- Email: [email protected]
Reconfiguration of distribution networks to minimize loss and disruption costs using genetic algorithms
Electric Power Systems Research 80 (2010) 53–62
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Reconfiguration of distribution networks to minimize loss and disruption costs using genetic algorithms Juan Carlos Cebrian, Nelson Kagan ∗ Department of Electrical Engineering, University of São Paulo, Escola Politécnica, Av. Prof. Luciano Gualberto, travessa 3 n◦ 380 - CEP - 05508-970 – São Paulo, Brazil
a r t i c l e
i n f o
Article history: Received 13 October 2008 Received in revised form 12 August 2009 Accepted 13 August 2009 Available online 19 September 2009 Keywords: Distribution networks Genetic algorithms Power quality Voltage sags Monte Carlo simulation
a b s t r a c t In this paper a computational implementation of an evolutionary algorithm (EA) is shown in order to tackle the problem of reconfiguring radial distribution systems. The developed module considers power quality indices such as long duration interruptions and customer process disruptions due to voltage sags, by using the Monte Carlo simulation method. Power quality costs are modeled into the mathematical problem formulation, which are added to the cost of network losses. As for the EA codification proposed, a decimal representation is used. The EA operators, namely selection, recombination and mutation, which are considered for the reconfiguration algorithm, are herein analyzed. A number of selection procedures are analyzed, namely tournament, elitism and a mixed technique using both elitism and tournament. The recombination operator was developed by considering a chromosome structure representation that maps the network branches and system radiality, and another structure that takes into account the network topology and feasibility of network operation to exchange genetic material. The topologies regarding the initial population are randomly produced so as radial configurations are produced through the Prim and Kruskal algorithms that rapidly build minimum spanning trees. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Most of the overhead electric power distribution systems must operate in radial configurations. Normally open and normally closed switches are located along the network in strategic points. By altering the topology to a different radial configuration, one might obtain losses reduction, improvement in the network voltage profile, and betterment of reliability indices. When a fault occurs, the fault block can be isolated and a new radial configuration might restore the rest of the load. The reconfiguration problem that determines the best possible configuration considering the above criteria and constraints can be modeled as a non-linear mixed integer problem, what produces an enormous search space due to the combinatory nature of the problem. Many approaches have been considered to address such a problem, particularly conventional integer programming algorithms, heuristic algorithms and mixed heuristiccombinatory algorithms to better explore the search space [1–11]. In this paper, the reconfiguration of distribution networks, considering power quality indices, is formulated as a non-linear mixed integer programming problem. The adopted solution technique
∗ Corresponding author. Tel.: +55 11 3091 5318; fax: +55 11 3091 5317. E-mail addresses: [email protected] (J.C. Cebrian), [email protected] (N. Kagan). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.08.005
consists in an evolutionary algorithm (EA) that considers a decimal base codification scheme. The EA makes use of recombination and mutation operators specially developed to maintain alternative radial configuration networks. Fitness functions are related to network losses and penalizations regarding voltage and loading constraints. The model described in this paper also handles power quality costs, namely long duration interruptions (greater or equal to 1 min) and customer process interruptions due to voltage sags, herein simply named as disruptions. An efficient power flow method [3] and [12] determines network losses, equipment loading and bus voltages for different network topologies. As for the power quality indices, a Monte Carlo simulation method [13] was implemented and incorporated into the optimization model. Other relevant aspects which are discussed in this paper regard the implementation of solutions that directly affect the efficiency of the evolutionary algorithm, namely the generation of the initial population, the EA codification scheme and the recombination operator. Concerning the EA selection procedure, two possible alternatives are analyzed, namely tournament and elitism. The latter one considers the saturation of the elite population over the generation cycles. The effects of the EA control parameters in the performance of the evolutionary algorithm, such as the size of the population and the recombination and mutation rates, are carefully studied and analyzed.
54
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
In order to produce the initial population, radial configurations are randomly generated through heuristic techniques based on minimum spanning tree algorithms, using criteria to establish the loop values associated with the impedance and resistance of the distribution network [14]. The Prim and Kruskal algorithms were implemented so as to produce the initial population. The mechanism of recombination utilizes topology information contained in the individuals subject to recombination. The algorithm identifies the branches to be exchanged based on the corresponding power flow values, in order to generate two new radial configurations. 2. Formulation of the reconfiguration problem including long interruptions and voltage sags The reconfiguration problem in distribution networks can be formulated as a mixed integer programming problem, i.e. it contains real and integer variables. The solution of such a formulation determines that one configuration that optimizes an objective function subject to a set of constraints. When considering long duration interruptions and customer process disruptions due to voltage sags, the reconfiguration problem can be formulated as follows:
Fig. 1. (a) Radial 14 bus network (tree) subject to islanding problems; (b) representation of a chromosome (genetic chain).
NCons
Min fa(i) = CtoPer xPer(i)xHyear +
(CostInt(k)(i) ) + CostDis(k)(i) )
(1)
necting vertices with lower cost branches until a complete tree is formed. A D set is formed by a vertex Vi ∈ V and a branch Ei ∈ E, for which V, E ∈ G, where G represents a given graph. Initially D is formed by only a vertex Vi ∈ V. In order to produce a minimum spanning tree, an i − j branch is chosen in each iteration so as its cost Cij is the lowest and the union D∪(i − j) favors the formation of a tree and a new sub-graph D ← D∪(i − j). The algorithm ends when the sub-graph D forms a complete tree.
The combinatorial nature of problem (1) and the radiality constraints make the solution of such a problem extremely difficult by conventional optimization techniques. As for the reconfiguration problem, all branches are supposed to have either normally closed or opened switches.
3.1.2. Kruskal algorithm Let G = (V,E) be an n bus graph. A set D is formed by V and Ei ∈ E. At first Ei = 0, i.e. D comprises n isolated buses. Let Ea be the set of network branches in crescent order of their costs Cij . Originally the minimum cost branch i − j in Ea is selected. In case such a branch connects two different buses, it is included in D to form a new set D, i.e. D ← D∪(i − j). If however it forms a loop, such branch is rejected. One proceeds until D represents a complete tree of G. In this case, D constitutes a solution.
k=1
Subject to -
Load flow constraints; radiality constraints; voltage constraints; reliability constraints; maximum power flow (loading) in feeders.
3. Evolutionary algorithm to the reconfiguration problem Evolutionary algorithms can be considered as metaheuristics, which natural selection, mutation and recombination processes are based on analogies with genetics. EAs represent a generic class of algorithms that includes the well known genetic algorithms (GA). The main differences between EAs and GAs rely on the size of the population, crossover and mutation operators, and on the codifications adopted. 3.1. Generation of the initial population In order to obtain feasible and well fitted configurations (radial, low network losses, adequate voltage profile), the paper proposes the use of heuristics based on the generation of minimum spanning trees. This type of heuristics sets out on an intelligent search, producing a set of configurations in the search space. This work uses the Prim and Kruskal algorithms for the generation of minimum spanning trees. The criteria considered to compare loops in the generation of minimum spanning trees in distribution networks are based on the resistance, reactance and impedance values of the network branches. 3.1.1. Prim algorithm The Prim algorithm initially selects one vertex of the network graph. The algorithm then initiates the tree generation by con-
3.2. Genetic structure In order to represent each possible system configuration, a chromosome structure is adopted, based on the number of interconnection switches, corresponding to network branches, which are needed to maintain the network radial and to avoid islanding. The structure of the genetic chain using the interconnection branches is illustrated in Fig. 1. The number of interconnection branches is generally only a small parcel of the total number of system branches. The numbering is fixed during the network reconfiguration, regardless the operations of branch exchange or position of interconnection branches in the genetic chain. 3.3. Fitness function While evaluating the quality of generated configurations in each generation, this paper considers aspects concerning the objective function and the constraints represented in formulation (1). Therefore the following attributes must be taken into account by the fitness function: (a) network energy losses, (b) long duration interruption costs, (c) customer process disruption costs due to voltage sags, (d) voltage profile and (e) branch capacity limits.
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Attributes (a), (d) and (e) can be readily determined by a load flow calculation for each network configuration. Attributes (b) and (c) regarding power quality costs can be determined by the methodology to be exposed in Section 4 on this paper. For a given configuration i, the fitness function can be achieved by the following mathematical formulation:
55
solutions, Etk , is considered as the local elite. During the iterative process of the algorithm, those better quality configurations are preserved to form the set of global elite ETG . The global elite set receives configurations from good quality configurations in each local elite set. It then corresponds to an enlarged set of incumbent solutions.
