Optimal allocation of distributed generation and remote control switches for reliability enhancement of a radial distribution system using oppositional differential search algorithm Saheli Ray, Aniruddha Bhattacharya, Subhadeep Bhattacharjee Department of Electrical Engineering, National Institute of Technology, Agartala, Tripura 799046, India E-mail:
[email protected] Published in The Journal of Engineering; Received on 1st July 2015; Accepted on 16th July 2015
Abstract: Reliability enhancement of power distribution system has attained much significance in the present competitive electricity market. Accordingly, methodologies to assess and improve distribution system reliability are also gaining much importance. This study proposes oppositional differential search (ODS) algorithm to solve reliability optimisation problem of radial distribution system. The objective of this study is to obtain the optimum number, size and location of distributed generation as well as optimal number and location of remote control switch simultaneously in radial distribution system in order to improve system reliability at a compromised cost. A multi-objective function has been formulated here. Differential search (DS) algorithm imitates the seasonal migration behaviour of an organism in search of efficiency of food areas. Opposition-based DS (ODS) algorithm has been used here to improve the quality of solution in minimum time. The proposed opposition-based DS (ODS) algorithm utilises opposition-based learning for Superorganism initialisation and also for iteration wise update operation. Simulation results obtained by ODS algorithm have been compared with that of DS algorithm and differential evolutionary algorithm. Simulation results reveal that ODS algorithm provides considerably superior performance, in terms of quality of solution obtained and computational efficiency.
1
Introduction
The deregulation of power system and opening up of competitive power market has led to increased demand for uninterrupted quality power. Distribution system has the greatest contribution to the interruption of supply to a consumer [1]. Hence, improving distribution system reliability is key consideration of power system operation. Improvement of reliability involves some preventive and corrective measures and hence, always incurs additional cost. So, the reliability enhancement methods are to be adopted keeping in view the cost involved in the process. Failure rate, repair time and restoration time are some important parameters that characterise reliability. Lowering the values of one or more of the above parameters can improve reliability to a significant extent. The present authors have adopted optimal placement of distributed generation (DG) and remote control switch (RCS) in the radial distribution network in order to improve reliability. DGs are small-scale generation located at the distribution site itself and supplying power locally. RCSs are devices which can connect or isolate a section of a network. Suitable locations of DGs and RCSs in a network may reduce the time to restore power and thus improve reliability. However, installation, operation and maintenance of DG and RCS have their own cost. Hence, a compromise is required, to find optimal allocation of RCSs and DGs. During adopting the present work, a number of literatures related to the present work have been reviewed. Some of these are briefly discussed here. An artificial intelligence technique with multi-agent system has been adopted by Bouhouras et al. [2] for performing a cost/worth assessment of reliability improvement in distribution networks. Switch allocation problems have been a matter of research interest since decades and many works have been carried out [3–5]. With the recent trend of automation, RCSs are attaining importance in reliability improvement studies. Some studies have been performed in order to develop strategies for RCS without covering allocation of switches [6, 7]. Allocation of RCS has been considered in various studies [8–11]. Viotto Romero et al. [12] proposed a dedicated Taboo search algorithm for optimal switch allocation for J Eng 2015 doi: 10.1049/joe.2015.0097
automatic load transfer in distribution systems. Abiri-Jahromi et al. [13] adopted mixed integer linear programming for optimal placement of sectionalising switches. An iterated sample construction with path relinking method has been adopted by Benavides et al. [14] and applied to switch allocation in electrical distribution networks. Assis et al. [15] proposed an optimisation methodology to allocate switches on radially operated distribution networks considering sectionalising and tie switches of different capacities, with manual or automatic operation schemes. Many studies have focused on network reconfiguration [16–20]. Kavousi-Fard and Akbari-Zadeh [21] proposed a multi-objective improved shuffled frog leaping algorithm for network reconfiguration in order to optimise reliability indices and active power loss. An artificial immune systems optimisation has been utilised by Alonso et al. [22] for multi-objective distribution system reconfiguration. Zou et al. [23] utilised methods including feeder reconfiguration, recloser installation, recloser replacement and DG installation to minimise system average interruption duration index (SAIDI), an important reliability index. Zidan and Sadani [24] studied the effect of network configuration on maximum loadability and maximum allowable DG penetration in distribution systems. Elsaiah et al. [25] applied an intelligent particle swarm optimisation (PSO)-based method for feeder reconfiguration to improve reliability. Ali et al. [26] studied the effect of loading pattern on the performance of reconfigured distribution system. Samui et al. [27] presented a direct approach to optimal feeder routing problem. Kavousi-Fard et al. [28] adopted distribution feeder reconfiguration as a reinforcement strategy to enhance the reliability of the distribution systems; and also proposed a new modified optimisation method based on harmony search algorithm. Many studies have been carried out including DG in the distribution system. Brown et al. [29] used sequential feeder method and a multiobjective genetic algorithm (GA) together to solve the optimisation of feeder addition problem in an islanded distribution system with DGs. Abdi and Afshar [30] proposed a multi-objective methodology for optimal distributed generation allocation and sizing in distribution system using a hybrid method based on improved
