Nowadays computer technology makes possible the study of. both the actual and proposed electrical systems under any operating

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1 45 CHAPTER - 3 PLANT GROWTH SIMULATION ALGORITHM 3.1 INTRODUCTION Nowadays computer technology makes possible the study of both the actual and proposed electrical systems under any operating condition without connecting them physically. Any kind of problems, such as system stability, unbalanced load distribution, impact of expansion on the existing systems, availability of short circuit capacity etc., can be solved intelligently and economically. The simulation results of various computer techniques can be correctly interpreted in the process of real time execution. But it is essential the basic understanding of power system engineering for the interpretation. Computer programs and/or simulation packages have been developed and are readily available to use them for studying, analyzing, planning, controlling, monitoring and designing the electrical systems. Unfortunately, the scope of getting erroneous results from simulation programs is more. The simulation programs require careful assembling and checking of system data used in the programs. In practice, engineering judgment is essential when reviewing the simulation results. In computer studies selecting the program based on the familiarity with the possible solution method is helpful when interpreting the results. Earlier the methods such as Gauss elimination, Laplace expansion, Gauss-Siedal, Cramer s rule and Newton - Raphson are widely used in computer based programs.

2 46 Various types of algorithms have been proposed for optimization of radial distribution systems. There are some drawbacks in the existing evolutionary algorithms. The issue of setting the values of various parameters of an evolutionary algorithm is crucial for good performance. Choosing the right initial value for the parameter can itself be a hard task. The proposed Plant Growth Simulation Algorithm (PGSA) is a simple and rapid optimization algorithm that does not need tuning many parameters. One of the major differences between the Plant Growth Simulation Algorithm presented in this work and the previously proposed algorithms is the manner in which it looks at the feasible region to obtain global optimal solution. Tong et al. [79] proposed PGSA aiming at global optima to solve integer programming. Cournede et al. [85] presented new method to simulate the growth of a plant. A trunk grows in the growth process from the root of a plant and a number of branches grow from the nodes on the trunk. Some new branches grow from the nodes of a plant. Till a plant is formed this process repeats continuously. The PGSA [89] do not require derivatives etc and uses only the objective function value related information. The growth model of plant phototropism is the basis for PGSA [93]. It searches the feasible region of integer programming as the plant grows in an environment. According to the change of the objective function it finds the probabilities to grow a new branch on different nodes of a plant and then forms the complete model. Wang C. and H. Cheng proposed an integrated algorithm for optimum dynamic reactive power in the RDS

3 47 [94]. The bionic random algorithm is suitable for non linear integer programming [95]. Trees growth simulation algorithm is developed for reactive power optimization [96]. Inspired by the analogy with the plant growth an optimized network reconfiguration for large distribution system is proposed by Wang and Cheng [103]. 3.2 PLANT GROWTH LAWS The biological experiments on the plants proved the following facts. (i) The probability to grow a new branch on a node in the plant growth process is greater when its morphactin concentration is higher. (ii) The environmental information of any node determines the morphactin concentration of the node on the plant. The relative position of any node affects its environmental information. 3.3 PLANT GROWTH PROBABILITY MODEL A probability model is established by simulating the growth process of plant phototropism [79]. The environment of a node Y on a plant is described by introducing a function f(y) in the model. The environment of the node Y for growing a new branch is better when the value of f(y) is small. A trunk M develops from its root B 0 in the plant growth process. Assuming k nodes B M1, BM2,, BMk with better environment compared to root B0 on the trunk M, means the f(y) of the k nodes and B0 satisfy f(bmi) < f(b0) (i=1, 2, 3.k), then the equation (3.1) can be concentrations CM1, CM2, used, CMk. to calculate their morphactin

