Optimal Placement of Capacitors, Voltage Regulators, and Distributed Generators in Electric Power Distribution Systems

Codification

The codification of the SGA is illustrated in Fig. 2. An individual of the population (also known as solution candidate or configuration) is represented by a matrix of size nd X nt, where nd represents the load levels, and nt is equal to: the number of candidate buses (nb) where fixed CAs can be allocated, plus the number of candidate buses (nb) where switched CAs can be allocated, plus the number of candidate lines (nl) where VRs can be allocated, plus the number of candidate buses (nb) where DGs can be allocated.

Figure 2: Proposed codification of candidate solutions.

In the case of fixed CAs (see first section of Fig. 2), a value other than zero indicates the number of standard capacitor units installed at bus i. This number is the same for all load levels under consideration. The first section of Figure 2 shows the representation of one fixed CA installed at bus 50, and two fixed CAs installed at bus 100. If the reactive power of each fixed CA unit is 300 kVAr, the codification represents injections of 300 kVAr and 600 kVAr (Q^fx ₅₀ =300 kVAr, Q ^fx ₈₀ = 600 kVAr ), at buses 50 and 100, respectively. In the case of switched CAs (see second section of Fig. 2) a value other than 0 states that the number of standard capacitor units of switched capacitors can be different in each load level. According to Fig. 2 one switched capacitor is installed at bus 80. In this case, three load levels are considered (heavy, average, and light). If the reactive power of each switched CA unit is 300 kVAr, the proposed codification represents an injection of 900 kVAr for the heavy load level, 600 kVAr for the average load level, and 300 kVAr for the light load level.

The third section of Fig. 2 shows the codification of VRs. A value other than zero indicates the presence of a VR in branch ij. The value within the cell indicates the type of VR. This number is equal for all load levels. The Tap positions of the VRs are calculated according to Equation 17.

The fourth section of Fig. 2 shows the codification for DGs allocation. A value other than 0 in the vector indicates the presence of a DG unit installed at bus i. The value within the cell indicates the active power dispatch of such unit. For example, if the maximum active power of a DG is 1000 kW, and it is modeled with steps of 1 kW, the value in the cell, according to the proposed codification, can vary from 0 to 1000 steps. In the fourth section of Fig. 2 the value within the cell for heavy load level is 897, which indicates an active power injection of 897 kW. For the average and light load levels, the dispatch of such unit is 642kW and 355kW, respectively. A default power factor is used to calculate the reactive power injection of the DGs. The power supplied by them is a continuous variable. However, in the proposed codification, it is considered as discrete. This is not a drawback, since the proposed codification allows considering a finegrain discretization of this variable. For example, if the maximum active power of a given DG unit is 1000 kW with steps of 0,01 kW, the value in the corresponding cell varies from 0 to 100,000. If the value of the cell is 99,445 the active power dispatch is 994,45 kW.

Initial population

The GA requires an initial population that will be modified and improved over the next iterations. In the proposed methodology, the initial population is generated in a randomly controlled way. The number of individuals of the population is given by k _ps . Since every individual of the population is represented by a matrix of four components (fixed and switched CAs, VRs, and DGs), each component is treated differently (see Fig. 2).

For fixed capacitors, a random number between 0 and n_i ^fx,max is selected; for example, if this number is 3, then 3 buses are randomly selected to allocate fixed CAs. Then, a new random number between 0 and n_i ^fx,max is selected for every bus in which the fixed capacitors are allocated in order to assign their respective sizes. The same procedure is executed for the case of switched capacitors. For voltage regulators, a random number between 0 and n ^vr,max is selected; for example, if this number is 2, then 2 branches are randomly selected to allocate VRs. After that, a new random number is selected for each voltage regulator to account for its type. Finally, regarding distributed generation, a random number between 0 and n ^dg,max is selected, and these DG units are randomly allocated in the network. After that, for each bus with DGs, a new number between 0 and n_i ^dg,max is also selected to account for their respective capacity.