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PSO and Monte Carlo simulation. Arya et al. [31] used differential evolutionary (DE) algorithm for optimising failure rate and repair time of distribution system having DG in standby mode. Mirjalili et al. [32] proposed a new method for loss reduction based on simultaneous DG placement and network reconfiguration. Raoofat [33] adopted a GA-based method to allocate DGs and RCSs simultaneously with a view to improve reliability and reduce energy loss considering multilevel load. Zidan et al. [34] applied a GA-based method to study the effect of load model on distributed generation allocation and feeders’ reconfiguration in unbalanced distribution systems. Civicioglu [35] introduced a new algorithm named differential search (DS) algorithm which has proved to provide superior solutions compared to other algorithms when applied to benchmark functions. One important characteristic of DS algorithm is its ability to involve more than one individual simultaneously to update the solution sets at a particular iteration, unlike many other algorithms. This improves the exploration ability of DS algorithm. Also, DS algorithm has no inclination to search the optimum solution always in the periphery of present best solution. This helps to balance its exploitation ability. The number of control parameters in DS algorithm is only two, which makes the parameter tuning process much simple. All these characteristics make DS algorithm superior compared to many other optimisation techniques to solve complex multi-modal optimisation problems. Opposition-based learning (OBL) was introduced by Tizhoosh [36]. Ventresca and Tizhoosh [37] first applied OBL to improve learning and back propagation in neural networks. Since then, it has been applied to many evolutionary algorithms (EAs) such as PSO by Wang et al. [38], DE by Rahnnamayan et al. [39], and ant colony optimisation by Mailisia [40]. The basic principle of OBL is to exploit opposite numbers to approach the solution. The inventors of OBL claim that a number’s opposite is probably closer than a random number to a solution. Thus, by comparing a number to its opposite, a smaller search space is needed to converge to the solution. Ergezer et al. [41] proved that a quasi-opposite number is usually closer than a random number to the solution. In fact, it has also been proved that a quasi-opposite number is usually closer than an opposite number to the solution. Furthermore, a new opposition method named quasi-reflection has been introduced [41] which is proved to have the highest expected probability of being closer to the problem solution among all OBL methods. The improved computational efficiency of quasi-reflection-based learning concept has motivated the present authors to incorporate this concept in DS algorithm and develop oppositional DS (ODS) algorithm to accelerate the convergence speed of DS algorithm to a larger extent by comparing the fitness of a solution estimate to its opposite and keeping the fitter one in the randomly selected Superorganism set. In some of the previous works as regard of optimal placement of RCSs or DGs, the number of RCS and DG has been taken as fixed. Moreover, in many of the works, either an optimal allocation of RCS or that of DG has been considered. There are comparatively a less number of literatures where both optimal RCS and DG allocation have been taken into account simultaneously taking multiobjective problem formulation. In this paper, a multi-objective function has been formulated considering both number and position of RCS and number, position and size of DG as unknown variables. The outcome of the proposed technique has been compared with that of DS and DE algorithms. Section 2 of this paper provides a brief description of the significance of DG and RCS in radial distribution system and its impact on the reliability parameters. Section 3 describes the mathematical formulation of the problem. Section 4 presents the DS algorithm. Section 5 explains OBL and the steps involved to solve the present problem using ODS algorithm. Results of simulation are presented and discussed in Section 6. The conclusion is drawn in Section 7.
2
RCS and DG in context to reliability
2.1 RCS in radial distribution network and reliability indices RCS has proved to be very useful with the recent trends of automation of distribution networks, since its switching time is very less. RCS may be sectionalising-switch (normally closed) or tie-switch (normally open). The RCS considered for installation in the present work is normally closed type. In radial network, RCS can be operated to isolate a faulty section from the rest of the network. The location of RCS can contribute significantly to enhance the reliability of a network. The basic reliability indices that are frequently used are failure rate, repair time, restoration time and outage duration. Failure rate denotes the frequency of occurrence of failure. Repair time represents the time required to repair a faulty section after a fault occurs. Restoration time represents the time required to restore service by switching operation after an interruption occurs. Outage duration denotes the total time of outage and is given either by the product of failure rate and repair time or by the product of failure rate and restoration time, as applicable. Although RCS does not affect the failure rate, it can have a notable impact on the outage duration. Optimal placement of RCSs can remarkably reduce the outage duration and thus enhance the reliability. If there is a fault located downstream to the load point (bus), and if there is no RCS in between the fault and the load point (bus) of concern, time to resume power to the load point will be equal to the time needed to repair the fault, i.e. repair time. On the other hand, an RCS in between the load point of concern and the fault location (downstream to the load point), can reduce this time to the operating time of the RCS, i.e. restoration time, since the opening of RCS will isolate the faulty segment from the healthy portion and power can be restored to the healthy portion. Since these indices do not take into account the number of customers and connected load, therefore the severity of the fault is not revealed by these indices. To get a clear picture of the fault severity, customer oriented indices are derived from the basic indices. The most frequently used customer oriented indices are system average interruption frequency index (SAIFI), SAIDI, customer average interruption duration index (CAIDI), expected energy not supplied (EENS) and so on. SAIFI is the ratio of the total number of customer interruptions to the total number of customers served. The ratio of the sum of customer interruption durations to the total number of customers is known as SAIDI. CAIDI represents the ratio of the sum of customer interruption durations to the total number of customer interruptions. EENS is the expected energy not supplied, generally expressed on per year basis. In order to reflect the severity or significance of a system outage, the indices such as SAIFI, SAIDI, CAIDI and EENS are very much helpful. Among several customer oriented indices, EENS is the index of concern in this work which is given by EENS =
Lj Uj
(1)
where n
Uj =
j 0
i0 =1
n
li0 repi0 +
j 1
i1 =1
li1 resi1
(2)
li0 and repi 0 denote the failure rate and repair time of the i0th distributor segment, respectively, n j0 denotes the total number of segments where fault has occurred and power can be resumed to load connected to the jth bus only after repairing of the fault. li1 and resi1 denote the failure rate and switching time or restoration time of the i1th distributor segment, respectively, and nj1 denotes the total number of segments where fault has occurred and power can be restored to the load of the jth bus through switching operation
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J Eng 2015 doi: 10.1049/joe.2015.0097