4 48 C Mi = f B 0 - f BM i Δ i = 1, 2, 3...k 1 k where, Δ = f B - f B 1 0 Mi i=1 (3.1) The equation (3.1) gives the relationship between morphactin concentration of any node on the plant and its environmental information. It can be concluded that the morphactin concentration of any node on the plant depends on environmental information of other nodes along with its environmental information. Assuming q nodes Bm1, Bm2,..., Bmq that have better environment compared to root B0 on the branch m, their corresponding morphactin concentrations will be Cm1, Cm2,, Cmq. Now, the morphactin concentrations of the nodes on branch m and also nodes except BM3 (as the morphactin concentration of BM3 becomes zero after it grows the branch m) on trunk M need to be computed. The computation can be performed by using the equation (3.2) derived by adding the related terms of the nodes on branch m and discarding the related terms of the node BM3 in equation (3.1). C C Mi mj = = w here 0 - f BM i f B i = 1, 2, 3...k Δ +Δ f Bm j f B j = 1, 2, 3...k Δ +Δ 1 2 k Δ = f B -f B 1 i=1 0 Mi, i¹3 q Δ = f B - f B 2 0 m j j=1 (3.2)

5 49 It is also possible to derive q k CMi + CMj = 1 from i=1, i 3 j=1 equation (3.2). Now, a new morphactin concentration state space is formed with the nodes (except BM3) on trunk M and branch m. In a similar way a new preferential growth node that will grow a new branch can be gained in the next step. This process is repeated until the plant is formed. Fig.3.1 Morphactin concentration state space In view of optimization problems, the possible solutions can be represented by the nodes on a plant, objective function can correspond to f(y), the search region of possible solutions can correspond to the length of the trunk and the branch, the initial solution can be represented by the root of a plant and the preferential growth node represents the next basic point. 3.4 CONCLUSIONS The simulation of plant growth process can be used to solve the optimal problems of various systems in particular it can be implemented for radial distribution systems. This method is capable of addressing the modeling challenges required by the radial distribution system.

6 50 CHAPTER - 4 CONDUCTOR SELECTION 4.1 INTRODUCTION The distribution system delivers power to various consumers through feeders, distributors, and service mains. Feeders are nothing but conductors having large current capacity and they carry bulk current to the feeding points. The conductors account for major contribution to power loss in distribution system. The power loss in distribution system is significantly high because of heavily loaded and lengthy feeders. For most of the time the line conductors are loaded below their thermal limitations. Literature revealed that I2 R losses in the distribution system account for about 13% of total power produced. The reactive currents contribute to a portion of these losses in distribution systems. It is very essential to reduce the distribution power loss in the optimal distribution planning so as to make the power delivery efficient. The power utilities are compelled to adopt best practices to reduce the power losses especially at distribution level. In the optimal planning process an important practice is the selection of conductor for design upgrading of distribution systems. It is, therefore, necessary to obtain the adequate size of conductor according to its current carrying capacity. Sujit Mandal et al. [60] presented very practical techniques and any utility can implement these techniques very easily. The conductor cost versus minimization of the weighted area between a linear load

7 51 and conductor cost characteristics provides a suitable approach to solve optimal conductor problem. An evolutionary programming (EP) has been presented for optimal branch conductor selection of radial distribution feeders in [70]. A method is proposed [71] to determine maximum loading for various types of loads radial distribution feeders within maximum current carrying capacity of conductors. The electrical distribution feeder optimization using simulated annealing is proposed in [72] to minimize the sum of total power loss cost and total investment cost. A generalized model for optimal conductor selection in radial distribution system planning is proposed [97] by considering diversity in load peaks along the load points and cost of power in addition to other factors. It is proposed a formula for maximum loading of conductor for radial distribution network to compute the change in loading of the radial network. The most sensitive node and the node having the minimum voltage deviation are found and operated [107]. In this chapter, Plant Growth Simulation Algorithm (PGSA) is proposed for selection of optimal set of conductors in radial distribution systems. The various factors such as, effect of load growth, voltage limits etc., are considered to test the method. The optimal set of conductors selected by this method will satisfy the maximum current carrying capacity of the conductors and also maintain acceptable voltage profile. Besides, it achieves the minimum capital cost of conducting material and cost of the feeder energy loss.

