2ND INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS 2008, 1214 NOVEMBER 2008, CLUJNAPOCA, ROMANIA
Approach for Optimal Reconfiguration of a Distribution System Bogdan Tomoiaga#1, Mircea Chindris*2 #
National Power Grid Company “Transelectrica”, Operational Unit – National Dispatch Center 1, Taberei st., 400114 ClujNapoca, ROMÂNIA 1
[email protected]
*
Technical University of ClujNapoca 15, C. Daicoviciu st., 400020 ClujNapoca, ROMÂNIA 2
[email protected]
paradigm to formulate this problem using Pareto optimality concept. The paper presents an original algorithm, aiming to find out the optimal openlooped points of electrical distribution systems with the purpose of obtaining optimal radial configurations taken into account diverse criteria in a flexible and robust mode. The novelty of the paper consists in: (i) original expressed form (Pareto optimality) of the problem of reconfiguration allowing in consideration more criteria in an objective manner in the optimisation process, (ii) an original solution to solve the problem (using evolution strategies considering the main genetic operators as selection, crossover, mutation etc.) in a nonprohibitive time.
Abstract—The configuration management (reconfiguration) of an electric distribution system consists in exchanging of functioning links between its elements in order to improve different performances. Since 1975, when Merlin and Back introduced this concept for loss reduction, until present, a lot of methods and algorithms to solve this multicriteria problem have been developed. Based on different heuristics or evolution strategies, these methods solve the problem by reducing it as a monocriteria problem with restrictions. This represents a major inconvenient because there are a lot of indices that must be taken into account in the optimisation process, especially in modern power systems, where distributed generation is predicted to play an important position and customers ask more and more reliability and quality in power supply. In this paper, authors propose a generalized paradigm to formulate this problem using Pareto optimality concept. In order to solve the problem, in the paper is also presented an original algorithm and its implementation based on evolution strategies.
II. RECONFIGURATION OF DISTRIBUTION SYSTEMS AS MULTICRITERIA PROBLEM The reconfiguration issue represents a multicriteria optimisation problem, where the solution is chosen after the evaluation of a group of indexes, named partial criteria, which represent multiple purposes.
I. INTRODUCTION The configuration management (reconfiguration) of an electric distribution system consists in exchanging of functioning links between its elements in order to improve different performances. In general, electrical distribution systems are operating in radial configurations. The reconfiguration problem is one of multicriteria optimisation, where the solution is chosen after the evaluation of a group of indexes, named partial criteria (e.g. active power losses, reliability, branches loads limits, voltages drops limits, etc.), which represent multiple purposes. Since 1975, when Merlin and Back [1] introduced the concept of distribution system reconfiguration for loss reduction, until present, a lot of methods and algorithms to solve this multicriteria problem have been developed. Based on different heuristics or evolution strategies, these methods (some with significant results) solve the problem by reducing it as a monocriteria problem with restrictions. This represents a major inconvenient because there are a lot of indices that must be taken into account in the optimisation process, especially in modern power systems, where distributed generation is predicted to play an important position and customers ask more and more reliability and quality in power supply. To eliminate the subjectivity and the rigidity of the above methods, authors propose a generalized
A. Criteria for Reconfiguration The most significant criteria, which must taken into account to operate a distribution system, are presented as follows. 1) Active Power Losses: For unbalanced and distorted systems the active power losses can be calculated using the following relationship: 4 ⎛ 50 2 ⎞ (1) ΔP = ∑∑ ⎜ ∑ Rijn ⋅ I ijn ⎟ ⋅ α ij ij∈E p =1 ⎝ n =1 ⎠ where: ΔP – active power losses; E – set of system lines (branches); Iijn – electric current through branch ij; Rijn – electric resistance of branch ij; n – number of harmonic component; p – phase number: three active phases and neutral conductor (if it exist); αij – binary variable, represents the status of a tie line (0 – open, 1  closed). To evaluate active power loss, it is necessary to perform power flow calculus. In radial electric networks with dispersed
65
2ND INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS 2008, 1214 NOVEMBER 2008, CLUJNAPOCA, ROMANIA
a.
generation operated with unbalance and distortion, a specific method, named backward/forward sweep, considers besides different types of node loads (constant powers, constant currents, constant admittances or a combination of the above), two types of distributed generators: generators which provide constant powers; generators that provide constant active powers and constant modules of voltages. The detailed algorithm is presented in [2].