before repairing of the fault. Uj denotes the annual outage duration for load connected to the jth bus and Lj is the average load connected at the jth bus. 2.2 Distributed generation With the introduction of DG in the distribution network, the so long passive distribution network becomes an active network. Hence, the characteristics of the distribution network undergo remarkable changes. DG may act as a supplementary source of power. In case of any unavailability of power from the source end, power may be supplied from the DG, which in turn increases the system reliability. Without DG in the distribution system, when interruption of power occurs in a load point due to a fault in the upstream of the load, time to resume power to that load would definitely be equal to the time needed to repair the fault, i.e. repair time. Presence of DG in the distribution system may change the scenario after fulfilment of the following conditions: (a) If fault occurs in a location upstream to a load point with RCS in between the fault and the load point. (b) If DG is present downstream to the fault, isolated from the fault with RCS. Under these circumstances, the time required to resume power to that load point may be equal to the restoration time, subject to the availability of power from DG. Here, EENS calculation can be done using (1) and (2). 3
Problem formulation
nbus
EENSj × C1 × CPV1
(3)
j=1
where EENSj corresponds to the EENS of the load of the jth bus, nbus stands for the total number of buses in the network, C1 stands for per unit cost of EENS ($/kWh) and CPV1 is the cumulative present value (CPV) of EENS. The CPV method converts all costs and benefits during the lifecycle of any equipment/device into the first year of operation. Thus, it helps to evaluate the total costs and benefits during the economic lifecycle of the equipments [33]. EENS is calculated using (1) and (2). Variables li0 , re pi0 , n j0 , li1 , resi1 and n j1 of (2) depend upon the number of RCSs (nRCS), position of RCS (RCSpos), number of DGs (nDG), position of DG (DGpos) and size of DG (DGsize). Hence, nRCS, RCSpos, nDG, DGpos and DGsize are the decision variables. CPV1 is calculated as follows CPV1 =
1 − (PV1 )EL 1 − PV1
(4)
where PV1 =
(1 + Iinf )(1 + LG) (1 + Iint )
(5)
where EL is the economic life time of the equipments, Iinf is the inflation rate, Iint is the interest rate and LG is the load growth rate. J Eng 2015 doi: 10.1049/joe.2015.0097
(6) F2 = nRCS × (Ci + Cm × CPV2 ) where nRCS denotes the total number of RCSs present in the system. Cistands for the installation cost and Cm stands for the maintenance cost of each RCS. CPV2 is the CPV of operation and maintenance cost of the equipments (applicable to both RCS and DG) which is expressed as follows CPV2 =
1 − (PV2 )EL 1 − PV2
(7)
where (1 + Iinf ) (1 + Iint ) The objective function to reduce DG cost is presented as PV2 =
F3 =
nDG j1 =1
((DGsize j × Cins ) + (DGsize j × Co × CPV2 )) 1
(8)
(9)
1
where DGsize j denotes the size of DG placed at the j1th bus, j1 1 denotes a bus with DG installed and nDG denotes the total number of DGs present in the system. Cins stands for the installation cost and Co stands for the operation and maintenance cost of DG per unit size. The overall normalised objective function to represent multiobjective formulation is expressed as F1 − F1min F2 − F2 min + w2 ∗ F1 max − F1 min F2 max − F2 min F − F 3 3 min (10) + w3 ∗ F3 max − F3 min where w1, w2 and w3 are weightage values assigned to the individual objectives, in between 0 and 1, such that w1 + w2 + w3 = 1, in order to find the best compromising solution. F1min, F2min, F3min are the minimum values of three objective functions when minimised individually. F1max is the maximum value of the EENS cost, when no RCS or DG is present in the network. F2max and F3max are the maximum values of the two objective functions, when RCSs and DGs are present in all positions, respectively. The objective functions are to be solved subject to the following constraints F = w1 ∗
The objective of this paper is to obtain the optimum number, size and location of DG as well as optimal number and location of RCS simultaneously in radial distribution system. A multi-objective formulation is developed with a view to find a compromised solution which improves the reliability (by reducing equivalent cost of EENS) and simultaneously reduces the cost associated with RCSs and DGs. The objective function to reduce EENS is given by F1 =
The objective function to reduce RCS cost is represented as
(i)
DGmin ≤ DGsize (k) ≤ DGmax ; (ii) (iii)
k = 1, 2, 3, . . . , nDG (11)
0 ≤ nRCS ≤ RCSmax 0 ≤ nDG ≤ DGmax
(iv) 0 ≤ RCSpos (i) ≤ 1; (v) 0 ≤ DGpos (j) ≤ 1;
no no
(12) (13)
i = 1, 2, 3, . . . , ns
(14)
j = 1, 2, 3, . . . , nbus
(15)
where DGmin and DGmax represent the minimum and maximum limits on the size of DG, repectively. RCSmax_no denotes the maximum number of RCSs that may be installed in the network considering one RCS at each segment. DGmax_no denotes the maximum number of DGs that may be installed in the system considering one DG at each bus. RCSpos and DGpos are the discrete variables, i.e. RCSpos and DGpos can have discrete values of either 0 or 1. 0 represents no RCS in a distribution segment of the network and 1 represents the presence of RCS in a distribution segment of the network. Similarly, for DG position, 0 represents no DG and 1 represents the presence of DG in a bus. ns denotes the total number of distribution segments in a network. The number of variables for DGsize will be same as the total number of variables for DGpos. The total number of 1s present in the RCSpos variable
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will give the number of RCSs (nRCS) of the network. Similarly, total number of 1s present in the DGpos variable will give the number of DGs (nDG) of the network. These constraints are valid for all the objective functions (3), (6), (9) and (10) mentioned above. 4
DS algorithm
The DS algorithm simulates the Brownian-like random-walk movement used by an organism to migrate [35]. With the periodical climatic changes, many organisms exhibit seasonal migration behaviour where they migrate from one habitat to a more efficient one with respect to capacity and efficiency of food areas. In the course of migration, the species undergoing migration forms a Superorganism comprising a large number of individuals. The Superorganism as a whole changes its position towards more fruitful areas. Each set of Superorganism (i.e. Xi, i = 1, 2, 3, …, N) contains D number of elements (Xij , j = 1, 2, 3, . . . , D), where D is the number of unknown variables of the problem and N denotes the number of Superorganism sets. Initial position of each member of Superorganism set is given by Xij = rand u pj − lowj + lowj
(16)
equation Stopversiteij = rand × (upJ − lowJ ) + lowJ
The DS algorithm has only two control parameters. To get optimum solution using DS algorithm, it is necessary to get proper values of parameters p1 and p2. p1 and p2 are expressed as p1 = c1 × rand and p2 = c2 × rand, where c1 and c2 are control parameters. The software code of the DS algorithm can be found in [42]. 5
Oppositional-based learning
OBL was developed by Tizhoosh [36] to improve computational efficiency and accelerate the convergence rate of various optimisation techniques. OBL considers current population as well as its opposite population at the same time. Many researchers successfully applied this learning process to different soft computing techniques [43–46]. Here, opposite and quasi-opposite numbers have been defined in one-dimensional space. These definitions can easily be extended to higher dimensions. If x be any real number between [qa, qb], its opposite number xo can be defined as xo = qa + qb − x
where upj and lowj are the upper and lower limits of search space, respectively. Therefore, Superorganism may be defined as ⎡
x11 ⎢ x21 ⎢ ⎢ X = ⎢ x31 ⎢. ⎣ .. xN 1
x12 x22 x32 .. . xN2
x13 x23 x33 .. . xN3
x14 . . . x24 . . . x34 . . . .. . xN4 . . .