NCons
Min fa(i) = CtoPer xPer(i)xHyear +
(Costint(k)(i) ) + CostDis(k)(i) )
3.5. Recombination
k=1
+
Nb
pen(Vj ) +
j=1
Where: pen (Vj ) =
⎧ ⎨0
(2)
⎫
(Vj − Vmax )2
0 |fl | − fl,max
pen(fl )
i=1
⎩ (V − V )2 min j
pen (fl ) =
NR
Vmin ≤ Vj ≤ Vmax ⎬ Vj > Vmax ⎭ Vj < Vmin
|fl | ≤ fl,max |fl | > fl,max
(3)
(4)
3.4. Selecting the best configurations When applying the evolutionary algorithm to such a minimization problem (2), the lower the fitness function assigned to a chromosome is, the better it is fitted or adjusted to the problem. This implies that such chromosome has a greater chance to survive for future generations. And in the following generation(s), it will have the skill to produce even better descendants. For the network reconfiguration problem the best technical configurations are selected by the tournament and elitism procedures. 3.4.1. Tournament The configurations are selected by means of tournaments. The number of applied tournaments is equal to the size of the population, what makes this procedure significantly different from the proportional selection, normally used for the Canonical GA. At each tournament, a set of Tk configurations is randomly chosen and the winning configuration is that one with better fitness function. A total of n tournaments generate a population of n individuals. The value of Tk is generally small, typically Tk ∈ {2, 3, 4}. 3.4.2. Elitism The elitism procedure is carried out before the recombination procedure. In each generation, a percentage of the best determined
Before performing the recombination, the best configurations (Etk ), i.e. a part of the population based on the corresponding fitness functions, form the local elite set. Each recombination operation takes individuals from the list, where one is randomly chosen from the subset Etk , and the other one is randomly chosen from the rest of the population (disregarding the subset Etk ). Fig. 2 shows two configurations, in which recombination is applied according to the following steps: Step 1: One of the parents is chosen to have predominance of the genetic load for children generation. This genetic predominance translates into the fact that for each one of the parents, the generated topologies will keep a higher number of branches from the predominant parent (genetic predominance). Step 2: From the predominant parent 1, a percentage of branches PLB-1 % superior to 50% (PLB-1 % = 70% for example) is chosen, in decreasing order of power flows. From another parent 2 a percentage of branches (PLB-2 % = 100% − PLB-1 %, PLB-2 % = 30% for example) is chosen, in decreasing order of power flows, especially the branches which are closer to the ones derived from the predominant parent. In order to complete the resulting configuration, additional lines are selected in a random way for the connection of all loads that might result islanded, preserving the radiality constraint. Regarding the generation of the second descendant, the proposed criteria and percentage of branches transferred are the same. In this case however the parent 2 becomes now the predominant one whereas parent 1 becomes less predominant. A lower number of branches is taken from the predominant parent 1 by the percentage PLB-2 % whereas a percentage PLB-1 % is taken for parent 2, both chosen in decreasing order of load flows. In order to complete the resulting configuration, additional lines are selected in a random way to avoid islanding, i.e. to connect isolated buses. The choice of the parameter PLB-1 % is empirical, though it is recommended to range from 60 to 90%. For lower values (below 60%),
Fig. 2. Two configurations submitted to recombination.
56
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Fig. 3. Generation of descendant individuals resulting from recombination.
this implies in low information transfer from parents to descendants, what leads the algorithm to take more iterations to reach the best solution. On the other hand, for higher values (above 90%), this parameter implies in a high amount of information transferred to descendants, thus taking to a premature convergence, which can be expressed by the determination of local optima. Fig. 3 shows the branches considered for recombination. The first descendant inherits the branches: 1–4, 4–6, 6–7, 7–16, 4–5, 1–8, 8–9, 9–12 and 8–10 from the predominant parent 1, and it inherits branches: 1–13, 13–15, 13–14 from the non-predominant parent 2. In order to complete the descendant network configuration, branch 5–11 is randomly chosen in such a way that all loads are supplied in a radial configuration. The second descendant, on the other hand, takes 70% of the branches from the second parent, inheriting the branches: 1–13, 13–15, 15–16, 16–7, 1–8, 8–9, 9–12, 1–4, 4–5 and 30% from the first parent, inheriting branches 4–6, 8–10, 10–14. Branch 11–9 is then randomly chosen to complete the radial and connected configuration. Two new configurations are then devised, for which new fitness function values will be assigned. 3.6. Mutation A mutation process was devised to maintain the radial structure of the network. An interconnection branch in the genetic chain is randomly selected to be closed. This operation generates a loop, with two supply sources. A closed switch is randomly chosen to be opened, replacing the former interconnection switch. 3.7. Control parameters in evolutionary algorithms Variable recombination and mutation rates are dynamically varied during the generations. The dynamic variation of these parameters, in the evolutionary reconfiguration algorithm, is dependant upon the permanence of the incumbent solution through the generations. In other words, in the beginning, if the process finds a new value of the incumbent solution, the recombination rate assumes its highest value and the mutation rate assumes its lowest value. During the final generations, when the incumbent solution does not alter after a pre-established number of generations, the value of the recombination rate assumes its lowest value, whereas the value of the mutation rate assumes its highest value. The following equations show how the described criteria are applied: R = R max −
Nc × (R max − R min ) Nc max
(5)
M = M min +
Nc × (M max − M min ) Nc max
(6)
3.8. Criterion for diversification An unfavorable characteristic of evolutionary algorithms is its strong attraction for local optima, provoked by a population submitted to the selection mechanisms. The lack of diversity is one of the principal factors for this event. Observing this problem, a mechanism was adopted to preserve a minimum degree of diversity in the population for each generation. The diversification of a population can be determined by the following equation: Div (%) = 100 −
#CMig Pop
× 100
(7)
3.8.1. Diversification during the recombination In order to maintain this diversity before carrying out the proposed recombination, half of the local elite set, Etk , is exchanged by another subset of the global elite set, ETG , every time saturation is achieved, i.e. when configurations in the local elite set present the same fitness function values. 3.8.2. Diversification during the mutation Another mechanism that favors an efficient way out from saturation is applied before mutation. Similarly to the process used in the recombination, the recombined configurations are analyzed. If the number of equal configurations surpasses half the total number of configurations, i.e. Div is greater than a pre-established percentage (x%), then the mutation rate is assigned a high value, so as to improve the diversity of the configurations. This conversely tends to increase the average fitness function of the population. The mutation rate returns to its original value, after saturation is no longer present in the population. 3.9. Stop criterion The stop criterion is based on the global incumbent solution (SIG ) obtained in each generation. If its value does not improve for a given maximum number of generations (Ncmax ), the process is assumed as converged. 4. Assessing the costs of long duration interruptions and process disruptions due to voltage sags For a given network configuration, the Monte Carlo simulation method is used to estimate long duration interruptions and cus-
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
tomer process disruptions due to voltage sags in different network locations. This is based on short-circuit simulations, on statistical data related to failure rates of the network components, fault locations, fault impedance, amongst others [15–18]. Each short-circuit condition, randomly established during the Monte Carlo simulation method, might cause short duration voltage variations (SDVV), namely voltage sags and swells, as well as short and long duration interruptions. The magnitudes of voltage sags and swells are determined by the short-circuit analysis of the network, given its topology and electric parameters of the system components (branches, transformers, generators, etc.). Their time durations can be estimated based on the natural fault extinction time and operating times of protective devices, normally related to the fault current. As for the determination of the number of events, a statistical counting is carried out for specific ranges of voltage sags magnitudes and durations. This counting is carried out for all short-circuit simulations.
57
Table 1 Maximum fault impedance values. Fault type
Maximum impedance
Three-phase Double-phase
10 20
Double-phase-to-ground Ground impedance Phase impedances
20 10
Phase-to-ground
30
4.1.4. Natural extinction time The natural extinction time is a random variable represented by the cumulative probability function of its duration, as shown in the Fig. 5.