8 LOAD FLOW METHOD FOR CONDUCTOR SELECTION The load flow technique for optimal selection of conductor is based on the power flow equations described in 2.2. The voltage magnitude at node i+1 with k type conductor is given below. 1/2 (P 2 + Q2 ) i, k i, k V = V 2-2(P r +Q x )+(r 2 + x 2 ) i+1, k i, k i, k j, k i, k j, k j, k j, k 2 V (4.1) i, k The active and reactive power losses of branch j with k type conductor can be calculated using, P lo ss (j, k ) = r j, k Q lo s s (j, k ) = x Pi,2 k + Q 2i, k j, k V2 i, k Pi,2 k + Q 2i, k V2 i, k (4.2) (4.3) The total active and reactive power loss of radial distribution system with k type conductor can be calculated as nb TPL = r j=1 j, k P i,2 k + Q 2i, k V2 i, k P 2 + Q2 nb i, k i, k TQ L = x 2 j=1 j, k V i, k (4.4) (4.5) 4.3 OBJECTIVE FUNCTION The plant growth simulation algorithm is presented to solve the optimization problem of optimal type of conductor selection. The objective function for selection of optimal conductor for branch j with k type conductor is formulated as follows: Min. F (j,k) = CL (j,k) + CC (j,k) (4.6)

9 53 i) Cost of energy Losses (CL): The annual cost for the loss in branch j with k type conductor is, CL (j, k) = Peak Loss (j, k) K p + K e Lsf 8760 (4.7) where, Kp = Annual cost of demand due to power loss (Rs./ kw) Ke = Annual energy loss cost (Rs./ kwh) Lsf = Loss factor Peak loss (j, k) = Active power loss of branch j with k type conductor under peak load conditions ii) Depreciation on Capital Cost (CC): The capital cost per annum for branch j with k type conductor is, CC (j, k) = Cost k Len j (4.8) where, = Interest and depreciation factor Cost (k) = Cost of k type conductor (Rs./ km) Len (j) = Length of branch j (km) Loss factor (Lsf) is defined as ratio of energy loss in the system during a given time period to the energy loss that could result if the system peak loss had persisted throughout that period. In British experience, Loss factor is expressed in terms of the Load factor (Lf) as, Lsf = 0.2 Lf Lf 2 (4.9) where, Lf = Load factor 4.4 IMPLEMENTATION OF PGSA TO CONDUCTOR SELECTION The complete algorithm for the proposed PGSA method for conductor selection is discussed below.

10 54 Step 1: Input the line and load data of the distribution system, constraints limits etc. Step 2: Form the search domain by taking the number of branches in the system as decision variables which corresponds to the length of the trunk and the branch of a plant. Step 3: Give the initial solution Xo (Xo is a vector with length equal to no. of branches/decision variables) which corresponds to the root of a plant, and calculate the initial value of objective function (Total cost). Step 4: Let the initial value of the basic point Xb, which corresponds to the initial preferential growth node of a plant, and the initial value of optimization Xbest equal to Xo, and let Fbest that is used to save the objective function value of the best solution Xbest be equal to f (Xo), namely, Xb = Xbest = Xo and Fbest = f(xo). Step 5: Initialize iteration count, count=1. Step 6: Search for new feasible solutions starting from basic point Xb = [x1b, x2b,, xib,, xnbb], where Xb corresponds to the initial conductor type of each branch. (nb is number of branches). Step 7: For k=1 to m where m is the number of available conductor types, let Xp be a new solution obtained by replacing jth decision variable by kth conductor type. Step 8: For the found solution Xp check it out the constraints, go to next step if it satisfies; otherwise abandon the possible solution Xp. Step 9: Calculate the objective function f (Xp ) and check f(xp )< f(xb ), if it does not satisfy abandon the possible solution Xp and increment k then go to step 7.

11 55 Step 10: Save the best possible solution from obtained feasible solutions. Step 11: If count >Nmax go to step 15; otherwise go to next step. Step 12: Calculate the probabilities C1, C2,., Ck of feasible solutions X1, X2,., Xk by using equation (3.1), which corresponds to determining the morphactin concentration of the nodes of a plant. Step 13: Calculate the accumulating probabilities C1, C2,, Ck of the solutions X1, X2, Xk. Select a random number β from the interval [0 1], β must belong to one of the intervals [0 C1], [ C1, C2],.,[ Ck-1, Ck], the accumulating probability of which is equal to the upper limit of the corresponding interval, and it will be the new basic point Xb for the next iteration, which corresponds to the new preferential growth node of a plant for next step. Step 14: Increment count by count+1 and return to step 6. Step 15: Output the results and stop. 4.5 ILLUSTRATIVE EXAMPLES Two examples consisting of 26 node and 32 node practical radial distribution systems are presented to illustrate the effectiveness of the proposed algorithm. The conductor data is given in table B Example 1 The fig. 4.1 shows single line diagram of a 26-node practical radial distribution system. The line and load data of 26-node radial distribution system are given in table B.2 [62]. The optimal conductor