xqo = rand(qc, xo )
The Superorganism sets migrate towards global minimum and during this process the elements of Superorganism sets search for some randomly selected positions suitable for temporary stop over, known as Stopoversite. It is calculated using the following formula
(17)
where donor is the target towards which randomly selected individuals of Superorganism move. Donor is created by random shuffling of Superorganism i.e. [Xrandom-shuffling(i)]. The extent to which the change in position of the individuals occurs, is controlled by a scale value, which is calculated by Scale = randg × 2 × rand1 × rand2 − rand3
(20)
If x be any real number between [qa, qb], its quasi-opposite point, xqo can be defined as
⎤ x1D x2D ⎥ ⎥ x3D ⎥ ⎥ .. ⎥ . ⎦ xND
Stopoversite = Superorganism + Scale
× donor − Superorganism
(19)
(18)
where randg is a random number generated using gamma distribution; rand1, rand2 and rand3 are three random numbers generated in the range of [0, 1] using uniform distribution. If position of individual elements of a Superorganism set at Stopoversite, is proved to be better than the previous position, the Superorganism set that made such discovery, immediately settles to the new position and continue its migration from that position onward. The individual elements of a Superorganism set that participate in the search process are selected by a random process. If any element goes beyond the limits of search space, that particular element is randomly shifted to another position using the following
(21)
where qc is the centre of the interval [qa, qb] and can be calculated as (qa + qb)/2 and rand(qc, xo) is a random number uniformly distributed between qc and xo. The same logic may be applied to reflect the quasi-opposite point xqo, and thus to obtain its quasi-reflected point xqr. If x be any real number between [qa, qb], the quasi-reflected point, xqr is defined as xqr = rand(qc, x)
(22)
where rand(qc, x) is a random number uniformly distributed between qc and x. 5.1 Pseudocode of ODS algorithm See Fig. 1. The basic flowchart of ODS algorithm is given in Fig. 2. 5.2 Steps of ODS algorithm as applied to the present problem The sequential steps of the ODS algorithm applied to find optimum number and location of RCS of a radial distribution system are as follows: Required: N: the size of Superorganism, where i = {1, 2, 3, …, N} D: the dimension of the problem G: the number of maximum iterations Step 1: Read input data: Li, repi, resi, li , the ODS algorithm parameters like control parameters c1 and c2 and so on. Step 2: Initialise the value of w1 = 0 and w2 = 0, where w1, w2 are weightage factors. Step 3: Initialise each Superorganism set using (16), where upj and lowj are the upper and lower limits of the variables as explained in (11)–(15) of Section 3.
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J Eng 2015 doi: 10.1049/joe.2015.0097
Fig. 1 Continued
Step 4: Create quasi-reflected Superorganism (QRS) set using (22) after satisfying all the constraints of upper and lower limits as in Step 3. Step 5: Evaluate the objective functions F1, F2 and F3 for each Superorganism set and QRS set using (3), (6) and (9), respectively. For the multi-objective function, evaluate the overall objective function for each set of initially generated Superorganism and QRS using (10). Step 6: Set a new Superorganism size (N ) by comparing each objective function value using (3), (6) and (9) for the three singleobjective functions and using (10) for the multi-objective functions, for Superorganism set and QRS set. Step 7: For cycle 1: G Step 8: Calculate Stopoversite and update Stopoversite by a random process, details of which is given in Fig. 2. Step 9: If Stopoversite crosses the limits, set Stopoversite = rand (upj − lowj ) + lowj, where upj and lowjare the upper and lower limits on the variables as given in (11)–(15) of Section 3. Step 10: Evaluate objective functions of Stopoversite, FStopoversite using F1, F2 and F3 as given by (3), (6) and (9) for the single objectives and using (10) for the multi-objective formulation, as performed in Step 5. Step 11: Replace objective functions of Superorganismi, FSuperorganismi , by objective functions of Stopoversitei, FStopoversitei , if objective function value of Stopoversitei is less than that of Superorganismi. Step 12: Replace Superorganismi by Stopoversitei Stopoversitei if objective function value of Stopoversitei is less than that of Superorganismi.