In each simulation, the event duration is determined by the lowest value between the fault natural extinction time and the protection time. The natural extinction time is randomly generated as shown in Fig. 5. The protection time is determined by the protection devices involved along the path of the short-circuit current and that duration corresponding to the minimum actuating device time. When the protection device acts, a partial or global interruption takes place in the feeder. This situation is carefully assessed by the model, by checking the status of each bus in accordance to the operation of protection devices. Power quality indices related to long duration interruptions and customer process disruptions due to voltage sags can thus be estimated for each requested location or customer under analysis. Even when a long duration interruption is registered for buses downstream in relation to the protection device, upstream buses are normally submitted to voltage sags or swells, according to their magnitude and duration. Moreover, by comparing the severity of the voltage sag or swell and the customer sensitivity curve, the model determines a possible process disruption, i.e. an interruption in the customer process. In other words, disruptions take place when a voltage sag (or swell) is more severe than the equipment or process immunity. Each equipment or process can be characterized by a sensitivity curve similar to the ones well known in the literature, such as the CBEMA and ITI curves [13]. An illustrative sensitivity curve to assess the effects of voltage sags is shown in Fig. 6, which is similar to the lower part of the ITI curve. This curve should be process or equipment specific, and can be represented by an equivalent sensitivity curve for each customer. The Monte Carlo simulations must be executed for a given number of times. This number can be dynamically determined, i.e. when the power quality indices in customers do not suffer any significant alterations in each additional simulation. Effects of short-circuit events in the transmission and supply networks are not considered in this work, though all possible distribution network configurations tend to be affected in the same way. Nevertheless, it is well known that most voltage sags and swells
Fig. 4. Distribution of occurrences according to the fault type.
Fig. 5. Cumulative probability distribution for fault natural extinction time.
4.1. Fault simulation By using the Monte Carlo method, the following parameters are randomly generated in each simulation: 4.1.1. Fault location The fault location is determined according to the probability distribution of fault occurrences. Different probability distributions might be considered, and the diversity of areas might be represented by assuming different failure rates associated to each branch. When the failure rate (#faults/km/year) is the same for all branches, a uniform probability distribution is assumed. 4.1.2. Type of faults The method considers the following fault types: phase-toground, double-phase, double-phase-to-ground and three-phase. Different probabilities are assigned for each fault type, as shown in the Fig. 4. 4.1.3. Fault impedance The method considers a range of fault impedance values for phase-to-phase and phase-to-ground faults. It ranges from 0 to a given maximum value, according to the fault type, as shown in Table 1.
58
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Fig. 6. Representative sensitivity curve.
Fig. 7. Representative protection curve.
in an electric power distribution system are generated by faults in this very distribution network. 4.2. Determining the frequency of long duration interruptions and voltage sags disruptions Monte Carlo simulations provide fault system states and the corresponding power quality indices for each studied location. The frequency of long duration interruptions and voltage sag disruptions vary practically according to the number of simulations. It is necessary to refer these indices to a given base, e.g. number of events per year. In order to obtain the yearly based indices, the following equations are applied: nyears
Nprot = f × dAL
Fig. 8. Diversity of the population without the criterion in the 14 bus network.
(8)
nintk /yr =
n intk nyears
(9)
ndisk /yr =
n disk nyears
(10)
The fault occurrence rate, per km, per year (f ) can be obtained in practice from data related to interruption occurrences provoked by the various possible network failures. 4.3. Determining long duration interruption and disruption costs In order to calculate the customer costs related to long duration interruptions and disruptions due to voltage sags, the following formulation is proposed: CostDis(k) = CtoDis(k) · Dempeak(k) · ndisk /yr
(11)
CostInt(k) = CtoInt(k) · Demavr(k) · NHoursInt(k) · nintk /yr
(12)
5. Results In order to illustrate the methodology herein proposed, simulations were carried out in two distribution networks. The first network is a well known 23 kV distribution system [2], which comprises 14 buses and 3 interconnection switches. The second case is a 135 bus network [21] with 21 interconnection switches, located in Tres Lagoas, Mato Grosso State, Brazil. In both cases, a single load level is considered. The control parameters for the evolutionary algorithm, considering dynamic recombination and mutation
rates as well as diversity control, are shown in the Table 2. The computational system used in all the tests is a 1.4 GHz to 128 MB RAM Pentium 4 CPU. The initial configurations were produced in the following way: 40% of the population was randomly generated, 30% by using the Prim’s minimum spanning trees and the remaining 30% by using the Kruskal’s minimum spanning trees. Three types of costs (resistance, reactance and impedance) were used to generate the Prim and Kruskal based subpopulations. In all the tests, voltage values should be within the range between 0.95 and 1.05 pu. The diversity limit Div was set to 50%. The stop criterion uses Ncmax = 30. In order to illustrate the methodology in both case-studies, an overcurrent protection device was assumed to be installed at the Substation for each feeder, bearing a time–current curve as shown in Fig. 7. For the evaluation of the impact of network losses, the unit cost is set to 0.145 $/kWh, which is the assumed value to determine the cost of losses in each network configuration. For illustrative purposes, the failure rate for each branch in the 14 bus system is assumed 1 failure/km, whereas at the 135 bus system the failure rate per branch is assumed 0.1 failure/km. In both cases, a total of 1000 short-circuit simulations are carried out in each configuration. All customers are industrial ones, and power quality costs are assumed to be CtoInt = 1.27 $/kWh and CtoDis = 3.0 $/kW. The interruption unit cost is taken as the average value of the ones shown in [19]. As for the unitary disruption cost, its value was based on [20]. Long duration interruptions are assumed to take NHoursInt = 3 h in both cases.
Table 2 Control parameter for the evolutionary algorithm. Number of system buses 14 135
Population
R min
R max
M min
M max
Tk
ETk (%)
ETG (%)
Ncmax
50 200
0.1
0.9
0.01
0.5
2
30
40
30
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
59
Table 3 Best configurations and network losses. Number of system buses
Configuration no.
Interconnection switches
Losses (kW)
14
01 02
7-8-15 7-8-4
466.12 479.29
135
01
7-51-53-84-90-96106-118-126-128-137138-139-141-144-145147-148-150-151-156
280.22
Fig. 9. Diversity of the population using the criterion in the 14 bus network.
Fig. 10. The best configurations: (a) two best configurations for the 14 bus network and (b) best configuration for the 135 bus network.
60
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Table 4 Network and power quality costs. Number of system buses
Configuration no.
Losses costs (k$/year)
Power quality costs (k$/year)
Total costs (k$/year)
14
01 02
592 608
1798 1796
2390 2405
135
01
355
557
912
Table 5 Comparing the proposed method to previously published methods (135 bus network). Reference
Losses (kW)
Losses costs (k$/year)
Interruption costs (k$/year)
Disruption costs (k$/year)
Total costs (k$/year)
Mantovani et al. [21] Carreno et al. [22] Proposed method
285.50 280.17 280.22
362 355 355
128 127 128
430 430 429
920 912 912
5.1. Genetic algorithm performance Figs. 8 and 9 show the influence of the diversity criteria for each iteration. One should notice that keeping the diversity index above a given pre-defined percentage allows for a better distribution of the genetic material, thus enlarging the search space so that local optima are avoided. The 14 bus system demanded a total of 34 evolution cycles (or generations) resulting in 0.5 s computation time. The 135 bus system took 216 evolution cycles in a 60 min total computation time. Best configurations are shown in Table 3 for both cases studied. Fig. 10 presents the corresponding network configurations.
5.2. Value of the costs for long duration interruptions and process disruptions Table 4 shows cost values related to power losses and power quality for the best configurations illustrated at Fig. 10. Figs. 11 and 12 respectively illustrate indices related to voltage sags and the number of disruptions due to voltage sags in the customers bearing the highest energy consumption, for both cases. Such frequency indices show the number of sags per year, classified according to magnitude and duration. The number of disruptions due to voltage sags is obtained by the use of the sensitive curve shown in Fig. 6.
Fig. 11. Histograms for the larger customer in the 14 bus network: (a) frequency of voltage sags and (b) frequency of disruptions due to voltage sags.
Fig. 12. Histograms for the larger customer in the 135 bus network: (a) frequency of voltage sags and (b) frequency of disruptions due to voltage sags.