Objective function

In the proposed SGA, the objective function f _o consists of a fitness (f) and a penalty (P) function, as indicated by Equation (33). The fitness function f corresponds to the annualized investment cost of the new elements installed in the network (CAs, VRs, and DGs) plus the cost of power losses (operational cost), and it is given by Equation (1). The second component of the objective function is a penalization applied to non-feasible solution candidates.

In this case, P is a penalty used for those individuals or solution candidates that violate constraints (11)-(16) and (20)-(28). If a given solution candidate does not violate these constraints, then the penalty is 0. P is given by Equation (34), where µ is the penalization factor, x _i is the decision variable, and x_i and represent its minimum and maximum limits.

Once the SGA generates a candidate solution (see Fig. 2) for the allocation of CAs, VRs, and/or DGs, the MINLP problem given by Equations (1)-(32) is turned into a nonlinear programming (NLP) problem. That is because the integer variables of the model are coded within the candidate solution of the SGA. The equivalent NLP problem is solved by means of a power flow. In this work, the backward/forward sweep power flow algorithm presented by Cheng and Shirmohammadi ^[41] was implemented. The power flow is computed to determine the stationary state of the network and determine voltages, currents, and power losses. Once the power flow has been calculated, constraints (11)-(16) and (20)-(28) are checked for feasibility. If any of these constraints are not held, the proposed candidate solution is considered unfeasible, and it is thus penalized as indicated by Equation (34).

Selection

In this step, a reduced number (k _sr) of candidate solutions (or configurations) of the current population, are randomly chosen to compete with each other and choose one configuration known as parent. The configuration with the best objective function of the reduced list of configurations is chosen. This procedure is executed twice to select two parents to be used in the recombination mechanism.

Recombination

In this step, the two parents chosen in the selection process are considered. Recombination consists of exchanging information between two arrays (parents) to create two new arrays named offspring. In this case, the recombination is performed by type of element, since fixed or switched CAs cannot combine with VRs or DGs and vice-versa. Once the recombination process is complete, two new individuals are created. The next step consists on evaluating the objective function of each individual and selecting the best one to continue with the next steps of the SGA (mutation, local search, quality improvement, and acceptation criteria).

Mutation

The mutation process is the main mechanism that introduces diversity in the algorithm. In this paper, the mutation mechanism is used to add or remove a single unit of CAs, VRs, or DGs. This procedure considers account a mutation rate k _mr corresponding to a pre-specified percentage of the total number of buses (in the case of allocating CAs or DGs) or branches (in the case of VRs) of the network.

Local search

In order to improve the quality of the current solution, a local search is performed to verify whether placing CAs or DGs at neighboring buses, or VRs in neighboring branches, improves the Fitness Function without introducing penalties. If this happens, the CA, VR, or DG is located at the new bus or branch; otherwise, it remains in its original location.

Quality improvement phase

Some elements (CAs, VRs, and/or DGs) of the proposed offspring may increase the total investment cost. Therefore, such elements must be removed from the solution. The unnecessary elements are those that can be discarded without modifying the penalty function (that is, without violating constraints) and represent a cost reduction. To eliminate such elements, they are sorted in descending order of their investment cost, and they are removed one by one. Every time an element is removed, a power flow is calculated to verify the fitness and penalty functions. At the end of this process, only the elements that do not modify the penalty function and worsen the fitness function are part of the solution proposal.

Acceptation criteria

In the proposed SGA, a new offspring is created in each iteration, so it is incorporated into the current population if its objective function z is better than the individual with the worst objective function in the current population. Furthermore, the new offspring must be different from all individuals of the current population; otherwise, the individual is discarded.

Stopping criteria

There are two stopping criteria for the SGA: the algorithm stops either when a maximum of iteration is reached, or when there has been a predetermined number of iterations (k _of) without improvement of the objective function.