Superorganismi =
Fig. 1 Pseudocode of ODS algorithm J Eng 2015 doi: 10.1049/joe.2015.0097
Stopoversitei ,
if Fstopoversite , Fsuperorganismi
i
Superorganismi , else
Step 13: Select a new parameter ‘jumping rate’ (Jr) within [0, 1]. Form QRS set from the newly developed Superorganism set as generated in the previous steps using Steps 37–42 of Section 5.1. This is an open access article published by the IET under the Creative Commons Attribution-NoDerivs License (http://creativecommons.org/licenses/by-nd/3.0/) 5
Fig. 2 Basic flowchart of ODS algorithm
Check constraints of upper bound and lower bound of newly created QOS. The constraints of upper bound and lower bound must be satisfied. Step 14: Calculate objective functions F1, F2 and F3 for each Superorganism set and QRS set using (3), (6) and (9), respectively. For the multi-objective function, evaluate the overall objective function for each set of initially generated Superorganism and QRS using (10). Step 15: Set a new Superorganism size (N ) by comparing objective function value for Superorganism set and QRS set. Step 16: Store the best objective function value among all the Superorganism sets.
Step 17: If maximum iteration G is reached, go to Step 18, else go to Step 8. Step 18: In case of the multi-objective problem, increment the values of w1 and w2 in the steps of 0.1 in such a way that for each value of w1, w2 should change its value from 0 to 1. w3 is selected for each case by subtracting the sum of w1 and w2 from 1, provided w3 remains positive . Perform steps 3–17 until the value of w1 reaches 1. Step 19: Best compromised solution – the algorithm described above generates the non-dominated set of solutions known as Pareto-optimal solutions. The decision-maker (power system operator) may have imprecise or fuzzy goals for each objective function.
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Fig. 2 Continued J Eng 2015 doi: 10.1049/joe.2015.0097
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6
Results and discussions
The ODS algorithm has been implemented on a test system and its performance has been compared with DS algorithm and DE to verify its feasibility for solving distribution system reliability optimisation problems. The algorithms have been coded in MATLAB software (version 7.10.0) on a processor of specification Intel (R) Core (TM) i7-2600 CPU 3.40 GHz with 2 GB RAM. 6.1 Description of the test system Fig. 3 33-Bus radial distribution system
Table 1 Different load levels Load level
Duration, h
% of peak load
340 5500 2920
1 0.4 0.5
1 2 3
To aid the operator in selecting an operating point from the obtained set of Pareto-optimal solutions, the fuzzy logic theory is applied to each objective function to obtain a fuzzy membership function as given below [47] ⎧ ⎪ 1 ⎪ ⎪ ⎨ max Fi − Fi mJi = max ⎪ F − Fimin ⎪ ⎪ ⎩ i 0
if Fi ≤ Fi min if Fi min , Fi , Fi max
(23)
if Fi ≥ Fi max
The best non-dominated objective function can be found when magnitude of µk as derived from the following equation is a maximum, where the normalised sum of objective function values for all objectives is the highest Q k i=1 mJ mk = M Q i k=1
i=1
mkJi
(24)
where Q denotes the total number of individual objective functions in (10), M is the number of non-dominated solutions. After completion of the process, the best solution of the problem is obtained.
The technique has been implemented on a 33-bus test system as shown in Fig. 3, where there is a circuit breaker at the beginning of the network and fuses at the starting point of each lateral branch. The RCSs are placed at the beginning of any distribution segment. Numbering of distribution segments is done in the following manner: distribution segment preceding bus 2 is numbered as distribution segment 1, preceding bus 3 is numbered as distribution segment 2 and so on. RCSs are numbered same as the distribution segment number on which they are located. DGs are numbered according to the bus number at which they are located. Failure rate, repair time and restoration time of distribution segments and the loads have been assumed as in [33]. Three different load levels are assumed as shown in Table 1. In the present work, the EENS has been calculated for all those load levels over a total period of 8760 h. Per kilowatt (kW) cost of EENS is taken as 8$ for all the load levels. The installation cost and maintenance cost per year of one RCS is 18 000$ and 2000$, respectively. The interest rate, inflation rate and load growth rate are 0.05, 0.08 and 0.05, respectively. Economic life time of the equipments is taken as 15 years. The data mentioned above are based on [33]. The DG size ranges between 0 and 1000 kW. The installation cost and operation & maintenance cost per year of DG have been taken as 600 and 300 $/kW, respectively. It is assumed that DG can supply power to a load point only if it is capable to meet the whole demand of that load point. Also, the DG supplies power in a direction downstream to its location after fulfilling the need (if any) of the load point at which it is located. It can supply power to upstream loads only after fulfilling the needs of all the load points located downstream. 6.2 Comparative study 6.2.1 Solution quality: Table 2 presents the best results for minimising EENS cost as applied to the 33-bus system. The results show that EENS cost and corresponding RCS cost obtained by ODS, DS and DE are same, but the DG cost is more in case of the DE and DS algorithm compared to that obtained by the ODS algorithm. Hence, the total cost obtained by the ODS algorithm is less. Table 3 gives a comparison of solutions and Fig. 4 shows the convergence characteristics for minimising EENS cost. The
Table 2 Best results for EENS cost minimization obtained using different methods levels No. of RCS and position (distribution segment no.)