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
A comparison with the methodologies presented in [21] and [22] is shown in Table 5 for the 135 bus network. Cost of losses, interruptions and disruptions were based on the best configurations found in [21] and [22] and compared to the best configuration in Fig. 10(b), obtained by the proposed method. The comparison shows similar costs of the proposed method and the ones obtained in [22]. In relation to the method in [21], the proposed method shows better results. 6. Conclusion This paper has shown a new method to determine the best configurations by simply changing the status of some few network switches. These actions might generate benefits both for the utility and for the customers, since aspects related to network losses, voltage regulation, capacity loading and power quality indices related to long duration interruptions and voltage sags are incorporated into the model. The computational program for the determination of optimal and sub-optimal configurations is based on an evolutionary algorithm specially developed to produce, along the generations, radial configurations that can better explore a huge search space. The main aspects concerning the performance of the EA were explored, and relate to the generation of the initial population, recombination and mutation rates, and selection criteria, amongst others. Each network configuration is evaluated by a load flow algorithm and by a Monte Carlo simulation method, which produce the fitness functions. The Monte Carlo simulation method demands a substantial computational effort. The authors are working towards very efficient methods in order to improve such computation times. This work represents a contribution for distribution companies to better understand how operation procedures, particularly reconfiguration schemes, can improve the overall performance of the system, taking into account network and customer related costs.
61
nintk /yr ndisk /yr NCons Nb NR Nc
number of interruptions per year for consumer k number of disruptions per year for consumer k total number of customers number of network buses number of network branches number of generations in which the incumbent solution is invariant maximum number of generations Ncmax Nprot number of simulations in which some device protection sets off number of years Nyears NHoursInt(k) number of interruption hours per event, for customer k pen(Vj ) penalty regarding the voltage limit at bus j pen(Vl ) penalty regarding the capacity limit at branch l Per(i) total network losses for configuration i (kW) PLB-1 % percentage of branches in parent 1 that are to be transferred to the descendant PLB-2 % percentage of branches in parent 2 that are to be transferred to the descendant Pop total number of configurations in the population SIG global incumbent solution Tk set of configurations to be randomly chosen in the tournament process Vmax maximum allowable voltage Vmin minimum allowable voltage Greek letters R recombination rate minimum recombination rate Rmin Rmax maximum recombination rate M mutation rate Mmin minimum mutation rate maximum mutation rate Mmax f network yearly failure rate per km
Appendix A. List of symbols References CtoPer unit cost of network losses ($/kWh) CostInt(k)(i) long duration interruption cost for customer k, configuration i ($/year) CostDis(k)(i) process disruption costs for customer k, configuration i ($/year) CtoDis(k) unit disruption cost at customer k ($/kW) #CMig maximum number of equal configurations CtoInt(k) unit interruption cost for customer k ($/kWh) dAL extension, in km, for the analyzed network Div diversification (%) Dempeak(k) peak demand for customer k (kW) Demavr(k) average demand for the customer k (kW) Etk set of configurations in the local elite set of configurations in the global elite ETG fa(i) objective function considering long duration interruptions and process disruption costs for a configuration i fl power flow in branch l maximum capacity for branch l fl,max Hyear number of hours in one year i network configuration index or EA individual j bus index k customer index l branch index n intk number of simulations in which long duration interruptions occur for consumer k n disk number of simulations in which voltage sag disruptions occur for consumer k
[1] J.J. Young, K.J. Chul, Jin-O. Kim, J.-O. Joong-Rin Shin, K.Y. Lee, An efficient simulated annealing algorithm for network reconfiguration in large-scale distribution systems, IEEE Transactions on Power Delivery 17 (4) (2002) 1070–1078. [2] S. Civanlar, J.J. Grainger, H. Yin, S.S.H. Lee, Distribution feeder reconfiguration for loss reduction, IEEE Transactions on Power Delivery 3 (3) (1988) 1217–1223. [3] M.E. Baran, F.F. Wu, Network reconfiguration in distribution systems for loss reduction and load balancing, IEEE Transactions on Power Delivery 4 (2) (1989) 1401–1407, April. [4] A.L. Morelato, A.J. Monticelli, Heuristic search approach to distribution system restoration, IEEE Transactions on Power Delivery 4 (4) (1989) 2235–2241, October. [5] V. Borozan, D. Rajicic, R. Ackovski, Improved method for loss minimization in distribution networks, IEEE Transactions on Power Delivery 10 (3) (1995) 1420–1425, August. [6] D. Shirmohammadi, H.W. Hong, Reconfiguration of electric distribution networks for resistive line losses reduction, IEEE Transactions on Power Delivery 4 (2) (1989) 1492–1498, April. [7] G. Peponis, M. Papadopoulos, Reconfiguration of radial distribution networks: application of heuristic methods on large-scale networks, IEE Proceedings—Generation, Transmission and Distribution 142 (November (6)) (1995) 631–638. [8] Q. Zhou, D. Shirmohammadi, W.-H.E. Liu, Distribution Feeder Reconfiguration For Operation Cost Reduction, IEEE Transactions on Power Systems 12 (2) (1997) 730–735, May. [9] S. Bahadoorsingh, J.V. Milanovic, G. Yan Zhang, C.P. Gupta, J. Dragovic, Minimization of Voltage Sag Costs by Optimal Reconfiguration of Distribution Network Using Genetic Algorithms, IEEE Transactions on Power Delivery 22 (4) (2007) 2271–2278, Oct. [10] F.V. Gomes, J.R. Carneiro, J.L.R. Pereira, A New Distribution System Reconfiguration Approach Using Optimal Power Flow and Sensitivity Analysis for Loss Reduction, IEEE Transactions on Power Systems 21 (4) (2006) 1616–1623, November. [11] G.K.V. Raju, P.R. Bijwe, An Efficient Algorithm for Loss Reconfiguration of Distribution System Based on Sensitivity and Heuristics, IEEE Transactions on Power Systems 23 (3) (2008) 1280–1287, August.
62
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
[12] M. Baran, F.F. Wu, Optimal sizing of capacitors placed on a radial distribution system, IEEE Transactions on Power Delivery 4 (1) (1989) 735–743. [13] N. Kagan, H. Arango, N.M. Matsuo, G. Vasconcelos, U. Castellano, S.X. Duarte, Evaluating the Risk of Equipment Disruption Related to Voltage Sags, in: ICHQP Conference on Harmonics and Quality of Power, Lake Placid, New York, USA, 2004. [14] M.C. Golbarg, H.P.L. Luna, Otimizac¸ão Combinatórial e Programac¸ão Linear, Editora CAMPUS, Rio de Janeiro, 2000. [15] R. Billinton, W. Wangdee, Impact of utilising sequential and nonsequential simulation techniques in bulk-electric-system reliability assessment, IEE Proceedings—Generation, Transmission and Distribution 152 (September (5)) (2005) 623–628. [16] Geun-Joon Lee, M.M. Albu, G.T. Heydt, A power quality index based on equipment sensitivity, cost, and network vulnerability, IEEE Transactions on Power Delivery 19 (July (3)) (2004) 1504–1510. [17] J.V. Milanovic, C. Myo Thu Aung, C.P. Gupta, The influence of fault distribution on stochastic prediction of voltage sags, IEEE Transactions on Power Delivery 20 (January (1)) (2005) 278–285.
[18] J.V. Milanovic, C.P. Gupta, Probabilistic assessment of financial losses due to interruptions and voltage sags-part I: the methodology, IEEE Transactions on Power Delivery 21 (April (2)) (2006) 918–924. [19] A.G. Massaud, M.T. Schilling, J.P. Hernandez, Electricity restriction costs, IEE Proceedings—Generation, Transmission and Distribution 141 (4) (1994) 299–304. [20] E.M. Motoki, Procedure for mitigation and reduction of power quality impact in textile industry, in: Msc in Electrical Engineering, University of São Paulo, São Paulo, 2007. [21] J.R.S. Mantovani, F. Casari, R.A. Romero, Reconfiguration of radial distribution systems using the voltage drop criteria, Revista Controle e Automac¸ão, Sociedade Brasileira de Automática, SBA 11 (02) (2000) 150– 159. [22] E.M. Carreno, N. Moreira, R. Romero, Distribution network reconfiguration using an efficient evolutionary algorithm, in: IEEE Power Engineering Society General Meeting, June 24–28, 2007, pp. 1–6.