n
SAIFI≈
i =1
ti
⋅ Ni
N
⋅T
(3)
ij∈E
where n is the number of electric system nodes and p is the number of connected components. In graphs theory terms, for a system with one source (p = 1) we talk about an optimal tree and for a system with more feeders (p > 1) we talk about an optimal forest with a number of trees (connected components) equal to that of source nodes. b. Safeguard of power supplies for all customers The attached graph of electric system should be connected (a tree or a forest of trees). c. It must be possible to impose and/or interdict the availability of an electric line d. Maximum number of admitted maneuvers e. Branch loads limits through lines
2) Reliability of electric system: The essential attributes of interruptions in the power supply of the consumers are the frequency and the duration. While the durations are predominantly influenced by the electrical system structure and the existent automations, the frequency is mainly influenced by the adopted working scheme; they can be minimized by the suitable choice of the loop opening points. For any fault that will lead to the interruption of power supply from the main distribution system, the existing distributed generators will be switching off because of a variety of reasons. We will mention only two of them: an operation of a power island purely with dispersed generators is considered, usually, unacceptable; it is important to create conditions for autoreclose of main circuit breaker, if this equipment is used in the distribution system. The system reliability can be considered from two points of view: that of a particular consumer; knowing failure rates of the equivalent element of the attached reliability block diagram (where restoring of supply is performed after the fault repair and/or the fault isolation) we can estimate different indices as well as the number of interruptions or MTBF (mean time between failures); possible restrictions in the reconfiguration problem – some consumers can impose maximal/minimal limits in contracts; that of the entire supply system; in this case, we propose to consider SAIFI (system average interruption frequency index), defined as: Total number of customer interruptions (longer than 3 minutes) / Total number of customers served [3]. Knowing the failure rates for each supplied node, we can estimate SAIFI using the relationship:
∑λ
Radial configuration of the system ∑αij = n − p
I ij ( phase ) ≤ I max,ij ( phase ) ; ∀ij ∈ E
(4)
I ij ( neutral ) ≤ I max,ij ( neutral ) ; ∀ij ∈ E
(5)
and f.
Voltage drops limits max U min j ( phase) ≤ U j ( phase) ≤ U j ( phase) ; ∀j ∈ X
(6)
where X is the network nodes set and phase = r , s , t . B. Pareto Optimality Problem Formulation The presented criteria (partial criteria) can be grouped in two different categories: criteria that must be minimized/maximized (objective functions); criteria that must be included within some bounds (restrictions). Therefore, the reconfiguration issue represents one of multicriteria optimisation problems. In addition, these criteria are incompatible from the point of view of measurement units and often they are conflicting criteria. Many solutions for onset the multicriteria optimisation problems can be found in the literature; among them, very used are: the main criteria method (εconstraint): optimisation of one objective while treating the other objectives as constraints bound by some allowable range εi; aggregation method (making a sum of objective functions, weighted or no): To create this synthesis function it is necessary to convert all partial criteria in the same measurement units; a very used method is to convert them in costs (usually, a tricky and often inaccurate operation). It is of interest to exist a model, which permits to take into account, in a same time, more objective functions and restrictions. Thus, we can use Pareto optimality, concept that defines a dominate relation among solutions. In Pareto optimisation, central concept is named undominated solution, if it satisfies the following two conditions: there exists no other solution that is superior at least in one objective function;
(2)
where: N – total number of customers served; Ni – total number of supplied customers from node i; T – the reference period [year]; n – the number of load nodes of the system; λti – total failure rate of the equivalent element, to the attached reliability block diagram, of node i [year1]. The detailed procedure is presented in [4] 3) Other criteria: These criteria are, usually, considered as restrictions and they arise from operation and technological considerations:
66
2ND INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS 2008, 1214 NOVEMBER 2008, CLUJNAPOCA, ROMANIA
mechanism that encourages apparition of some subpopulations corresponding to different optimum points. For this goal, a partitioning function, defined between two chromosomes must be defined. We propose the relationship:
it is equal or superior with respect to other objective function values. The set of Pareto solutions form Paretofront associated to a problem.
