A new intelligent method for optimal allocation of D-STATCOM with uncertainty

Abstract

According to the standards, optimal allocation and sizing of D-STATCOM should be implemented accurately to make the maximum efficiency. Thus, this paper suggests a new approach based on social spider optimization (SSO) algorithm to deal with the optimal allocation and sizing of D-STATCOM in the distribution systems. The problem is formulated in a multi-objective framework based on scenario production and SSO algorithm to capture the uncertainties of active and reactive loads suitably. The proposed scenario based method generates a number of possible scenarios with different probabilities to model the uncertainty effects. The feasibility and satisfying performance of the proposed method are examined using an IEEE standard distribution test system.

Keywords

D-STATCOM allocation SSO algorithm scenario production multi-objective optimization

1 Introduction

Distribution power system is the last link in the electrical grids for delivering energy to the consumers. As a result, any failure in this part has a direct effect on the consumers which can reduce the reliability of the system severely [1, 2]. In this situation, any planning strategy that can reinforce the distribution system should be regarded carefully. Some of the most significant techniques in the voltage level of distribution systems are network automation, adding protective devices, improving the accuracy of the available protection methods, reclosing and switching, fast fault prediction techniques, fast team to accelerate the repair process and using flexible AC transmission system (FACTs) devices [3 –5]. Among the FACTs devices, distribution static compensator (D-STATCOM) is the most famous devices that can be incorporated in the distribution system to inject active/reactive power locally to the system [6]. In comparison with the other well-known compensators such as capacitors and reactors, DSTATCOM device does not have switching issues such as resonance or transient harmonic [7]. In addition, D-STATCOM improves the power quality of the grid by improving the flicker suppression, voltage regulation and voltage balancing [8]. In fact, D-STATCOM can clean up the voltage from any unbalance or harmonic distortion [9]. The literature has introduced other benefits for the use of D-STATCOM in comparison with other compensators including low harmonic production, low power losses, high regulatory ability and small size [10]. These characteristics root from the especial feature of D-STATCOM in automatic local compensation. Nevertheless, best use of D-STATCOM is achieved by the optimal allocation and sizing of this strategic device in the power system. It is shown in the literature that optimal allocation of D-STATCOM can affect the amount of power losses in the distribution feeders [11 –13]. By reducing the power losses, the voltage level of the buses can be amended and thus the power quality of the electric services is enhanced. In other words, reducing the power losses and improving the voltage profile have direct effects on the operational costs directly. However, to the best of knowledge of the authors, there are yet very little works done on the D-STATCOM allocation and sizing problem. In [14], the effect of using D-STATCOM on these two targets is assessed without considering the uncertainty issues. Therefore, output results are prone to bias. In [15], particle swarm optimization (PSO) is used for optimal allocation of D-STATCOM as well as DGs in the radial networks. Here, neglecting the uncertainties of the problem has reduced from the efficiency of the work. In [16], optimal location and sizing of D-STATCOM is employed for reducing the power loss, improving the current and voltage profile in distribution networks using the Immune Algorithm. In [17], Distribution Static Synchronous Series Compensator (DSSSC) as a FACTS device is used for optimizing the power losses and voltage deviation in the radial distribution grids. The simulation results on the 12-bus and 69-bus test systems show the positive effect of this device on the above targets. In [18], differential evolution algorithm is used for optimizing the location and size of D-STATCOM in the radial systems. Also, the reconfiguration technique is solved in this paper to improve the network condition more than before. While each of these works have provided good results, but neglecting the uncertainty of the new smart grids is a big deficiency in front of the reliable outputs.

According to the above discussions, this paper investigates the optimal allocation and sizing of D-STATCOM devices in the distribution systems considering the uncertainties of active and reactive loads. In this regard, a stochastic framework based on scenario generation is introduced to model the uncertainty effects [19]. Regarding the optimization process, social spider optimization (SSO) algorithm is suggested to solve the problem optimally [20]. The SSO was proposed in 2012 for modeling the supportive behavior of social spiders in the colony. In comparison with other methods such as genetic algorithm (GA), PSO, KH [21], CSA [22, 23], HS [24], SFLA [25], FA [26, 27, 26, 27], ANN [28] and HBMO [29, 30] and TLA [31] and BA [32, 33] the high search ability, local search and simple concept can be named as the main features of SSO. In short, the main contributions of this paper can be summarized as follows: 1) Utilization of the SSO algorithm for the first time to solve the optimal allocation and sizing of D-STATCOM, 2) Introduction of a new stochastic framework based on scenario generation to handle the uncertainties of the active and reactive loads in the D-STATCOM problem and 3) constructing a multi-objective framework for assessing the effect of D-STATCOM on the feeder resistive losses and voltage profile of the buses simultaneously. The feasibility and efficient performance of the proposed stochastic framework is shown on the IEEE 69-bus test system. According to the simulation results, installation of D-STATCOM in the appropriate location in the system can improve the voltage of buses when reducing the network losses. Nevertheless, according to the high cost of D-STATCOM, it was seen that the benefits of adding extra D-STATCOM is not economical. Therefore, an initial analysis is necessary to reach a balance between the cost and benefits for considering D-STATCOM.