Contents lists available at ScienceDirect
Electric Power Systems Research journal homepage: www.elsevier.com/locate/epsr
Reconfiguration of distribution networks to minimize loss and disruption costs using genetic algorithms Juan Carlos Cebrian, Nelson Kagan ∗ Department of Electrical Engineering, University of São Paulo, Escola Politécnica, Av. Prof. Luciano Gualberto, travessa 3 n◦ 380 - CEP - 05508-970 – São Paulo, Brazil
a r t i c l e
i n f o
Article history: Received 13 October 2008 Received in revised form 12 August 2009 Accepted 13 August 2009 Available online 19 September 2009 Keywords: Distribution networks Genetic algorithms Power quality Voltage sags Monte Carlo simulation
a b s t r a c t In this paper a computational implementation of an evolutionary algorithm (EA) is shown in order to tackle the problem of reconfiguring radial distribution systems. The developed module considers power quality indices such as long duration interruptions and customer process disruptions due to voltage sags, by using the Monte Carlo simulation method. Power quality costs are modeled into the mathematical problem formulation, which are added to the cost of network losses. As for the EA codification proposed, a decimal representation is used. The EA operators, namely selection, recombination and mutation, which are considered for the reconfiguration algorithm, are herein analyzed. A number of selection procedures are analyzed, namely tournament, elitism and a mixed technique using both elitism and tournament. The recombination operator was developed by considering a chromosome structure representation that maps the network branches and system radiality, and another structure that takes into account the network topology and feasibility of network operation to exchange genetic material. The topologies regarding the initial population are randomly produced so as radial configurations are produced through the Prim and Kruskal algorithms that rapidly build minimum spanning trees. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Most of the overhead electric power distribution systems must operate in radial configurations. Normally open and normally closed switches are located along the network in strategic points. By altering the topology to a different radial configuration, one might obtain losses reduction, improvement in the network voltage profile, and betterment of reliability indices. When a fault occurs, the fault block can be isolated and a new radial configuration might restore the rest of the load. The reconfiguration problem that determines the best possible configuration considering the above criteria and constraints can be modeled as a non-linear mixed integer problem, what produces an enormous search space due to the combinatory nature of the problem. Many approaches have been considered to address such a problem, particularly conventional integer programming algorithms, heuristic algorithms and mixed heuristiccombinatory algorithms to better explore the search space [1–11]. In this paper, the reconfiguration of distribution networks, considering power quality indices, is formulated as a non-linear mixed integer programming problem. The adopted solution technique
∗ Corresponding author. Tel.: +55 11 3091 5318; fax: +55 11 3091 5317. E-mail addresses: [email protected] (J.C. Cebrian), [email protected] (N. Kagan). 0378-7796/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.epsr.2009.08.005
consists in an evolutionary algorithm (EA) that considers a decimal base codification scheme. The EA makes use of recombination and mutation operators specially developed to maintain alternative radial configuration networks. Fitness functions are related to network losses and penalizations regarding voltage and loading constraints. The model described in this paper also handles power quality costs, namely long duration interruptions (greater or equal to 1 min) and customer process interruptions due to voltage sags, herein simply named as disruptions. An efficient power flow method [3] and [12] determines network losses, equipment loading and bus voltages for different network topologies. As for the power quality indices, a Monte Carlo simulation method [13] was implemented and incorporated into the optimization model. Other relevant aspects which are discussed in this paper regard the implementation of solutions that directly affect the efficiency of the evolutionary algorithm, namely the generation of the initial population, the EA codification scheme and the recombination operator. Concerning the EA selection procedure, two possible alternatives are analyzed, namely tournament and elitism. The latter one considers the saturation of the elite population over the generation cycles. The effects of the EA control parameters in the performance of the evolutionary algorithm, such as the size of the population and the recombination and mutation rates, are carefully studied and analyzed.
54
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
In order to produce the initial population, radial configurations are randomly generated through heuristic techniques based on minimum spanning tree algorithms, using criteria to establish the loop values associated with the impedance and resistance of the distribution network [14]. The Prim and Kruskal algorithms were implemented so as to produce the initial population. The mechanism of recombination utilizes topology information contained in the individuals subject to recombination. The algorithm identifies the branches to be exchanged based on the corresponding power flow values, in order to generate two new radial configurations. 2. Formulation of the reconfiguration problem including long interruptions and voltage sags The reconfiguration problem in distribution networks can be formulated as a mixed integer programming problem, i.e. it contains real and integer variables. The solution of such a formulation determines that one configuration that optimizes an objective function subject to a set of constraints. When considering long duration interruptions and customer process disruptions due to voltage sags, the reconfiguration problem can be formulated as follows:
Fig. 1. (a) Radial 14 bus network (tree) subject to islanding problems; (b) representation of a chromosome (genetic chain).
NCons
Min fa(i) = CtoPer xPer(i)xHyear +
(CostInt(k)(i) ) + CostDis(k)(i) )
(1)
necting vertices with lower cost branches until a complete tree is formed. A D set is formed by a vertex Vi ∈ V and a branch Ei ∈ E, for which V, E ∈ G, where G represents a given graph. Initially D is formed by only a vertex Vi ∈ V. In order to produce a minimum spanning tree, an i − j branch is chosen in each iteration so as its cost Cij is the lowest and the union D∪(i − j) favors the formation of a tree and a new sub-graph D ← D∪(i − j). The algorithm ends when the sub-graph D forms a complete tree.
The combinatorial nature of problem (1) and the radiality constraints make the solution of such a problem extremely difficult by conventional optimization techniques. As for the reconfiguration problem, all branches are supposed to have either normally closed or opened switches.
3.1.2. Kruskal algorithm Let G = (V,E) be an n bus graph. A set D is formed by V and Ei ∈ E. At first Ei = 0, i.e. D comprises n isolated buses. Let Ea be the set of network branches in crescent order of their costs Cij . Originally the minimum cost branch i − j in Ea is selected. In case such a branch connects two different buses, it is included in D to form a new set D, i.e. D ← D∪(i − j). If however it forms a loop, such branch is rejected. One proceeds until D represents a complete tree of G. In this case, D constitutes a solution.
k=1
Subject to -
Load flow constraints; radiality constraints; voltage constraints; reliability constraints; maximum power flow (loading) in feeders.
3. Evolutionary algorithm to the reconfiguration problem Evolutionary algorithms can be considered as metaheuristics, which natural selection, mutation and recombination processes are based on analogies with genetics. EAs represent a generic class of algorithms that includes the well known genetic algorithms (GA). The main differences between EAs and GAs rely on the size of the population, crossover and mutation operators, and on the codifications adopted. 3.1. Generation of the initial population In order to obtain feasible and well fitted configurations (radial, low network losses, adequate voltage profile), the paper proposes the use of heuristics based on the generation of minimum spanning trees. This type of heuristics sets out on an intelligent search, producing a set of configurations in the search space. This work uses the Prim and Kruskal algorithms for the generation of minimum spanning trees. The criteria considered to compare loops in the generation of minimum spanning trees in distribution networks are based on the resistance, reactance and impedance values of the network branches. 3.1.1. Prim algorithm The Prim algorithm initially selects one vertex of the network graph. The algorithm then initiates the tree generation by con-
3.2. Genetic structure In order to represent each possible system configuration, a chromosome structure is adopted, based on the number of interconnection switches, corresponding to network branches, which are needed to maintain the network radial and to avoid islanding. The structure of the genetic chain using the interconnection branches is illustrated in Fig. 1. The number of interconnection branches is generally only a small parcel of the total number of system branches. The numbering is fixed during the network reconfiguration, regardless the operations of branch exchange or position of interconnection branches in the genetic chain. 3.3. Fitness function While evaluating the quality of generated configurations in each generation, this paper considers aspects concerning the objective function and the constraints represented in formulation (1). Therefore the following attributes must be taken into account by the fitness function: (a) network energy losses, (b) long duration interruption costs, (c) customer process disruption costs due to voltage sags, (d) voltage profile and (e) branch capacity limits.
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Attributes (a), (d) and (e) can be readily determined by a load flow calculation for each network configuration. Attributes (b) and (c) regarding power quality costs can be determined by the methodology to be exposed in Section 4 on this paper. For a given configuration i, the fitness function can be achieved by the following mathematical formulation:
55
solutions, Etk , is considered as the local elite. During the iterative process of the algorithm, those better quality configurations are preserved to form the set of global elite ETG . The global elite set receives configurations from good quality configurations in each local elite set. It then corresponds to an enlarged set of incumbent solutions.