s ⎛⎜ d ( x , x ) ⎞⎟ = i j ⎠ ⎝
III. MULTICRITERIA RECONFIGURATION USING EVOLUTION STRATEGIES
In literature there are a lot of proposed methods to solve the reconfiguration problems. Because of its combinatorial nature, the most of them are based on different heuristic or metaheuristic (branch exchange, branch and bound, simulated annealing, tabu search, etc.) approaches. On the other hand, some authors have developed attractive methods based on evolution strategy, in particular on genetic algorithms; some significant results are presented in [5]. However, an important drawback of these methods is the fact that they solve the reconfiguration problems as single objective problems. The method proposed in this paper solves these problems as multicriteria problems using evolution strategies. Taking into account the fact that the graphs attached to electric systems are rare graphs (for instance, the system presented in figure 1 [6]), the representation using the branches lists was preferred because a binary codification of the problem – binary chromosome with fixed length can be obtained. Binary values of the chromosome will indicate the status of any electric line: 0 – open, 1  closed).
2 1 + e d ( xi , x j )
(7 )
where: s – partitioning function; d(xi, xj) – distance between two chromosome xi and xj; in binary codification; this can be the Hamming distance (the number of different positions between chromosomes). A partitioning function must satisfy the following conditions: (i) it must be an increasing function, (ii) s(0) = 1 and (iii) lim s ( d ) = 0 [7]. d→∞ Considering the partitioning function, we can calculate the partitioning fitness function for a chromosome (as criterion for selection): * f (x ) = i
f (x ) i
n
s⎛⎜ d ( x , x ) ⎞⎟ ∑ i j ⎠ j =1 ⎝
(8)
where: f* is the partitioning fitness function; f – fitness function; n – number of chromosomes from an ecological niche. 2) Crossover: The selection of the number and positions of cut points for crossover operator depends on the system topology. If these points are selected in an inadequate mode we will obtain “bad” chromosomes: (i) unconnected systems with isolated nodes or (ii) connected systems with loops. To avoid these situations, we propose that the number of cut points must equal the cyclomatic number (of fundamental circuits) corresponding to attached graph: l = m – n + p (where: m – number of branches, n – number of nodes, p – number of connected components).
Fig. 1. Threefeeder distribution system
3) Mutation: By simply altering the value of a chosen gene, from a radial scheme we cannot obtain another radial configuration. Hereby, we use this operator only in the case when, performing crossover operator, it results nonradial configurations.
A. Genetic Operators Below different genetic operators used to solve the problem of reconfiguration are presented.
4) Inversion (permutation): It is the most important operator used after performing crossover and mutation (if it is necessary). This operator makes some branchexchanges, repairing existing “bad” chromosomes, and increases the diversity of a population.
1) Selection: The goal of selection operator is to assure more chances to replicate for the best chromosomes of a population. The selection is performed taking into account the fitness of chromosomes; it chooses the chromosomes with the best fitness. Most used selection methods for monoobjective problems are Monte Carlo and tournament. For multimodal function optimisation we can use the ecological niche method. To detect more optimum points (ecological niches), a genetic algorithm (GA) must have a supplementary
B. Pareto Optimisation Using Genetic Algorithms Srinivas and Deb [8] have given a genetic algorithm for Pareto optimisation using the ecological niche method. Based on this algorithm, authors have developed an original algorithm dedicated for electric distribution systems reconfiguration (SIGRECO/Pareto, with a population containing maximum ten chromosomes). This algorithm allows considering diverse criteria in a flexible mode (as objectives or restrictions) in a manner presented in Table I.
67
2ND INTERNATIONAL CONFERENCE ON MODERN POWER SYSTEMS MPS 2008, 1214 NOVEMBER 2008, CLUJNAPOCA, ROMANIA
TABLE I CRITERIA CONSIDERED IN SIGRECO/PARETO ALGORITHM
Concerning System
Criteria
ΔP SAIFI Number of manoeuvres Voltage drops Number of interruptions MTBF Loads limits through lines Unavailable electric lines
Load nodes
Branches
For unbalanced and harmonic polluted case, the software is in a test process and results will be published in future works.