2 D-STATCOM model

As mentioned before, static compensator (D-STATCOM) belongs to the category of FACTS devices and is connected to the grid in the shunt mode. STATCOM plays the role of a power regulator in the transmission systems while the distribution version is called distribution STATCOM (D-STATCOM). D-STATCOM can produce/consume reactive power by the use of internal capacitors/reactors automatically. But, it can produce active power just if equipped with a power supply. At the point of common coupling of D-STATCOM with the grid, the voltage level of this point will determines the direction of power flow either upward/downward. The voltage level is determined by the use of DC capacitors. If the voltage level of the D-STATCOM source is higher than the coupling point, then power flow is toward the grid (in the capacitor mode). On the opposite point, if the voltage magnitude of the D-STATCOM is lower than the common coupling point, then the power flow would be toward the D-STATCOM (reactor mode). In order to consider D-STATCOM in the system, its complete model is required. In the steady state analysis, the literature suggests the model of STATCOM as an acceptable model for D-STATCOM [14]. This model is shown in Fig. 1. In this model, the DSTATCOM is considered as a PV bus (bus j) which is connected to the bus i through a series impedance. The series impedance can simulate the power losses of the transformer and the inverter as R _T + jX _T [14]. In this figure, the active and reactive loads are shown by P _Li + jQ _Li. In the model, the bus j as the PV bus has a fixed voltage which is controlled by the D-STATCOM automatically using the internal shunt capacitor or reactors.

3 Problem formulation

As mentioned before, the proposed problem is an optimization problem with two objective functions. It is clear that there are also some operational and security constraints that should be considered.

3.1 Objective functions

- Improving voltage profile: This target is achieved by reducing the maximum voltage deviation of the buses from 1 pu as follows: $f_{1} (X) = d_{volt} (X) = max {| 1 - V_{min} |, | 1 - V_{max} |}$ (1) where V _min and V _max are the minimum and maximum voltage values buses, respectively.

- Reducing the active power losses: The most attractive target in the view of the utilities is the active power losses function which is calculated by summation of the resistive losses of feeders as follows: $f_{2} (X) = P_{loss} (X) = \sum_{i = 1}^{N_{br}} R_{i} \times {| I_{i} |}^{2}$ (2)

In the above equation, X is the control vector, R _i is the resistance of ith branch, I _i is the current of ith branch and N _br is the number of branches.

3.2 Operational constraints

- Bus voltage limits: the voltage level of the buses must be preserved in the predetermined ranges as follows: $V_{min} \leq V_{i} \leq V_{max}$ (3)

- Maximum power flow: the maximum power flow capacity of the feeders should be preserved: $| P_{ij}^{Line} | < P_{ij, max}^{Line}$ (4) where $P_{ij}^{Line}$ is the power flow in the feeder ij and $P_{ij, max}^{Line}$ is the maximum power flow in the feeder ij.

- Distribution power flow equations: The active and reactive power flow equations can be considered as equality constraints: $\begin{matrix} P_{i} = \sum_{i = 1}^{N_{bus}} V_{i} V_{j} Y_{ij} cos (θ_{ij} - δ_{i} + δ_{j}) \\ Q_{i} = \sum_{i = 1}^{N_{bus}} V_{i} V_{j} Y_{ij} sin (θ_{ij} - δ_{i} + δ_{j}) \end{matrix}$ (5) where V _i is the voltage magnitude of ith bus, θ _ij is the admittance angle between the buses i and j, δ _i is the voltage angle of ith bus, Y _ij is the admittance magnitude between the buses i and j, N _bus is the number of buses. P _i and Q _i are the net active and reactive power injection to the ith bus.

