents, in the distribution net design, a novelty with respect to other developed research. In order to be able to consider two or more conductor sizes for the construction of the future lines, we have used (instead of a binary alphabet with value 0 and 1) different real numbers 0 and 1 in the positions of the string that represents a definite topology (routes between nodes of the electric power distribution system), a different real number for each size of conductor. For example the following string: 3010201001 represents a configuration of an electric net, with 5 electric lines of a total of 10 possible lines and they are built with the size of conducting number 1, 2 and 3. This codification, therefore, permits use a great number of different conductor sizes in the electric line constructions and also of several substation sizes. The size of the population is a very important factor to obtain efficiency. The length of the strings affect directly on the size of the population. For the evaluation of the solutions of the optimal design of distribution systems problem, it is desired to precisely know the variable cost associated to the topology that is represented by an individual in the population. After using the habitual operators, crossover and mutation rates, some of the solutions will disappear and other new ones will appear, this leads to a new population and finishes a generation (iteration) of the evolutionary algorithm. Premature convergence is the big problem in evolutionary algorithm techniques, so we try to solve it by permitting that infeasible solutions contributing in the generation while penalizing them. IV. MULTIOBJECTIVE OPTIMIZATION Here the possibility of evaluating the reliability function of a distribution net is shown, as well as the global economic cost of the electric power distribution system. “Fig. 1,” shows three possible solutions with their values of the cost functions and the reliability function. The solutions 1 and the 3 are non-dominated; if a given solution is “dominated” at least by some other, then it is named dominated solution. To obtain the solutions we use genetic algorithms that meets a non-dominated solution and also that they present a radial topology in exploitation (but with additional “reserve lines”). The topologies of the distribution nets obtained are not radial in their entirety, but they will be radial in exploitation.
Reliability Cost Fig.1. Representations of solutions of cost function and reliability. The other thing that makes our work different to the previous researches is the treatment of the infeasible solution. As mentioned, we have both feasible and infeasible solutions that both of them contemplate in the population, the algorithm does not ignore the infeasible solutions. Our method replaces an infeasible solution with the nearest feasible solution. The infeasibility of a solution is a problem when some nodes (demand centers) are not connected to any line, then results a verification using the information of the strings that they represent. V- COMPUTATIONAL RESULTS The new evolutionary algorithm has been applied intensively to the multi objective optimal design of several real size distribution systems. A compatible PC has been used (CPU Pentium 1400 MHz and 128 Mb of RAM) with the operating system WinXp to run MATLAB. Table I. shows the characteristics of the example of optimal design, indicating the dimensions of the used distribution system and the mathematical aspects. Most of the distribution networks data has been provided by Kerman electric utility. Notice the number of 0-1 variables of the distribution network indicating that the complexity of the optimization and the dimensions of this network is significantly larger than most of the ones usually described in technical papers [7]. In this study we used a 20 kV electric net, which was a portion of Kerman’s power distribution net and the network is represented in the “Fig. 2,”.The complete net, the existing
feeders (thickness lines) and the proposed feeders (other lines). The existing net presents two substations at nodes 1 and 17 with the capacitance of 40 MVA. The sizes of proposed substation are: 40 MVA and 10 MVA. The existing lines of the net possess the following conductor sizes: 3x75Al, 3x118Al, 3x126Al and 3x42Al. Also in the design of the net it is proposed 4 different conductor sizes for the construction of new lines: 3x75Al, 3x118Al, 3x126Al and 3x42Al. We have used a multiobjective toolbox that is written to work in MATLAB environment, which is called MOEA. In Table II we have shown the results of the multiobjective optimization program that have leaded us to the final non dominated solutions curve. This table provides the objective function values (“cost” in millions of Rialls, and “Reliability”, function of expected energy not supplied, in kWh) of the ideal solutions, the number of generations (Gen.) and the objective function values of the best topologically meshed network solution for the distribution system. The multiobjective optimization finishes when the stop criteria is reached, that is when the upper limit of the defined generation number reached. The used crossover rate is 0.3 and the mutation rate is 0.02. The population size is 180 individuals and the process uses roulette wheel selection method; it reuses the previous population in the new generation. “Fig. 3,” shows the non-dominated solution
TABLE I CHARECTERISTIC OF THE DISTRIBUTION SYSYTEM
No. of existing demand nodes
43
No. of proposed demand nodes
121
No. of total demand nodes
164
No. of existing lines
43
No. of proposed lines
133
No. of total lines
176
No. of existing substations
2
No. of proposed substations
0
No. of total substations
2
No. of total 0-1 variables
532
Fig.2. Existing and proposed feeders of the distribution network.
Fig.3. Evolution of the non-dominated solution curve
Fig.4. Solution of multi objective optimal design model curve during the optimization design. The trade off between two solutions is very easy and subjective. The curve shows that when the cost improves the reliability decreases and vice versa. VI. CONCLUSIONS We have developed the optimization model of mixedinteger non linear programming for the optimal design of electric power distribution system. The multiobjective optimization model contemplating the following aspects: Non linear objective function of economic costs associated to the electric distribution net and a new objective function representing the reliability of the electric net. These objective functions are subject to the mathematical restrictions that were mentioned. Localizations and optimal sizes of electric lines and substations, proposed for the planner, for the optimal design. The developed model is easily applicable to n objectives (without increasing the conceptual complexity of the corresponding algorithm). Reliability objective of the electric power distribution system evaluated by means of a fitting function that indicates a radial topologies, keeping in mind, in general, faults (errors) of any order in the electric lines (studious for the first-order faults in the obtained computational results). TABLE II RESULTS OF THE MULTIOBJECTIVE OPTIMIZATION PROGRAM
Gen.
COST
RELIABILITY
75
42613.4367
198.17357
We developed a new genetic algorithm for the optimal design of distribution systems. In the previous researches, a binary alphabet was used to codify the solutions of optimal design. Here we improved an important aspects of the genetic algorithm employing a new non binary alphabet (using real numbers) so they could include more information like several conductor sizes of the lines and of the substations, and also they are incorporated with reserve electric lines to improve the reliability of the distribution net. Here we have the treatment of the algorithm to the infeasible solutions; instead of ignoring them we have used a method to just penalize them. Comparison between the genetic algorithm and a classic branch and bound algorithm, the genetic algorithm comparatively wastes a few calculation times in the process of optimal design of distribution systems. Practical usefulness of reaching a curve for non-dominated solutions permits that the planner disposes of an abundant number of optimal multiobjective solutions among the selected solutions. REFERENCES [1]
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