Objec tive X X
Restriction 
Ignore 


X

X


X



X

X



X
V. CONCLUSION The paper addresses the problem of electric distribution system reconfiguration in an open electricity market environment. In this context, dispersed generation is predicted to perform an important position and the customers ask more and more quality in power supply and reliability. Taking into account the multiobjective nature of reconfiguration problem, the introducing of Pareto optimality concept assures an objective and robustness result. Hereby, it can be eliminated the weak points of usual methods: (i) errors caused by the conversion of objective functions in the same measurements units and (ii) the subjectivities caused by introducing of weights for different criteria. The existent reconfiguration methods used nowadays either demands prohibitive execution times or results in nonoptimal solutions (in the case of most common heuristics). One of the most promising directions is the use of evolution strategies, representing specific methods of artificial intelligence (in particular, genetic algorithms), which provide optimal solutions in reduced times. These kinds of methods, with the equilibrium between the exploration of the potential solutions space and the exploitation of obtained information, offer a robust frame in order to solve reconfiguration problems in the case of large real systems.
IV. TEST RESULTS The proposed algorithm has been implemented in the visual environment programming language C++ Builder. Considering 16 buses system from [6], we introduced two DG units: G7 and G14 (figure 1). These DGs are considered constant PU units, with following parameters: G7: Pg7=0.02p.u., Qg7MAX=±0.005p.u., Ug7=1p.u; G14: Pg14=0.015p.u.. Performing reconfiguration, as Pareto problem (considering balanced and sinusoidal regime), we obtain functioning schemes optimising two criteria: active power losses and SAIFI (table II). Shown in table III are the failure rates of branches considered in reconfiguration calculus. We can observe that, in the analysed case (table II), Paretofront is reduced to a single solution
ACKNOWLEDGMENT The authors wish to thank National Power Grid Company – Transelectrica for the financial support. REFERENCES [1] A. Merlin and H. Back, “Search for a MinimalLoss Operating Spanning Tree Configuration in an Urban Power Distribution System”, in Proc. Fifth Power Systems Computer Conference (PSCC), Cambridge, 1975, p.p. 118. [2] M. Chindris, B. Tomoiaga, P.C. Taylor and L. Cipcigan, "The Load Flow Calculation in Radial Electric Networks with Distributed Generation under Unbalanced and Harmonic Polluted Regime", in Proc. 42nd International Universities Power Engineering Conference – UPEC 2007, University of Brighton, September 46, 2007, pp.9091 (abstract), paper on CDROM. [3] IEEE Guide for Electric Power Distribution Reliability Indices, IEEE Standard 13662003, Dec. 2003. [4] M. Chindriş, B. Tomoiagǎ, C. Bud, "Algoritm bazat pe logica fuzzy pentru estimarea frecvenţei întreruperilor întro reţea electrică", Conferinţa Naţională şi Expoziţia de Energetică – CNEE 2007, Sinaia, 79 noiembrie, 2007, pag. 827833.. [5] M. A. N. Guimarães, C. A. Castro, R. Romero, "Reconfiguration of distribution systems by a modified genetic algorithm", in Proc. IEEE Power Tech Conference, Lausanne, Switzerland, July 15, 2007, paper on CDROM. [6] S. Civanlar, J. J. Grainger, H. Yin, S. S. H. Lee, "Distribution Feeder Reconfiguration for loss reduction", IEEE Trans. Power Delivery, vol. 3, no. 3, p.p. 12171223, July 1988. [7] D. E. Goldberg, J. Richardson, "Genetic algorithms with sharing for multimodal function optimization", in Proc. The 2nd International Conference on Genetic Algorithms, New York, 1987, pp. 4149. [8] N. Srinivas, K. Deb, "Multiobjective function optimization using nondominated sorting genetic algorithms", Evolutionary Computing, no. 2, pp. 221248, 1995.
TABLE II
RESULTS OF PARETO RECONFIGURATION WITH TWO OBJECTIVES Tie lines 716, 810, 911
Active power losses
SAIFI
CPU time
0.004183 p.u.
0.2241
610ms
TABLE III
FAILURE RATES OF NETWORK BRANCHES FROM FIGURE 1 Branch
i
j
λ [year1]
Branch
i
j
λ [year1]
1
1
4
0.6525
9
9
12
0.696
2
4
5
0.696
10
3
13
0.957
3
4
6
0.783
11
13
14
0.783
4
6
7
0.348
12
13
15
0.696
5
2
8
0.957
13
15
16
0.348
6
8
9
0.696
14
5
11
0.348
7
8
10
0.957
15
10
14
0.348
8
9
11
0.957
16
7
16
0.783
68