DG position ( Load point no.) and size (kW)
Objective function EENS cost ($)(A)
RCS cost ($)(B)
DG cost ($)(C)
Total Cost ($)(A+B+C)
Total No. of iterations
DE
31(2-32)
904878
1699100
90673000
93276978
108
DS
31(2-32)
4(414), 5(1000), 6(968), 9(418), 11 (872), 12(832),13(711), 14(194) 15 (956), 16(1000), 17(1000), 18(735),20 (1000), 21(1000) 22(763), 25(1000), 26 (756), 28(193), 33(1000) 12(1000), 13(512), 14(956), 15(178), 16(1000), 17(1000), 18(957), 22(1000), 25(952), 33(1000) 12(975), 13(618), 14(364), 15(380), 16 (1000), 17(1000), 18(754), 21(97), 22 (1000), 25(934), 33(992)
904878
1699100
52370000
54973978
88
904878
1699100
49671000
52274978
75
Method
ODS
31(2-32)
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Table 3 Comparison of solution for EENS cost minimisation among different methods after 50 trials Methods
Minimum, $
Maximum, $
Average, $
Simulation time, s
No. of hits to optimum solution
904 878 904 878 904 878
908 436 904 878 904 878
906 799 904 878 904 878
14.2002 13.1250 12.0312
23 50 50
DE DS ODS
Table 4 Best results for RCS cost minimization obtained using different methods Method DE DS ODS
No. of RCS
RCS position
Objective function RCS cost ($) (B)
EENS cost ($) (A)
DG cost ($)(C)
Total Cost ($)(A+B+C)
0 0 0
-
0 0 0
8271100 8271100 8271100
-
8271100 8271100 8271100
Table 5 Comparison of solution for RCS cost minimisation among different methods after 50 trials Methods Minimum, $
DE DS ODS
0 0 0
Maximum, $
Average, $
Simulation time, s
No. of hits to optimum solution
0 0 0
0 0 0
9.3012 7.0031 6.9823
50 50 50
Fig. 4 Convergence characteristics of EENS cost minimisation obtained by DE, DS and ODS
ODS algorithm reaches minimum solution in 75 iterations while the DS algorithm and DE take 88 and 108 iterations, respectively, to reach minimum solution. Tables 4 and 5, respectively, display the best results and comparison of solutions when RCS cost is minimised. The best value of minimising RCS cost leads to no RCS installation and hence no improvement in reliability. Since without RCS, any faulty portion cannot be isolated from healthy part, installing DG does not help to reduce failure rate or repair time. So, no DG is installed in this case. The above mentioned scenario is not acceptable as there is no improvement in reliability. Convergence characteristics of Fig. 5 for minimising RCS cost show that ODS reaches minimum solution at around 28 iterations whereas DS and DE take more than 40 and 50 iterations, respectively, to reach minimum solution. Tables 6 and 7 exhibit the results and comparison of solutions for minimising DG cost. In this case, minimising DG cost leads to no DG installation, but RCS installation can influence the system reliability. In the present case of minimising DG cost, corresponding EENS cost and RCS cost together obtained by ODS algorithm is less than that obtained by the other algorithms. Thus, the total cost is less using ODS algorithm. Convergence characteristics of Fig. 6 for minimising DG cost exhibits that ODS algorithm takes about 10 iterations to reach minimum solution, whereas DS algorithm and DE reach minimum solution at around 35 and 55 iterations, respectively. Figs. 4–6 reveal a smoother convergence of ODS as compared to DS and DE. Table 8 presents the results obtained for the multiobjective formulation. It may be seen from the results that both RCS cost and DG cost obtained is less, but the EENS cost obtained is more using ODS algorithm than using DS algorithm. However, the total cost is notably less using ODS algorithm. As compared to DE also, the total cost obtained by ODS algorithm is remarkably J Eng 2015 doi: 10.1049/joe.2015.0097
Fig. 5 Convergence characteristics of RCS cost minimisation obtained by DE, DS and ODS
less. The corresponding SAIDI values for the configurations are also obtained. It may be observed that the when EENS cost is less, SAIDI is less and when EENS cost is more, SAIDI is also more. This is because of both EENS and SAIDI depend on the outage duration of the load points. A comparison of solutions showing minimum, maximum and average values for the multiobjective formulation obtained by the different algorithms is presented in Table 9. Figs. 7–9 give the Pareto-optimal front for the multi-objective problem obtained using DE, DS and ODS algorithms, respectively. It may be seen from the results of multi-objective formulation, although the EENS cost is more than that obtained by singleobjective function of minimising EENS cost, but the RCS cost and DG cost is much less than the corresponding RCS cost and DG cost obtained during the single-objective function of EENS cost minimisation. This makes the solution more feasible by lowering the total cost to a remarkable extent. Minimisation of EENS cost lowers the EENS cost to 904 878$, but the total cost becomes 52 274 978$. Whereas, in the multi-objective formulation, the
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Table 6 Best results for DG cost minimization obtained using different methods Method
DE DS ODS
No. of DG
DG position
Objective function DG cost ($)(C)
EENS cost ($)(A)
RCS position
RCS cost ($)(B)
Total Cost ($)(A+B+C)
0 0 0
-
0 0 0
7028500 6999200 6768300
3,5,6,7,8,10,11,12,13,16,19,20,21,22,23,26,28,31,32 3, 6, 7, 11, 14, 15, 16, 20, 22, 23, 24, 27, 28, 29, 32 2,3,4,5,9, 12, 13, 15, 17, 20, 21, 23, 24, 25, 27, 28, 29, 31, 32
1041400 822162 1041400
8069900 7821362 7809700
Table 7 Comparison of solution for DG cost minimisation among different methods after 50 trials Methods