4 Stochastic framework

4.1 Scenario-based method

This paper uses a stochastic method based on scenario generation for modeling the uncertainties of active and reactive loads. In this method, each uncertain parameter is replaced by a probability density function (PDF). In the scenario method, each PDF is discretecised. Figure 2 shows a normal PDF with seven probability levels.

After determining the PDFs, the possible scenarios of the problem can be produced. This process is done using the Roulette wheel mechanism (RWM). Figure 3 shows a RWM with seven probability levels.

Suppose that our problem has m uncertain parameters. For each uncertain variable, a random value rand in the range [0,1] is produced. According to its value, rand will trap in one of the slices of the RWM. This slice refers to one of the load levels in Fig. 2. This process is repeated m time to generate a complete scenario. Therefore, each scenario has a structure as below: $R_{s} = [r_{1}^{s}, r_{2}^{s}, r_{3}^{s} \dots, r_{m}^{s}]_{1 \times m}$ (6) where $r_{j}^{s}$ is the value of the jth uncertain variable in sth scenario. By considering the probability of each random value $r_{j}^{s}$ as $α_{j}^{s}$ , the probability of a scenario is calculated as below: ${Pro}^{s} = α_{1}^{s} \times α_{2}^{s} \times α_{3}^{s} \times \dots \times α_{m}^{s}$ (7)

Initially, a high number of scenarios (here 2000) are produced. But, then the scenario reduction process is done based on 1) omitting similar scenarios and 2) omitting scenarios with low probability. Finally, N _s most probable and dissimilar scenarios are stored. The problem is solved for each scenario and the objective function is calculated f _s.

After using the scenario reduction technique, the remained scenarios are applied to the system and the objective functions are evaluated. Each scenario indicates a possible deterministic framework for the problem investigated. In fact, the proposed stochastic framework will convert the stochastic problem to a specific number of deterministic problems with different probabilities. After solving the problem in the framework which is determined by a scenario, an optimal solution is found for the problem. However, it is expected that finally we will have a single optimal solution for the stochastic problem than a set of solutions. Here the necessity of aggregation process is clear. By the use of the aggregation process, not only the interpretation of each single scenario become possible, also the united structure of the original stochastic problem is preserved. Therefore, the final solutions which are found by each scenario are aggregated according to their probability values as below: $f = \sum_{s = 1}^{N_{Sr}} π_{s}^{norm} \times f_{s}$ (8) where f _s is the optimal value of the objective function in sth scenario. Also, $π_{s}^{norm}$ is calculated follows: $π_{s}^{norm} = \frac{{Pro}^{s}}{\sum_{s = 1}^{N_{Sr}} {Pro}^{s}}$ (9)

4.2 SSO Algorithm

SSO algorithm was first introduced in 2012 by imitating the supportive behavior of social spiders in the colony [16]. The SSO algorithm is constructed based on the male and female spiders in the mating process. In this algorithm, the female spiders attract other spiders regardless of their gender. A more attractive spider is one with higher weight and less distance from the relevant spider.

On the other hand, the behavior of male spiders among themselves depends on their size. The bigger spiders are assumed as dominant spiders and vice versa. Dominant males mate have more chance to mate with the female spiders. The non-dominate spiders are used to avoid the generation of any abnormal brood. Suppose that we have N _F female and N _M male spiders. Generally, the number of female spiders constructs about 65% –90% of the whole population. Then, the fitness function is calculated for all the population. According to the fitness function value, a weighting factor (w _i) is designated to the spiders as follows: $w_{i} = \frac{f_{w} - f_{i}}{f_{w} - f_{b}}$ (10)

In the above equation, f _b is the best spider and f _w is the worst spider (their fitness function value). Then, by mimicking the females’ behavior, their position is updated using the below equation: $\begin{matrix} X_{i, F}^{Iter + 1} = X_{i, F}^{Iter} \pm β_{1} w_{c} e^{- d_{ic}^{2}} (X_{c} - X_{i, F}^{Iter}) \\ \pm β_{2} w_{b} e^{- d_{ib}^{2}} (X_{b} - X_{i, F}^{Iter}) + (β_{3} - 0.5) \end{matrix}$ (11) where β ₁, …, β ₈ are random numbers, X _c is the nearest spider, X _b is the best solution, d _i c is Euclidian distance of ith individual to her closest spider and d _i b is Euclidian distance of ith individual from the best solution.