Description of tests

Thirteen tests were performed on this system. The first twelve tests were performed considering three load levels that were obtained with load factors of S ₀ =1,0 (heavy), S ₁ =0,8 (average), and S ₂ =0,5 (light), with a duration of T ₀ =1.000 h, T ₁ =6.760 h and T ₂ =1.000 h. The energy cost for all load levels was: c _0,1,2 =0,06 US$/kWh. The minimum voltage magnitude was considered to be 0,95 p.u. The last test was performed considering three load levels that were obtained by using load factors S ₀ =1,25, S ₁ =1,00, and S ₂ =0,625, with a duration of T ₀ =1.000 h, T ₁ =6.760 h, and T ₂ =1.000 h. The energy cost was c ₀ =2,95 US$/kWh, c ₁ =1,78 US$/kWh, and c ₂ =0,7 US$/kWh; and the value of V ^min =0,9 p.u and V ^max =1,05 p.u. These values are the same as those considered in ^[36]. The control parameters of the SGA were as follows: the initial population size (k _ps ) was 50, the rate by tournament (k _sr ) was 3, the mutation rate (k _mr ) is 3%, the maximum number of iterations (k _Max ) was 10.000, and the maximum number of iterations without improvement of the objective function (incumbent) (k _of ) was 1.000.

The thirteen tests performed in this system were labeled according to the elements to be optimally allocated (see first column of Table II). Test 1 corresponds to the initial state (IS) of the distribution network. Tests 2 and 3 (labeled as CA and VRA) correspond to the optimal allocation of capacitors and voltage regulators, respectively. Test 4 corresponds to the optimal allocation of capacitors and voltage regulators (CA-VRA). Test 5 was designed to compare the data with ^[1]. To this effect, a DG unit of 1.000 kW and a power factor of 0,95 was introduced in the initial system. Tests 6 to 8 were designed for the optimal allocation of capacitors (CA), voltage regulators (VRA), and both (CA-VRA), considering the previous DG unit introduced in test 5. In test 9, the optimal allocation of capacitors, voltage regulators and distributed generators (CA-VRA-DGA) was performed jointly. In Test 10, the optimal allocation of a single DG unit was performed (n ^dg,max =1), with a maximum active power of 1.000 kW and a power factor of 0,95 lagging. The total installation cost (C_i,g ^dg) was US$ 50.000. The active power was discretized in steps of 0,01 kW, each with an operation cost (C_i,g,d ^dg) of 0,5 US$ for all load levels. The solution of Test 10 was used as the initial state (IS-DGA) for Tests 11 to 13, which consider the optimal allocation of CA, VRA and CA-VRA, respectively.

Table II: Results summary of the IEEE 69-bus EDN

Result analysis

Table II shows a summary of the results. For comparative purposes, the initial state (IS) of the network is indicated in the first row of the table. In all tests, the minimum voltage magnitude of the network for all load levels was greater than the permitted minimum value of 0,95 p.u, and power losses were lower than those in the IS. In Test 2 (CA), three switched capacitors were allocated, with a total investment cost of US$ 65.294 (loss cost plus investment cost). In Test 3 (VRA), one voltage regulator of 200 A was allocated in branch 45-46, with a total investment cost of US$ 76.646. In Test 4 (CA-VRA), one fixed capacitor, one switched capacitor, and one voltage regulator of 200 A were allocated, with a total investment cost of US$ 60.627. In this case, the VR was allocated in branch 46-47. These first three tests showed better results than those reported in ^[1].

In Test 6 (CA), one fixed capacitor and two switched capacitors were allocated with a total investment cost of US$ 52.773. In this case, power losses dropped sharply when compared to the initial case (Test 5 IS-DG1). In Test 7 (VRA), one voltage regulator of 200 A was allocated in branch 45-46, with a total investment cost of US$ 71.452. In Test 8 (CA-VRA), the SGA did not allocate any VRs, and the algorithm found the same solution of Test 6. In Test 9 (CA-VRA-DGA), the solution did not include any DGs; the algorithm found the same solution of Test 4 due to the high investment cost of DGs. In Test 10 (IS-DGA), a single DG was allocated to bus 50. The active power injected for each load level is 1.000 kW for level 1 (heavy), 1.000 kW for level 2 (average), and 656,23 kW for level 3 (light). Note that the active power losses of this system drastically decrease due to the presence of the DG unit. In Test 11 (CA), one switched capacitor was allocated with a total investment cost of US$ 21.255. In Test 12 (VRA), one voltage regulator of 200 A was allocated in the branch 49-50, with a total investment cost of US$ 55.838. In Test 13 (CA-VRA), the SGA did not allocate VRs, and the algorithm found the same results of Test 11.