Minimum, $
Maximum, $
Average, $
Simulation time, s
No. of hits to optimum solution
0 0 0
0 0 0
0 0 0
11.0001 9.7132 8.0721
50 50 50
DE DS ODS
Table 9 Comparison of solution among different methods after 50 trials (multi-objective function) Methods
Minimum, $
Maximum, $
Average, $
DE DS ODS
34 255 150 34 024 960 32 342 350
34 263 890 34 031 320 32 344 720
34 257 248 34 025 342 32 342 445
ODS and DS. These are presented in Tables 3, 5 and 7. The results prove superior computational efficiency of ODS algorithm. Fig. 6 Convergence characteristics of DG cost minimisation obtained by DE, DS and ODS
total cost reduces to 32 342 350$, but the EENS cost increases to 2 660 300$. Therefore, a reduction of 19 932 628$ total cost can be achieved by multi-objective formulation. As it has already been proved that minimising RCS cost, DG cost or EENS cost individually may increase the corresponding cost of the others to a large extent, hence, the multi-objective function can prove to be more realistic as it provides a compromised solution. 6.2.2 Computational efficiency: Time taken by ODS algorithm to reach the best solution for three different single-objective formulations are 12.0312, 6.9823 and 8.0721 s. For same objective functions, time required by DS algorithm to reach best solution is 13.1250, 7.0031 and 9.7132 s, respectively. Time taken by DE to reach the minimum solution is even more than that taken by both
6.2.3 Robustness: Performance of any stochastic algorithm cannot be judged by the results of a single run. A number of trial runs are required to have a useful conclusion about the performance of the algorithm. As ODS algorithm is a stochastic optimisation technique, randomness is obvious and many trials should be made to reach the optimum result. In this study, 50 trial runs have been carried out to obtain each result. An algorithm is said to be robust, if it gives consistent results during these trial runs. The results of Table 3 prove ODS algorithm to be better in terms of robustness as compared to DE and equally robust as compared to DS. In case of RCS cost minimisation and DG cost minimisation as provided in Tables 5 and 7, respectively, the algorithms exhibit similar performance in terms of robustness. However, ODS outperforms DS and DE in terms of simulation time. Table 9 shows that, for the multi-objective formulation, the average value of objective function obtained by ODS algorithm is close to the minimum value to a greater extent as compared to the average and
Table 8 Best results for multi-objective function obtained using different methods Method
DE DS ODS
Variables
Objective function
Total ($)(A +B+C)
SAIDI ((hours/ customer/year)
No. of RCS & position
No. of DG & position
DG size (kW)
EENS cost ($) (A)
RCS cost ($)(B)
DG cost ($) (C)
5 (5, 15, 23, 29, 31) 6 (5, 14, 23, 24, 27, 30) 5 (3, 15, 24, 28, 30)
6(12, 16, 17, 18, 25, 32) 6(12, 16, 17, 18, 25, 33) 6 (12, 16, 17, 18, 25, 33)
779, 1000, 1000, 522, 888, 938 778,1000, 1000, 504, 874, 953 903, 1000, 1000, 546, 424, 931
2595100
274050
31386000
34255150
7.62594
2421100
328860
31275000
34024960
7.13706
2660300
274050
29408000
32342350
7.73343
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J Eng 2015 doi: 10.1049/joe.2015.0097
Fig. 7 Pareto-optimal front for multi-objective problem obtained by DE
minimum values of DS algorithm and DE. This proves ODS to be better in terms of robustness. Therefore, the above results establish the enhanced consistency of DS algorithm to achieve superior quality solutions, in a computationally efficient and robust way. 6.3 Parameter tuning To get optimum solution using ODS algorithm, it is necessary to get proper values of jumping rate Jr, along with parameters c1 and c2. Moreover, the value of the cost function may vary with the Superorganism size also. For different values of these parameters, the minimum value of cost function is evaluated for the multi-
objective function. For a single value of one parameter, other parameters have been varied for their all possible combinations. However, presenting all the results in a tabular form will occupy a huge space. Hence, a brief summarised result is provided in Table 10. Too large or small values of Superorganism size may not be capable to get the minimum value of cost. For each Superorganism size of 20, 50, 100 and 200; 50 trials have been run. After a rigorous tuning procedure, it has been found that the best value of objective function is obtained with c1 = 0.3 and c2 = 0.3 and Jr = 0.3 with a Superorganism size 50. For Superorganism size more than 50, there is no improvement in the result. Moreover, beyond Superorganism size of 50,
Fig. 8 Pareto-optimal front for multi-objective problem obtained by DS J Eng 2015 doi: 10.1049/joe.2015.0097
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Fig. 9 Pareto-optimal front for multi-objective problem obtained by ODS Table 10 Influence of ODS parameters on multi-objective function value ($) after 50 trials Superorganism size
20
Jumping rate
0.2
0.3
0.5
0.8
50
0.2
0.3
0.5
c1
c2
0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9
0.2
0.3
0.5
0.8
0.9
32 832 980 32 800 142 32 835 170 32 834 327 32 834 327 32 801 824 32 790 410 32 805 492 32 800 912 32 801 278 32 835 420 32 815 420 32 825 432 32 825 567 32 825 492 32 836 623 32 815 432 32 826 024 32 825 783 32 825 542 32 345 824 32 345 824 32 346 872 32 346 743 32 346 750 32 343 032 32 342 720 32 342 520 32 342 675 32 342 712 32 345 932 32 345 712 32 346 024 32 346 122 32 346 210