NCons
Min fa(i) = CtoPer xPer(i)xHyear +
(Costint(k)(i) ) + CostDis(k)(i) )
3.5. Recombination
k=1
+
Nb
pen(Vj ) +
j=1
Where: pen (Vj ) =
⎧ ⎨0
(2)
⎫
(Vj − Vmax )2
0 |fl | − fl,max
pen(fl )
i=1
⎩ (V − V )2 min j
pen (fl ) =
NR
Vmin ≤ Vj ≤ Vmax ⎬ Vj > Vmax ⎭ Vj < Vmin
|fl | ≤ fl,max |fl | > fl,max
(3)
(4)
3.4. Selecting the best configurations When applying the evolutionary algorithm to such a minimization problem (2), the lower the fitness function assigned to a chromosome is, the better it is fitted or adjusted to the problem. This implies that such chromosome has a greater chance to survive for future generations. And in the following generation(s), it will have the skill to produce even better descendants. For the network reconfiguration problem the best technical configurations are selected by the tournament and elitism procedures. 3.4.1. Tournament The configurations are selected by means of tournaments. The number of applied tournaments is equal to the size of the population, what makes this procedure significantly different from the proportional selection, normally used for the Canonical GA. At each tournament, a set of Tk configurations is randomly chosen and the winning configuration is that one with better fitness function. A total of n tournaments generate a population of n individuals. The value of Tk is generally small, typically Tk ∈ {2, 3, 4}. 3.4.2. Elitism The elitism procedure is carried out before the recombination procedure. In each generation, a percentage of the best determined
Before performing the recombination, the best configurations (Etk ), i.e. a part of the population based on the corresponding fitness functions, form the local elite set. Each recombination operation takes individuals from the list, where one is randomly chosen from the subset Etk , and the other one is randomly chosen from the rest of the population (disregarding the subset Etk ). Fig. 2 shows two configurations, in which recombination is applied according to the following steps: Step 1: One of the parents is chosen to have predominance of the genetic load for children generation. This genetic predominance translates into the fact that for each one of the parents, the generated topologies will keep a higher number of branches from the predominant parent (genetic predominance). Step 2: From the predominant parent 1, a percentage of branches PLB-1 % superior to 50% (PLB-1 % = 70% for example) is chosen, in decreasing order of power flows. From another parent 2 a percentage of branches (PLB-2 % = 100% − PLB-1 %, PLB-2 % = 30% for example) is chosen, in decreasing order of power flows, especially the branches which are closer to the ones derived from the predominant parent. In order to complete the resulting configuration, additional lines are selected in a random way for the connection of all loads that might result islanded, preserving the radiality constraint. Regarding the generation of the second descendant, the proposed criteria and percentage of branches transferred are the same. In this case however the parent 2 becomes now the predominant one whereas parent 1 becomes less predominant. A lower number of branches is taken from the predominant parent 1 by the percentage PLB-2 % whereas a percentage PLB-1 % is taken for parent 2, both chosen in decreasing order of load flows. In order to complete the resulting configuration, additional lines are selected in a random way to avoid islanding, i.e. to connect isolated buses. The choice of the parameter PLB-1 % is empirical, though it is recommended to range from 60 to 90%. For lower values (below 60%),
Fig. 2. Two configurations submitted to recombination.
56
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Fig. 3. Generation of descendant individuals resulting from recombination.
this implies in low information transfer from parents to descendants, what leads the algorithm to take more iterations to reach the best solution. On the other hand, for higher values (above 90%), this parameter implies in a high amount of information transferred to descendants, thus taking to a premature convergence, which can be expressed by the determination of local optima. Fig. 3 shows the branches considered for recombination. The first descendant inherits the branches: 1–4, 4–6, 6–7, 7–16, 4–5, 1–8, 8–9, 9–12 and 8–10 from the predominant parent 1, and it inherits branches: 1–13, 13–15, 13–14 from the non-predominant parent 2. In order to complete the descendant network configuration, branch 5–11 is randomly chosen in such a way that all loads are supplied in a radial configuration. The second descendant, on the other hand, takes 70% of the branches from the second parent, inheriting the branches: 1–13, 13–15, 15–16, 16–7, 1–8, 8–9, 9–12, 1–4, 4–5 and 30% from the first parent, inheriting branches 4–6, 8–10, 10–14. Branch 11–9 is then randomly chosen to complete the radial and connected configuration. Two new configurations are then devised, for which new fitness function values will be assigned. 3.6. Mutation A mutation process was devised to maintain the radial structure of the network. An interconnection branch in the genetic chain is randomly selected to be closed. This operation generates a loop, with two supply sources. A closed switch is randomly chosen to be opened, replacing the former interconnection switch. 3.7. Control parameters in evolutionary algorithms Variable recombination and mutation rates are dynamically varied during the generations. The dynamic variation of these parameters, in the evolutionary reconfiguration algorithm, is dependant upon the permanence of the incumbent solution through the generations. In other words, in the beginning, if the process finds a new value of the incumbent solution, the recombination rate assumes its highest value and the mutation rate assumes its lowest value. During the final generations, when the incumbent solution does not alter after a pre-established number of generations, the value of the recombination rate assumes its lowest value, whereas the value of the mutation rate assumes its highest value. The following equations show how the described criteria are applied: R = R max −
Nc × (R max − R min ) Nc max
(5)
M = M min +
Nc × (M max − M min ) Nc max
(6)
3.8. Criterion for diversification An unfavorable characteristic of evolutionary algorithms is its strong attraction for local optima, provoked by a population submitted to the selection mechanisms. The lack of diversity is one of the principal factors for this event. Observing this problem, a mechanism was adopted to preserve a minimum degree of diversity in the population for each generation. The diversification of a population can be determined by the following equation: Div (%) = 100 −
#CMig Pop
× 100
(7)
3.8.1. Diversification during the recombination In order to maintain this diversity before carrying out the proposed recombination, half of the local elite set, Etk , is exchanged by another subset of the global elite set, ETG , every time saturation is achieved, i.e. when configurations in the local elite set present the same fitness function values. 3.8.2. Diversification during the mutation Another mechanism that favors an efficient way out from saturation is applied before mutation. Similarly to the process used in the recombination, the recombined configurations are analyzed. If the number of equal configurations surpasses half the total number of configurations, i.e. Div is greater than a pre-established percentage (x%), then the mutation rate is assigned a high value, so as to improve the diversity of the configurations. This conversely tends to increase the average fitness function of the population. The mutation rate returns to its original value, after saturation is no longer present in the population. 3.9. Stop criterion The stop criterion is based on the global incumbent solution (SIG ) obtained in each generation. If its value does not improve for a given maximum number of generations (Ncmax ), the process is assumed as converged. 4. Assessing the costs of long duration interruptions and process disruptions due to voltage sags For a given network configuration, the Monte Carlo simulation method is used to estimate long duration interruptions and cus-
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
tomer process disruptions due to voltage sags in different network locations. This is based on short-circuit simulations, on statistical data related to failure rates of the network components, fault locations, fault impedance, amongst others [15–18]. Each short-circuit condition, randomly established during the Monte Carlo simulation method, might cause short duration voltage variations (SDVV), namely voltage sags and swells, as well as short and long duration interruptions. The magnitudes of voltage sags and swells are determined by the short-circuit analysis of the network, given its topology and electric parameters of the system components (branches, transformers, generators, etc.). Their time durations can be estimated based on the natural fault extinction time and operating times of protective devices, normally related to the fault current. As for the determination of the number of events, a statistical counting is carried out for specific ranges of voltage sags magnitudes and durations. This counting is carried out for all short-circuit simulations.
57
Table 1 Maximum fault impedance values. Fault type
Maximum impedance
Three-phase Double-phase
10 20
Double-phase-to-ground Ground impedance Phase impedances
20 10
Phase-to-ground
30
4.1.4. Natural extinction time The natural extinction time is a random variable represented by the cumulative probability function of its duration, as shown in the Fig. 5.
