Discrete Particle Swarm Optimization for Optimal DG Placement in Distribution Networks

Transcription

1 Discrete Particle Swarm Otimization for Otimal DG Placement in Distribution Networs Panaj Kumar, Student Member, IEEE, Nihil Guta, Member, IEEE, Anil Swarnar, Member, IEEE, K. R. Niazi, Senior Member, IEEE Deartment of Electrical Engineering, Malaviya National Institute of Technology Jaiur, India Abstract This aer rooses a new PSO based method to solve otimization roblems having discrete variables. The conventional PSO rovides tentative solutions having continuous decision variables. The roosed discrete PSO (DPSO) method utilizes the inherent inability of PSO to solve roblems with discrete variables. In the roosed method each of these tentative solutions generates a set of solutions with discrete decision variables. Out of this set, the best fit solution is selected for the further comutation. Moreover, a reair algorithm is suggested to correct infeasible individuals whenever aeared in the set of tentative solutions. The roosed method is alied to solve the otimal distributed generation lacement roblem in radial distribution systems with the objectives to reduce annual energy losses and node voltage deviations. The roosed method is tested on 69 bus test distribution system and the alication results are romising when comared with recent techniques. Keywords distributed generation; article swarm otimization; distribution system; ower losses; voltage rofile C G c 1, c 2 F(i) gbest H i i itr itr max w N N G N L n P G,min P G,max P G n,i B PL i I. NOMENCLATURE Total number of candidate DGs Acceleration coefficients Objective function (Wh) Best article osition based on overall swarm exerience Time duration of ith load level (h) Load Level Current iteration Maximum iteration Exonent coefficient Set of system nodes Maximum number of DGs installed Set of Load Levels System node Minimum DG limit Maximum DG limit DG caacity at nth node Loss Base case DG PL Loss with DG i /14/$ IEEE best Best osition of th article achieved based on its own exerience r 1 ( ), r 2 ( ) Random numbers in the range [0, 1] s 1 s + V mins V maxs V n,i ΔV n,i v 1 v + w w max w min λ Position of th article at th iteration Position of th article at (+1)th iteration Minimum ermissible node voltage Maximum ermissible node voltage Voltage at nth node for the ith load level Node voltage deviation at nth node for the ith load level Velocity of th article at th iteration Velocity of th article at (+1)th iteration Inertia weight Maximum value of inertia weight Minimum value of inertia weight Penalty Function II. INTRODUCTION Distributed Generation (DG) are now becoming more oular owing to taing local renewable energy resources, energy loss reduction, voltage rofile imrovement, ea load shaving, deferring investments in transmission and distribution networs, substation caacity release, etc., aart from meeting the energy demand of ever growing loads. A significant amount of voltage is droed across distribution feeders which results in substantial amount of ower and energy losses [1]. The advantages associated with DG s deends greatly how otimally they are laced in distribution networs. The roblem of DG lacement involves the determination of its siting and sizing, therefore it becomes a mixed integer, nonlinear, highly comlex combinatorial otimization roblem with certain equality and inequality constraints subjected to networ oeration. Recently many stochastic based otimization techniques have shown their otential to solve otimal DG allocation roblem while minimizing ower losses and imroving voltage rofiles and some of them are mentioned below. Moradi and Abedini [2] resented a combined genetic algorithm (GA)/article swarm otimization (PSO) aroach to allocate otimal DGs. In this method, sites of DGs are