32 814 570 32 798 990 32 823 765 32 823 824 32 823 824 32 791 892 32 771 432 32 806 721 32 800 127 32 801 143 32 815 721 32 817 725 32 819 870 32 819 843 32 819 723 32 815 892 32 812 097 32 819 923 32 819 924 32 819 877 32 344 934 32 344 934 32 345 390 32 344 509 32 344 438 32 342 842 32 342 350 32 342 408 32 342 522 32 342 540 32 343 480 32 343 380 32 344 012 32 345 037 32 345 542
32 827 321 32 810 523 32 821 620 32 822 642 32 822 642 32 800 124 32 785 421 32 812 453 32 812 242 32 813 795 32 827 765 32 828 632 32 828 910 32 829 025 32 829 133 32 828 096 32 827 863 32 828 957 32 829 120 32 829 280 32 347 853 32 347 853 32 347 729 32 347 894 32 347 549 32 343 020 32 342 532 32 342 590 32 342 624 32 342 645 32 343 750 32 344 127 32 345 280 32 345 302 32 345 421
32 831 521 32 821 746 32 821 824 32 822 094 32 822 094 32 800 432 32 801 022 32 810 073 32 811 295 32 812 308 32 833 264 32 831 875 32 832 376 32 832 439 32 831 042 32 833 312 32 832 098 32 832 399 32 832 600 32 832 050 32 346 572 32 346 572 32 347 023 32 347 320 32 347 414 32 343 189 32 342 590 32 342 784 32 342 804 32 342 790 32 345 389 32 345 245 32 345 304 32 345 385 32 345 496
32 832 052 32 822 950 32 822 175 32 822 208 32 822 208 32 815 043 32 809 214 32 816 702 32 819 980 32 820 034 32 832 942 32 833 025 32 833 657 32 834 006 32 834 120 32 832 930 32 833 172 32 833 421 32 834 258 32 834 795 32 345 523 32 345 523 32 346 092 32 346 350 32 346 430 32 343 200 32 342 976 32 342 810 32 342 827 32 342 836 32 345 420 32 345 472 32 345 450 32 345 410 32 345 503 Continued
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Table 10 Continued Superorganism size
Jumping rate
0.8
100
0.2
0.3
0.5
0.8
200
0.2
0.3
0.5
0.8
c2
0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9 0.2 0.3 0.5 0.8 0.9
c1 0.2
0.3
0.5
0.8
0.9
32 344 920 32 344 923 32 345 946 32 346 248 32 346 410 32 345 903 32 345 800 32 346 866 32 346 726 32 346 785 32 343 010 32 342 756 32 342 580 32 342 720 32 342 725 32 345 940 32 345 738 32 346 024 32 346 110 32 346 360 32 344 944 32 344 923 32 346 012 32 346 248 32 346 435 32 345 824 32 345 800 32 346 872 32 346 680 32 346 710 32 342 810 32 342 620 32 342 520 32 342 675 32 342 680 32 345 932 32 345 690 32 346 024 32 345 903 32 346 032 32 344 824 32 344 815 32 345 946 32 345 820 32 346 410
3 234 285 32 343 761 32 344 390 32 346 074 32 345 982 32 344 876 32 344 970 32 345 890 32 344 509 32 344 450 3 234 300 32 342 350 32 342 456 32 342 560 32 342 580 32 343 560 32 343 402 32 344 070 32 345 133 32 345 542 3 234 302 32 343 803 32 344 265 32 346 193 32 346 025 32 344 872 32 344 960 32 345 390 32 344 490 32 344 438 32 342 792 32 342 350 32 342 420 32 342 490 32 342 540 32 343 480 32 343 342 32 344 012 32 345 037 32 345 421 3 234 285 32 343 761 32 344 305 32 346 074 32 345 982
3 234 822 32 344 424 32 345 385 32 346 324 32 346 402 32 347 903 32 347 940 32 347 824 32 347 870 32 347 520 32 343 560 32 342 465 32 342 585 32 342 683 32 342 690 32 343 798 32 344 165 32 345 320 32 345 372 32 345 443 3 234 822 32 344 612 32 345 532 32 346 458 32 346 456 32 347 625 32 347 800 32 347 680 32 347 678 32 347 549 32 343 040 32 342 490 32 342 474 32 342 728 32 342 645 32 343 702 32 344 247 32 345 105 32 345 271 32 345 421 3 234 762 32 344 731 32 345 385 32 346 109 32 346 034
32 345 420 32 344 380 32 345 410 32 346 932 32 346 870 32 346 610 32 346 602 32 347 050 32 347 365 32 347 618 32 343 340 32 342 508 32 342 760 32 342 832 32 342 806 32 345 440 32 345 280 32 345 343 32 345 402 32 345 510 32 345 438 32 344 520 32 345 508 32 346 932 32 346 936 32 346 572 32 346 432 32 347 120 32 347 432 32 347 390 32 343 189 32 342 590 32 342 725 32 342 792 32 342 810 32 345 310 32 345 245 32 345 304 32 346 021 32 345 496 32 345 420 32 344 492 32 345 492 32 346 743 32 346 720
32 345 612 32 345 390 32 345 498 32 345 920 32 346 020 32 345 550 32 345 510 32 346 092 32 346 320 32 346 420 32 343 436 32 342 976 32 342 872 32 342 820 32 342 880 32 345 470 32 345 430 32 345 428 32 345 438 32 345 525 32 345 685 32 345 390 32 345 412 32 345 880 32 346 275 32 345 620 32 345 128 32 346 010 32 346 720 32 346 520 32 343 200 32 342 852 32 342 810 32 342 620 32 342 745 32 345 230 32 345 472 32 345 310 32 345 410 32 345 478 32 345 721 32 345 390 32 345 324 32 345 321 32 345 870
Table 11 Effect of Superorganism size on multi-objective function Superorganism size
Minimum, $
Maximum, $
Average, $
20 50 100 200
32 771 432 32 342 350 32 342 350 32 342 350
32 775 625 32 344 720 32 346 010 32 344 720
32 772 103 32 342 445 32 342 716 32 342 445
simulation time also increases. The variation of results obtained by ODS algorithm with variation in Superorganism size is shown in Table 11. 7
Conclusion
In this paper, ODS algorithm has been successfully implemented to find optimum number and location of RCS and optimum number, J Eng 2015 doi: 10.1049/joe.2015.0097
location and size of DG in a radial distribution network. Analyses of all the results reveal that the performance of ODS algorithm in all respect is better in comparison with the DS and DE algorithms. Thus, it may be concluded that ODS algorithm may act as an efficient tool to solve this type of reliability optimisation problems. In the present work, the RCSs and DGs are assumed as 100% reliable. In future extension of this work, the uncertainty of the proper functioning of RCSs and DGs may be introduced, to make the work more realistic. 8
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