In each simulation, the event duration is determined by the lowest value between the fault natural extinction time and the protection time. The natural extinction time is randomly generated as shown in Fig. 5. The protection time is determined by the protection devices involved along the path of the short-circuit current and that duration corresponding to the minimum actuating device time. When the protection device acts, a partial or global interruption takes place in the feeder. This situation is carefully assessed by the model, by checking the status of each bus in accordance to the operation of protection devices. Power quality indices related to long duration interruptions and customer process disruptions due to voltage sags can thus be estimated for each requested location or customer under analysis. Even when a long duration interruption is registered for buses downstream in relation to the protection device, upstream buses are normally submitted to voltage sags or swells, according to their magnitude and duration. Moreover, by comparing the severity of the voltage sag or swell and the customer sensitivity curve, the model determines a possible process disruption, i.e. an interruption in the customer process. In other words, disruptions take place when a voltage sag (or swell) is more severe than the equipment or process immunity. Each equipment or process can be characterized by a sensitivity curve similar to the ones well known in the literature, such as the CBEMA and ITI curves [13]. An illustrative sensitivity curve to assess the effects of voltage sags is shown in Fig. 6, which is similar to the lower part of the ITI curve. This curve should be process or equipment specific, and can be represented by an equivalent sensitivity curve for each customer. The Monte Carlo simulations must be executed for a given number of times. This number can be dynamically determined, i.e. when the power quality indices in customers do not suffer any significant alterations in each additional simulation. Effects of short-circuit events in the transmission and supply networks are not considered in this work, though all possible distribution network configurations tend to be affected in the same way. Nevertheless, it is well known that most voltage sags and swells
Fig. 4. Distribution of occurrences according to the fault type.
Fig. 5. Cumulative probability distribution for fault natural extinction time.
4.1. Fault simulation By using the Monte Carlo method, the following parameters are randomly generated in each simulation: 4.1.1. Fault location The fault location is determined according to the probability distribution of fault occurrences. Different probability distributions might be considered, and the diversity of areas might be represented by assuming different failure rates associated to each branch. When the failure rate (#faults/km/year) is the same for all branches, a uniform probability distribution is assumed. 4.1.2. Type of faults The method considers the following fault types: phase-toground, double-phase, double-phase-to-ground and three-phase. Different probabilities are assigned for each fault type, as shown in the Fig. 4. 4.1.3. Fault impedance The method considers a range of fault impedance values for phase-to-phase and phase-to-ground faults. It ranges from 0 to a given maximum value, according to the fault type, as shown in Table 1.
58
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Fig. 6. Representative sensitivity curve.
Fig. 7. Representative protection curve.
in an electric power distribution system are generated by faults in this very distribution network. 4.2. Determining the frequency of long duration interruptions and voltage sags disruptions Monte Carlo simulations provide fault system states and the corresponding power quality indices for each studied location. The frequency of long duration interruptions and voltage sag disruptions vary practically according to the number of simulations. It is necessary to refer these indices to a given base, e.g. number of events per year. In order to obtain the yearly based indices, the following equations are applied: nyears
Nprot = f × dAL
Fig. 8. Diversity of the population without the criterion in the 14 bus network.
(8)
nintk /yr =
n intk nyears
(9)
ndisk /yr =
n disk nyears
(10)
The fault occurrence rate, per km, per year (f ) can be obtained in practice from data related to interruption occurrences provoked by the various possible network failures. 4.3. Determining long duration interruption and disruption costs In order to calculate the customer costs related to long duration interruptions and disruptions due to voltage sags, the following formulation is proposed: CostDis(k) = CtoDis(k) · Dempeak(k) · ndisk /yr
(11)
CostInt(k) = CtoInt(k) · Demavr(k) · NHoursInt(k) · nintk /yr
(12)
5. Results In order to illustrate the methodology herein proposed, simulations were carried out in two distribution networks. The first network is a well known 23 kV distribution system [2], which comprises 14 buses and 3 interconnection switches. The second case is a 135 bus network [21] with 21 interconnection switches, located in Tres Lagoas, Mato Grosso State, Brazil. In both cases, a single load level is considered. The control parameters for the evolutionary algorithm, considering dynamic recombination and mutation
rates as well as diversity control, are shown in the Table 2. The computational system used in all the tests is a 1.4 GHz to 128 MB RAM Pentium 4 CPU. The initial configurations were produced in the following way: 40% of the population was randomly generated, 30% by using the Prim’s minimum spanning trees and the remaining 30% by using the Kruskal’s minimum spanning trees. Three types of costs (resistance, reactance and impedance) were used to generate the Prim and Kruskal based subpopulations. In all the tests, voltage values should be within the range between 0.95 and 1.05 pu. The diversity limit Div was set to 50%. The stop criterion uses Ncmax = 30. In order to illustrate the methodology in both case-studies, an overcurrent protection device was assumed to be installed at the Substation for each feeder, bearing a time–current curve as shown in Fig. 7. For the evaluation of the impact of network losses, the unit cost is set to 0.145 $/kWh, which is the assumed value to determine the cost of losses in each network configuration. For illustrative purposes, the failure rate for each branch in the 14 bus system is assumed 1 failure/km, whereas at the 135 bus system the failure rate per branch is assumed 0.1 failure/km. In both cases, a total of 1000 short-circuit simulations are carried out in each configuration. All customers are industrial ones, and power quality costs are assumed to be CtoInt = 1.27 $/kWh and CtoDis = 3.0 $/kW. The interruption unit cost is taken as the average value of the ones shown in [19]. As for the unitary disruption cost, its value was based on [20]. Long duration interruptions are assumed to take NHoursInt = 3 h in both cases.
Table 2 Control parameter for the evolutionary algorithm. Number of system buses 14 135
Population
R min
R max
M min
M max
Tk
ETk (%)
ETG (%)
Ncmax
50 200
0.1
0.9
0.01
0.5
2
30
40
30
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
59
Table 3 Best configurations and network losses. Number of system buses
Configuration no.
Interconnection switches
Losses (kW)
14
01 02
7-8-15 7-8-4
466.12 479.29
135
01
7-51-53-84-90-96106-118-126-128-137138-139-141-144-145147-148-150-151-156
280.22
Fig. 9. Diversity of the population using the criterion in the 14 bus network.
Fig. 10. The best configurations: (a) two best configurations for the 14 bus network and (b) best configuration for the 135 bus network.
60
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
Table 4 Network and power quality costs. Number of system buses
Configuration no.
Losses costs (k$/year)
Power quality costs (k$/year)
Total costs (k$/year)
14
01 02
592 608
1798 1796
2390 2405
135
01
355
557
912
Table 5 Comparing the proposed method to previously published methods (135 bus network). Reference
Losses (kW)
Losses costs (k$/year)
Interruption costs (k$/year)
Disruption costs (k$/year)
Total costs (k$/year)
Mantovani et al. [21] Carreno et al. [22] Proposed method
285.50 280.17 280.22
362 355 355
128 127 128
430 430 429
920 912 912
5.1. Genetic algorithm performance Figs. 8 and 9 show the influence of the diversity criteria for each iteration. One should notice that keeping the diversity index above a given pre-defined percentage allows for a better distribution of the genetic material, thus enlarging the search space so that local optima are avoided. The 14 bus system demanded a total of 34 evolution cycles (or generations) resulting in 0.5 s computation time. The 135 bus system took 216 evolution cycles in a 60 min total computation time. Best configurations are shown in Table 3 for both cases studied. Fig. 10 presents the corresponding network configurations.
5.2. Value of the costs for long duration interruptions and process disruptions Table 4 shows cost values related to power losses and power quality for the best configurations illustrated at Fig. 10. Figs. 11 and 12 respectively illustrate indices related to voltage sags and the number of disruptions due to voltage sags in the customers bearing the highest energy consumption, for both cases. Such frequency indices show the number of sags per year, classified according to magnitude and duration. The number of disruptions due to voltage sags is obtained by the use of the sensitive curve shown in Fig. 6.
Fig. 11. Histograms for the larger customer in the 14 bus network: (a) frequency of voltage sags and (b) frequency of disruptions due to voltage sags.
Fig. 12. Histograms for the larger customer in the 135 bus network: (a) frequency of voltage sags and (b) frequency of disruptions due to voltage sags.