12 56 selection for the system with and without load growth is obtained as follows Conductor Selection of 26 Node RDS without Load Growth The modifications of conductor type without considering load growth based on the proposed PGSA are presented in table 4.1. The reconductoring is necessary for all the branches except 17, 19, 23 and 25. The voltage profile of 26-node radial distribution system is shown in fig.4.2. It can be seen from the table 4.2, the minimum voltage is improved from pu to pu. The voltage regulation improvement is found as 2.81%. The total real power loss is reduced to kw after conductor selection as given in the table 4.3. The summary of results of 26-node system is given in table 4.4. The savings after conductor selection is Rs /-. The result of objective function is shown in fig.4.3. Fig.4.1 A 26-node practical radial distribution system

13 57 Table4.1. Conductor selection without load growth for 26 node system Existing Conductor Conductor selected by in base case proposed PGSA method (From) (To) 1 to 16, 18,20,24 Mink 17,19,23,25 21,22 Squirrel Branch Number Fig.4.2 Voltage profile for 26-node practical radial distribution system without load growth Fig.4.3 Objective function of 26-node system without load growth

14 58 Table4.2. Node voltages of 26-node system without load growth Node V (pu) before Conductor V (pu) after Conductor No. Selection Selection

15 59 Table4.3. Power loss of 26-node system without load growth Branch No. Before After conductor conductor selection selection by PGSA Ploss Qloss Ploss Qloss (kw) (kvar) (kw) (kvar) Total Loss

16 60 Table4.4. Results of 26-node system without load growth Description Conductor Selection Base Case by PGSA Min. Voltage (pu) Real Power Loss (kw) Total Cost (Rs.) Conductor Selection of 26 Node RDS with Load Growth There is load growth with time due to both new and existing customers in the distribution system. The conductor losses are higher if the rate of load growth is large. Assuming a rate of load growth of 0.07 and growth period of 12 years the effect of load growth is illustrated in this section. The modifications of conductor type with load growth obtained by PGSA are tabulated in table 4.5. It shows that the reconductoring is necessary for all the branches except 21. The node voltages are tabulated in table 4.6 and the voltage profile of the system is shown in fig.4.4. It is seen that minimum voltage is improved from pu to pu. The voltage regulation is improved by 7.51%. The power loss is given in table 4.7. The percentage of real power loss reduction after conductor selection is The summary of simulation results is tabulated in the table 4.8. The total cost reduction is Rs /- after conductor selection. The simulation result of objective function is shown in fig.4.5.

17 61 Table4.5. Conductor selection of 26 node system with load growth in 12 years Existing Conductor in Conductor modified by base case proposed PGSA method (From) (To) Mink 19 Rabbit 21 Branch Number 1 to 18, 20, 22 to 25 Fig.4.4 Voltage profile for 26-node radial distribution system with load growth in 12 years Fig.4.5 Objective function for 26-node system with load growth in 12 years

18 62 Table4.6. Node voltages of 26-node radial distribution system with load growth Node V (pu) before Conductor V (pu) after Conductor No. Selection Selection

19 63 Table4.7. Power loss of 26-node RDS with load growth Branch No. Before After conductor conductor selection selection by PGSA Ploss Qloss Ploss Qloss (kw) (kvar) (kw) (kvar) Total Loss

20 64 Table4.8. Summary of results of 26-node system with load growth Description Conductor Selection Base Case by PGSA Min. Voltage (pu) Real Power Loss (kw) Total Cost (Rs.) Fig.4.6 A 32-node practical radial distribution system Example-2 The single line diagram of a 32-node practical system in India is shown in fig.4.6. The line and load data of this system are given in table B.3 [62]. The optimal conductor selection for the 32- node system with and without load growth is obtained using PGSA Conductor Selection of 32 Node RDS without Load Growth The modifications of conductor type by PGSA are tabulated in table 4.9. The reconductoring is necessary for all the branches except