Fig. 3 depicts the voltage profile of the IEEE 69-bus EDN for the tests performed. Only 10 out of the 13 tests are illustrated since results of Tests 8, 9, and 13 are the same as those in Tests 6, 4, and 11, respectively (see Table II). It can be observed in Fig. 3 that differences in voltages are relatively small in nodes near the substation for all tests. However, for nodes further away from the substation, voltages are much more different for each test. It can be noted that, for the initial state (IS) and Test 5 (which is shown here for comparative purposes with ^[1]), the voltages in nodes 62 to 66 are below 0,92 p.u. Nonetheless, the sole inclusion of DG (Test 10) raised the voltage in these nodes over 0,94. As expected, the best voltage profile in these nodes was obtained when VRs were allocated in the network (Test 12).

Figure 3: Voltage profile of the IEEE 69-bus EDN for maximum load level.

Figure 4 depicts power losses in different branches for Tests 1 to 4. For the sake of simplicity, only branches with power losses higher than 2 kW are illustrated. Also, only 4 tests are shown since power losses for other tests are similar to the ones already illustrated (see Table II). It can be seen that the optimal allocation of CAs and VRs reduces power losses, even though these elements do not provide active power to the EDN.

Figure 4: Power losses of the IEEE 69-bus EDN for maximum load level.

Result comparison

The results obtained with the SGA were compared with a hybrid method proposed in ^[2], a particle swarm optimization (PSO) method proposed in ^[36], and a Tabu Search metaheuristic proposed in ^[43]. This comparative study is shown in Table III. The bases of comparison are: 1) total investment cost, 2) loss cost, 3) investment cost of CAs, 4) optimal allocation of elements, 5) reactive power, 6) minimum voltage for each load level, and 7) capacitor type (fixed and/or switched). The results are sorted in descending order of their total investment cost. Note that the total investment cost found by the proposed SGA is lower than those reported in ^[43] and ^[36], and equal to the solution found in ^[2]. No further comparison is presented, since the aforementioned studies do not consider the joint allocation of CAs, VRs, and DGs, as proposed in this paper.

Table III: Result summary of the IEEE 69-bus EDN (comparative study)

Table IV shows a comparative analysis of the investment proposals (cost and time) obtained in ^[1], and the ones found by the proposed SGA. In most of the tests, the proposed SGA was able to find better solutions than those reported in ^[1], and the computation time decreased drastically. For the test labeled as CA-VRADG1, the total investment cost of the SGA was slightly higher due to an increment in power losses resulting from the power flow. However, the solution found in these cases is the same as the one reported in ^[1] (the type of elements and location).

Table IV : Results and comparisons of investment proposals.

Fig. 5 illustrates the convergence of the SGA for different runs performed in Test 2. Note that the reproducibility of results is evident since the algorithm reached the same final value for all runs. Due to the diversity of solutions in the initial population, some runs took more iterations to converge. However, all of them reached the total cost (investment plus losses) reported by Test 2 in Table II.

Figure 5: Convergence of the SGA for different runs in Test 2 in the IEEE-69-bus EDN.

Fig. 6 depicts the allocation of DGs, VRs, and CAs resulting from Tests 2, 4, 6, and 11 for illustrative purposes. Note that not all of the devices are considered to be simultaneously allocated. Further details regarding the results of these tests can be consulted in Table II.

Figure 6: Optimal allocation of elements for different tests