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
A comparison with the methodologies presented in [21] and [22] is shown in Table 5 for the 135 bus network. Cost of losses, interruptions and disruptions were based on the best configurations found in [21] and [22] and compared to the best configuration in Fig. 10(b), obtained by the proposed method. The comparison shows similar costs of the proposed method and the ones obtained in [22]. In relation to the method in [21], the proposed method shows better results. 6. Conclusion This paper has shown a new method to determine the best configurations by simply changing the status of some few network switches. These actions might generate benefits both for the utility and for the customers, since aspects related to network losses, voltage regulation, capacity loading and power quality indices related to long duration interruptions and voltage sags are incorporated into the model. The computational program for the determination of optimal and sub-optimal configurations is based on an evolutionary algorithm specially developed to produce, along the generations, radial configurations that can better explore a huge search space. The main aspects concerning the performance of the EA were explored, and relate to the generation of the initial population, recombination and mutation rates, and selection criteria, amongst others. Each network configuration is evaluated by a load flow algorithm and by a Monte Carlo simulation method, which produce the fitness functions. The Monte Carlo simulation method demands a substantial computational effort. The authors are working towards very efficient methods in order to improve such computation times. This work represents a contribution for distribution companies to better understand how operation procedures, particularly reconfiguration schemes, can improve the overall performance of the system, taking into account network and customer related costs.
61
nintk /yr ndisk /yr NCons Nb NR Nc
number of interruptions per year for consumer k number of disruptions per year for consumer k total number of customers number of network buses number of network branches number of generations in which the incumbent solution is invariant maximum number of generations Ncmax Nprot number of simulations in which some device protection sets off number of years Nyears NHoursInt(k) number of interruption hours per event, for customer k pen(Vj ) penalty regarding the voltage limit at bus j pen(Vl ) penalty regarding the capacity limit at branch l Per(i) total network losses for configuration i (kW) PLB-1 % percentage of branches in parent 1 that are to be transferred to the descendant PLB-2 % percentage of branches in parent 2 that are to be transferred to the descendant Pop total number of configurations in the population SIG global incumbent solution Tk set of configurations to be randomly chosen in the tournament process Vmax maximum allowable voltage Vmin minimum allowable voltage Greek letters R recombination rate minimum recombination rate Rmin Rmax maximum recombination rate M mutation rate Mmin minimum mutation rate maximum mutation rate Mmax f network yearly failure rate per km
Appendix A. List of symbols References CtoPer unit cost of network losses ($/kWh) CostInt(k)(i) long duration interruption cost for customer k, configuration i ($/year) CostDis(k)(i) process disruption costs for customer k, configuration i ($/year) CtoDis(k) unit disruption cost at customer k ($/kW) #CMig maximum number of equal configurations CtoInt(k) unit interruption cost for customer k ($/kWh) dAL extension, in km, for the analyzed network Div diversification (%) Dempeak(k) peak demand for customer k (kW) Demavr(k) average demand for the customer k (kW) Etk set of configurations in the local elite set of configurations in the global elite ETG fa(i) objective function considering long duration interruptions and process disruption costs for a configuration i fl power flow in branch l maximum capacity for branch l fl,max Hyear number of hours in one year i network configuration index or EA individual j bus index k customer index l branch index n intk number of simulations in which long duration interruptions occur for consumer k n disk number of simulations in which voltage sag disruptions occur for consumer k
[1] J.J. Young, K.J. Chul, Jin-O. Kim, J.-O. Joong-Rin Shin, K.Y. Lee, An efficient simulated annealing algorithm for network reconfiguration in large-scale distribution systems, IEEE Transactions on Power Delivery 17 (4) (2002) 1070–1078. [2] S. Civanlar, J.J. Grainger, H. Yin, S.S.H. Lee, Distribution feeder reconfiguration for loss reduction, IEEE Transactions on Power Delivery 3 (3) (1988) 1217–1223. [3] M.E. Baran, F.F. Wu, Network reconfiguration in distribution systems for loss reduction and load balancing, IEEE Transactions on Power Delivery 4 (2) (1989) 1401–1407, April. [4] A.L. Morelato, A.J. Monticelli, Heuristic search approach to distribution system restoration, IEEE Transactions on Power Delivery 4 (4) (1989) 2235–2241, October. [5] V. Borozan, D. Rajicic, R. Ackovski, Improved method for loss minimization in distribution networks, IEEE Transactions on Power Delivery 10 (3) (1995) 1420–1425, August. [6] D. Shirmohammadi, H.W. Hong, Reconfiguration of electric distribution networks for resistive line losses reduction, IEEE Transactions on Power Delivery 4 (2) (1989) 1492–1498, April. [7] G. Peponis, M. Papadopoulos, Reconfiguration of radial distribution networks: application of heuristic methods on large-scale networks, IEE Proceedings—Generation, Transmission and Distribution 142 (November (6)) (1995) 631–638. [8] Q. Zhou, D. Shirmohammadi, W.-H.E. Liu, Distribution Feeder Reconfiguration For Operation Cost Reduction, IEEE Transactions on Power Systems 12 (2) (1997) 730–735, May. [9] S. Bahadoorsingh, J.V. Milanovic, G. Yan Zhang, C.P. Gupta, J. Dragovic, Minimization of Voltage Sag Costs by Optimal Reconfiguration of Distribution Network Using Genetic Algorithms, IEEE Transactions on Power Delivery 22 (4) (2007) 2271–2278, Oct. [10] F.V. Gomes, J.R. Carneiro, J.L.R. Pereira, A New Distribution System Reconfiguration Approach Using Optimal Power Flow and Sensitivity Analysis for Loss Reduction, IEEE Transactions on Power Systems 21 (4) (2006) 1616–1623, November. [11] G.K.V. Raju, P.R. Bijwe, An Efficient Algorithm for Loss Reconfiguration of Distribution System Based on Sensitivity and Heuristics, IEEE Transactions on Power Systems 23 (3) (2008) 1280–1287, August.
62
J.C. Cebrian, N. Kagan / Electric Power Systems Research 80 (2010) 53–62
[12] M. Baran, F.F. Wu, Optimal sizing of capacitors placed on a radial distribution system, IEEE Transactions on Power Delivery 4 (1) (1989) 735–743. [13] N. Kagan, H. Arango, N.M. Matsuo, G. Vasconcelos, U. Castellano, S.X. Duarte, Evaluating the Risk of Equipment Disruption Related to Voltage Sags, in: ICHQP Conference on Harmonics and Quality of Power, Lake Placid, New York, USA, 2004. [14] M.C. Golbarg, H.P.L. Luna, Otimizac¸ão Combinatórial e Programac¸ão Linear, Editora CAMPUS, Rio de Janeiro, 2000. [15] R. Billinton, W. Wangdee, Impact of utilising sequential and nonsequential simulation techniques in bulk-electric-system reliability assessment, IEE Proceedings—Generation, Transmission and Distribution 152 (September (5)) (2005) 623–628. [16] Geun-Joon Lee, M.M. Albu, G.T. Heydt, A power quality index based on equipment sensitivity, cost, and network vulnerability, IEEE Transactions on Power Delivery 19 (July (3)) (2004) 1504–1510. [17] J.V. Milanovic, C. Myo Thu Aung, C.P. Gupta, The influence of fault distribution on stochastic prediction of voltage sags, IEEE Transactions on Power Delivery 20 (January (1)) (2005) 278–285.
[18] J.V. Milanovic, C.P. Gupta, Probabilistic assessment of financial losses due to interruptions and voltage sags-part I: the methodology, IEEE Transactions on Power Delivery 21 (April (2)) (2006) 918–924. [19] A.G. Massaud, M.T. Schilling, J.P. Hernandez, Electricity restriction costs, IEE Proceedings—Generation, Transmission and Distribution 141 (4) (1994) 299–304. [20] E.M. Motoki, Procedure for mitigation and reduction of power quality impact in textile industry, in: Msc in Electrical Engineering, University of São Paulo, São Paulo, 2007. [21] J.R.S. Mantovani, F. Casari, R.A. Romero, Reconfiguration of radial distribution systems using the voltage drop criteria, Revista Controle e Automac¸ão, Sociedade Brasileira de Automática, SBA 11 (02) (2000) 150– 159. [22] E.M. Carreno, N. Moreira, R. Romero, Distribution network reconfiguration using an efficient evolutionary algorithm, in: IEEE Power Engineering Society General Meeting, June 24–28, 2007, pp. 1–6.