21 65 branch no. 30. The voltage profile is shown in fig.4.7. The node voltages and power loss obtained by PGSA are given in table 4.10 and 4.11 respectively. It is seen that minimum voltage is improved from pu to pu, i.e. improvement in the voltage regulation is 1.30%. The total real power loss is reduced after conductor selection to kw. The results of 32-node system are summarized and tabulated in table Total cost reduction after conductor selection is Rs /-. The objective function curve is shown in fig.4.8. Table4.9. Modifications of conductor for 32 node system without load growth Existing Modified Conductor Conductor in by proposed PGSA base case (From) method (To) 1 to 21 Rabbit Mink 22 to 28, 31 Mink 29 Rabbit 30 Branch Number Table4.10. Node voltages of 32-node system without load growth Node No. V (pu) before V (pu) after Conductor Selection Conductor Selection

22 Table4.11. Power loss of 32-node radial distribution system without load growth Branch Before conductor After conductor selection selection by PGSA No. Ploss (kw) Qloss (kvar) Ploss (kw) Qloss (kvar)

23 Total loss

24 68 Table4.12. Summary of results of 32-node system without load growth Conductor Selection Description Base Case by PGSA Min. Voltage (pu) Real Power Loss (kw) Total Cost (Rs.) Fig.4.7 Voltage profile of 32-node radial distribution system without load growth Fig.4.8 Objective function for 32-node radial distribution system without load growth

25 Conductor Selection of 32 Node System With Load Growth The effect of load growth, assuming a rate of load growth of 0.07 and growth period of 3 years, is illustrated for 32 node system in this section. The modifications of conductor type are given in table It can be seen that reconductoring is necessary for all the branches except branch no. 30. The node voltages are given in table 4.14 and shown in fig.4.9. The minimum voltage is improved from pu to pu and the regulation is improved by 1.62%. The power loss is tabulated in table The percentage of real power loss reduction after conductor selection is The summary of results is given in table The total cost reduction after conductor selection is Rs /-. The result of objective function for this system is shown in fig Table4.13. Conductor selection of 32 node system with load growth for 3 years Existing Conductor selected by Conductor in proposed PGSA method base case (From) (To) 1 to 21 Rabbit Mink 22 to 28, 31 Mink 29 Rabbit 30 Branch Number Table4.14. Node voltages of 32 node system with load growth in 3 years Node V (pu) before conductor V (pu) after conductor No. selection selection

26

27 71 Table4.15. Power loss before and after conductor selection with load growth in 3 years Branch No. Before conductor After conductor selection selection Ploss (kw) Qloss (kvar) Ploss (kw) Qloss (kvar)

28 Total Fig.4.9 Voltage profile for 32-node radial distribution system with load growth Fig.4.10 Objective function for 32-node radial distribution system with load growth

29 73 Table4.16. Summary of results of 32-node system with load growth for 3 years Conductor Selection Description Base Case by PGSA Min. Voltage (pu) Real Power Loss (kw) Total Cost (Rs.) / /- The comparison of results of 26-node and 32-node practical distribution systems obtained by PGSA with existing method [62] is given in table 4.17 and table 4.18 respectively. The PGSA for selection of optimal conductor is found better than the existing method. Table4.17. Comparison of results of 26 node system Before conductor After conductor selection selection Descript Existin Witho With Existin ion g ut load load g Withou With growth method t load load [62] growth growth method growth [62] by PGSA Min. Voltage ,32,80 5,04, ,56,97 2,46, ,75,82 2,57, (pu) Real Power Loss (kw) Total Cost (Rs.) Savings (Rs.)

30 74 Table4.18. Comparison of results of 32 node system Before conductor After conductor selection selection Descript Existin Without With Existin ion g load load g Withou With method growth growt metho t load load h d [62] growth growth [62] by PGSA Min. Voltage ,32, ,00, (pu) Real Power Loss (kw) Total Cost (Rs.) Savings (Rs.) 4.6 CONCLUSIONS The selection of optimal set of conductors with and without load growth in the design of a distribution system is a complex problem. The optimal set of conductors selected by the proposed PGSA method minimizes the sum of cost of energy losses and depreciation cost of conductor. The algorithm also satisfies the voltage and current constraints. The proposed method has been tested on practical 26node and 32-node radial distribution systems.