2 determined using GA and their sizes are otimized using PSO. Kansal et al. [3] emloyed PSO to determine otimal sizing and siting of different tye of DGs and validates results with the analytical method of Acharya et al. [4]. An evolutionary rogramming (EP) based technique was resented by Khatod et al. [5] where authors used robabilistic techniques to address uncertainty in load and generations. Rao et al. [6] reconfigured the distribution networ along with DG lacement using Harmonic Search Algorithm (HSA). Garcia and Mena [7] imlemented Modified Teaching-Learning otimization (MLTBO) algorithm to solve otimal DG allocation roblem and validated the obtained results with their roosed Brute Force algorithm (BFA). Kollu et al. [8] emloyed HSA with Differential oerator for solving DG installation roblem. On the basis of results obtained, authors concluded that sensitivity based aroaches generates inferior solutions to simultaneous solution for otimal site and size roblem (SSOSSP). Hooshmand and Mohami [9] alied hybrid aroach of Bacterial foreging and PSO (BF PSO). In this method whenever the direction random vector of BF stagnates, it is relaced by the velocity vector of the PSO to imrove convergence. Most of these aforementioned methods have emloyed search sace reduction using some node sensitivity based aroach and thereby restricted the list of candidate nodes for DG lacement. In general ractice, the high otential buses identified using loss sensitivity factors have roven less than satisfactory as loss sensitivity factors may not always indicate the aroriate lacement [10]. Therefore it may leads to inferior solution. PSO has several advantages over other meta-heuristic techniques in term of simlicity, convergence seed, and robustness [11]. It rovides convergence to the global or near global oint irresective of the shae or discontinuities of the cost function [12]. However, the erformance of the PSO greatly deends on its arameters and it often suffers from the roblems such as being traed in local otima due to remature convergence [13]. PSO is a oulation based metaheuristic otimization technique in which the movement of articles is governed by the two stochastic acceleration coefficients, i.e., cognitive and social comonents and the inertia comonent. Each article udates its revious velocity and osition vectors according to the (7)-(9). PSO is inherently designed to address otimization roblems with continuous variables. The otimal DG lacement roblem involves mixed-integer variables. The roven otential of PSO can be utilized to solve such otimization roblems if some suitable measures have been taen so that it could become cometent to discrete variables. In this aer, Discrete PSO (DPSO) method is roosed to solve otimal DG lacement roblem in distribution systems by minimizing energy losses and imroving voltage rofiles while considering variation in load demands. The roosed method emloys the adatability of PSO to efficiently handle with discrete decision variables associated with the siting of DGs and does not squeezing the roblem search sace. The DGs oerated at unity ower factor are only considered in this study. The roosed method is alied on standard 69-bus test distribution system and the alication results are romising when comared with other recent oulation based techniques. III. PROBLEM FORMULATION The lacement of DGs regulates active ower flow in the distribution networ and thus affecting feeder ower losses and node voltage rofiles. Therefore, in this wor the saving in energy losses and imrovement in node voltage rofile are considered as two objectives for develoing otimal DG allocation roblem formulation. However, these two objectives are not combined into a single objective using weighted sum aroach [14] or using fuzzy framewor [15], rather a enalty factor aroach is used to consider the second objective. The single objective function to be otimized can be exressed as below: B DG F() i = λ( PLiHi PLi Hi) ; i N (1) L λ = 1/(1+ Max( ΔV ); n N (2) is the enalty function. where, V mins/maxs V n,i ; V min < V n,i < V mins ΔV n,i = 0 ; V maxs V n,i V mins (3) a very large number ; else Subject to ni, P G,min P G n,i P G,max (4) C G N G (5) n m; nm, N (6) The equation (3), (4) and (5) refers to node voltage deviation, maximum & minimum bounds of DG oututs and number of DGs to be installed in the distribution networ, resectively. Equation (6) not ermitted reeated nodes in the solution. IV. PROPOSED DPSO PSO is a robust stochastic evolutionary comutation technique based on the movement and intelligence of swarms [16]. It was first introduced by Kennedy and Eberhart in Since then PSO have evolved and became a romising tool to solve comlex combinatorial otimization roblems. The ey feature of PSO is its simlicity in concet and imlementation. A swarm consists of number of articles that fly through the feasible solution sace to search otimal solutions [17]. Each article udates its osition based on its revious velocity vector, own best exerience best and the best exerience of the swarm gbest. The movement of each article naturally evolves to an otimal or near-otimal solution. The movement of article for the th iteration is exressed as er the following model:

3 v + ( best s ) + c r ( ) ( gbest s ) 1 = wv + c (7) 1r1 ( ) 2 2 s = s + v (8) where c 1 and c 2 are acceleration coefficient, r 1 ( ) and r 2 ( ) are randomly generated numbers in the range [0, 1]. The inertia weight w is given by wmax wmin w= wmax itr itr (9) max inertia weight varied linearly. Therefore, exonential decaying inertia weight is suggested in the roosed DPSO and is given by (10). The grahical comarison between linear and roosed exonential inertia weight function is shown in Fig.2. The coefficient w governs the rate of decay of inertia weight with iterations. The value of this coefficient is taen as 5. For this value, the inertia weight is about to ercetible at the end of search. The roosed DPSO offers more diversity and better control of article velocity and thus enhances exloration and exloitation otential of the PSO. This enable the swarm to exloits the region efficiently, where the global otima may exist and thus avoids local traings. w itr w = wmax ex itrmax (10) Fig.1 Demonstration of DPSO PSO is inherently designed to address otimization roblems having continuous decision variables. While dealing with otimal DGs allocation roblem, the sites are strictly discrete variables. Several variants of PSO to handle discrete variables have been reorted in literature, and only some of them are mentioned as; Reference [18], [19], [20] transformed continuous decision variables into binary numbers, by emloying Fuzzy Matrices [21], sigmoid function [22] or sequence metric [23] to comute velocity and udate osition of articles. However, many researchers adoted the most general way by rounding-off each candidate site to its nearest integral value. But, the candidate solutions so obtained are not those which are suggested by the search algorithm. In the roosed DPSO, a set of tentative solutions are generated from each candidate solution having continuous decision variable(s). These tentative solutions do not have any continuous decision variable. For this urose, all ossible combinations are generated by alying floor and ceiling rounding of each continuous decision variable, as shown in Fig. 1. Since DGs are normally installed at only few number of locations, the set of tentative solution obtained from each candidate solution is not very large. However, some of the tentative solutions may violate roblem constraints and thus become infeasible. They are corrected using a reair algorithm, as described later on. The fitness of all feasible tentative solutions is calculated and the candidate solution is then relaced by the best fit tentative solution. The inertia weight imarts momentum to articles and is crucial to decide movement of the swarm [24]. If the articles find the desirable area, where the global otima may exist, very early, it may tend to miss the global otima [25]. It may haen due to high velocity assigned to articles when the Fig.2 Comarison of linear and exonential inertia weights A. Reair Algorithm Whenever articles violate roblem constraints, they are not rejected, but are corrected using reair algorithm. If any two reeated sites observed, any one of them is randomly selected from N. If the site violates its bounds then a new site is generated by rolling over N. However, if the site is not an integer, it is corrected as exlained earlier. Similarly if sizes of DGs are violating their uer and lower limits, they are also set as er (4) by rolling over their bounded limits. B. DPSO Algorithm The Proosed DPSO algorithm for solving the otimal DG lacement roblem can be exressed by following stes: Ste 1: Inut system data. Ste 2: Calculate the ower losses for each load conditions for base case using load flow. Ste 3: Initialize DPSO by randomly generating the swarm of desired size in the feasible search sace. Ste 4: Evaluate fitness of each article and obtain best and gbest. Ste 5: Udate velocity and osition of articles as er control equations of PSO defined by (7), (8) and (10).

4 Ste 6: Correct infeasible article, if any, using reair algorithm. Ste 7: Evaluate fitness and udate best and gbest then go to ste 5. Ste 8: Sto when termination criterion fulfills and dislay gbest article. V. SIMULATION RESULTS AND DISCUSSIONS The effectiveness of the roosed DPSO method is investigated on 69-bus test distribution system taen from [26]. This system has 68 sectionalizing lines, 5 tie-lines and 8 feeders. The normally oen switches are The total real and reactive ower demand at nominal load level is W and VAr, resectively. The control arameters considered for the roosed method are resented in Table I and the load levels along with their duration considered from [26] are shown in Table II. The limit of DG size ranges from 0 to 2 MW for the each node as in [6]. The roosed method is rogrammed in MATLAB, and the simulations are carried on a comuter with Intel core i5, 3.2 GHz rocessor, 4 GB RAM. TABLE I. PSO PARAMETERS w min w max c 1 c 2 Swarm size itr max TABLE II. LOAD LEVEL AND LOAD DURATION Load Level 0.50 (light) 1.0 (nominal) 1.60 (ea) Load duration(h) The roosed DPSO method is now alied on this test distribution system. The rated substation voltage is considered as 1.0.u. The secified node voltage deviation is taen as +5% and 10%. The obtained results are resented and comared in Table III. The table shows that the roosed method rovides a solution with less ower loss and better voltage rofiles than other mentioned methods. It is noteworthy that the method of [6] has alied node sensitivity aroach for raning system nodes and select to three of them as candidate nodes for DG lacement. This aroach drastically reduces the search sace. However, it leads to oor solution as the otimal nodes were not included in the search sace. The solution rovided by the roosed method contains nodes 11 and 18, which are not in the list of candidate nodes of [6]. For the nominal load level, SSOSSP [8] shows the best solution among all other mentioned methods and the roosed DPSO method shows comarable results with this method in terms of ower losses and minimum node voltage. The voltage rofile of the system for the ea load level, before and after otimal DG lacement, is shown in Fig. 3. The figure shows a significant imrovement in the node voltage rofile by DG lacement with all nodes acquire voltages within ermissible secified range. The effect of linear and exonential modulations of inertia weight on the convergence of roosed DPSO is shown in Fig.4, and is also comared with that of conventional PSO. The figure shows that exloitation otential of DPSO is imroved using linear modulation of inertia weight. However, when exonential modulations are emloyed, it is observed that articles exloring and exloiting the search sace more comrehensively and thus roosed DPSO enables to achieve global or near global solution. Therefore, the suggested modifications in the roosed DPSO guide the swarm effectively. Base case SSOSSP [8] HSA [6] BFA [7] MTLBO [7] Proosed DPSO Methods TABLE III. COMPARISON RESULTS Caacity(nodes) Caacity(nodes) Caacity(nodes) Caacity(nodes) Caacity(nodes) Load level Light(0.5) Nominal(1.0) Pea(1.6) (11) (17) (61) (65) (65) (65) (64) (64) (64) (63) (63) (63) (11) (18) (61) (11) (18) (61) (11) (11) (17) (18) (18) (61) (61) (61) (64) The solution quality obtained using roosed DPSO method is resented in Table IV. The table shows best, average and the worst value of the fitness function along with the Coefficient of Variance (COV) of the samle of solutions obtained after 100 trials of DPSO. It can be observed from the table that solution quality obtained using roosed method is reasonably good, as the best and average fitness values are in close roximity and the COV is about 0.1. This shows robustness of the roosed method. The average CPU time

5 incurred by this method is 265s. This can be considered reasonable, though the execution time does not matter much in this case as the DG sizing and siting roblems lie in the lanning domain [3]. Fig. 3 Node voltage rofile for ea load condition Fig. 4 Convergence characteristics of conventional and roosed DPSO TABLE IV. SOLUTION QUALITY OF DPSO Fitness Function (Wh) Best Average Worst COV initialization, most of the articles are with zero fitness. But, at the end of comutational search they acquire their ositions in the close roximity of the best article. The roosed DPSO method enables to exlore all ossible combinatorial solutions which exist in the near vicinity of each candidate solution. Therefore, it may be intuitively redicted that the roosed DPSO method will be more effective for large dimension roblems containing a large number of closely situated local otima. VI. CONCLUSIONS Distributed or disersed generations (DGs) are now becoming significantly imortant for distribution systems as they are not only roduces green energy to match increased load demands but also rovides more reliable and better ower quality of electric suly at imroved energy efficiency. All these benefits deends how otimally they have been laced in the distribution networs. In this aer attemts have been made for otimal DGs allocation using roosed discrete PSO (DPSO). The roosed method is esecially designed to address otimization roblems having discrete or mixedinteger decision variables. The inherent drawbac of the conventional PSO has been fruitfully utilized to imrove its erformance and comutational efficiency. In DPSO, a set of all ossible tentative solutions, each with discrete variables, is roduced from the candidate solution. The best fit solution is selected out of all these solutions so obtained. This enhances the exloration and exloitation otentials of PSO and thus imroves its convergence characteristic. Further, the roblem search sace is not squeezed using any sensitivity based aroach and it has been observed that such aroaches may lead to generate sub-otimal solution. Moreover, the modulations of inertia weight are allowed to vary by roosing a new exonential function. This strategy regulates articles' velocities more recisely and thus hels in better exloitation of the search sace. This enables roosed DPSO to avoid multile local traings. The roosed method is tested on 69-bus test distribution system and the alication results are romising when comared with other recently established techniques available in the literature. Fig. 5 Initial and final osition of the swarm The initial and final osition of the swarm in the feasible search sace for a samle trial of the roosed DPSO is shown in Fig. 5. It can be deicted from the figure that during random REFERENCES [1] D. Q. Hung, N. Mithulananthan, and R. C. Bansal, Analytical exressions for dg allocation in rimary distribution networs, IEEE Trans. on Ener. Convers., vol. 25( 3), , Setember [2] M. H. Moradi, and M. Abedini, A combination of genetic algorithm and article swarm otimization for otimal DG location and sizing in distribution systems, Electr. and Energy Syst., vol. 34 (1), , January [3] S. Kansal, V. Kumar, and B. Tyagi, Otimal lacement of different tye of DG sources in distribution networs, Elect. and Energy Syst., vol. 53, , December [4] N. Acharya, P. Mahat, and N. Mithulananthan, An analytical aroach for DG allocation in rimary distribution networ, Electr. Energy Syst., vol. 28(10), , December [5] D. K. Khatod, V Pant, and J. Sharma, Evolutionary rogramming based otimal lacement of renewable distributed generators, IEEE Trans. on Syst., vol. 28(2), , May [6] R. S. Rao, K. Ravindra, K. Satish, and S. V. L. Narasimham, loss minimization in distribution system using networ reconfiguration in the

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