Similar process is done for male spiders. Here, first the spider population is sorted by the weighting factor. The middle spider is selected as the median individual. The spiders with higher weights are regarded as dominate males. The position of the ith dominant male X _i, DM is improved as follows:

$\begin{matrix} X_{i, DM}^{Iter + 1} & = X_{i, DM}^{Iter} + β_{5} w_{F} e^{- d_{ic}^{2}} (X_{c}^{F} - X_{i, DM}^{Iter}) \\ + (β_{6} - 0.5) \end{matrix}$ (12)

Similarly, the non-dominant spiders are attracted to the weighted mean of the male population (M _w) as below: $X_{i, NM}^{Iter + 1} = X_{i, NM}^{Iter} + β_{7} (M_{w} - X_{i, NM}^{Iter})$ (13) the mating process between the dominant males and female spiders is implemented using the RWM with the mating range (r _m). The mating range is calculated using variables upper (u _j) and lower (l _j) bounds: $r_{m} = \frac{1}{2 n_{v}} \sum_{j = 1}^{n_{v}} (u_{j} - l_{j})$ (14) where n _v is the number of control variables. The new broods will replace the worst solution in the spider population.

4.3 Application of SSO algorithm on the problem

The below steps are required to apply the proposed SSO algorithm on the problem:

Step 1: Input data: the input data includes the network data, the algorithm data and the D-STATCOM data, the termination criterion and the stochastic characteristics of the random parameters.

Step 2: Generate the initial population. Each individual in the population shows a promising solution for the problem and includes the optimal size and location of D-STATCOMs in the network.

Step 3: Calculate the fitness function. For each individual in the SSO population, the expected value of the objective function should be calculated using the proposed stochastic framework.

Step 4: Perform the mutualism interactions as described in Section 4.2 to improve the position of the population.

Step 5: Perform the commensalism interactions as described in Section 4.2 to improve the position of the population.

Step 6: Perform the parasitism interactions as described in Section 4.2 to improve the position of the population.

Step 7: Update the population and calculate the objective function values.

Step 8: Check the termination criterion. If it is satisfied finish the algorithm, other wise return to step 4 and repeat the steps.

5 Simulation results

The proposed method is examined on the IEEE 69-bus test system [19]. The single line diagram of the test system is shown in Fig. 4. The nominal voltage of the test system is 12.66 kV supplying 3,802.19 kW active and 2,694.59 kVar reactive loads. Initially, 2000 possible scenarios are produced that after scenario reduction are reduced to 20 scenarios. We have considered 100 iterations as the termination criterion for SSO algorithm with 10 male and 15 female spiders.

First, the analysis is done neglecting the uncertainty. Also, since the price of D-STATCOM is high, we have allowed just maximum two devices be installed in the system. Therefore, we make two different cases of 1) allocation of one D-STATCOM and 2) allocation of two D-STATCOMs. Table 1 shows the results of simulations. Here we have done both the single-objective and multi-objective optimizations. According to Table 1, the initial power loss is 225 kW that is reduced to 150 kW after optimization of the power losses. The amount of power loss reduction is less for other two cases. Also, the best voltage profile has been archived by single-objective optimization of the voltage deviation. Nevertheless, the power losses value is increased more than the initial value. Finally, the best comprise between the two targets is found in the multi-objective optimization. The optimal location and size of the D-STATCOM is shown in Table 2.

Now, the uncertainty effects are considered in the evaluations. Table 3 shows the comparative values of the objective functions. According to this table, considering uncertainty has resulted in incremental values in the objective functions.

Finally, in order to better understand the probabilistic behavior of the objective functions, Fig. 5 is shown at below. For better comparison, the cumulative functions are provided here. As it can be seen rom these PDFs, each target can take different values by different probability values. These figures show the stochastic behavior of the targets clearly.

In the second part, we suppose that two D-STATCOMs can be considered in the system. The simulation results are repeated and the results are shown in Tables 4–5. According to these results, the new additional D-STATCOM device has the most effect on the results of single-objective optimization. Nevertheless, the positive effect of adding the new device to the system in the multi-objective optimization is neglecting. In other words, it is not beneficial to install new D-STATCOM device in the system for this little improvement in the objectives. Finally,the PDF.

6 Conclusion

This paper introduced a new stochastic framework based on scenario production and SSO algorithm to deal with the optimal allocation and sizing of DSTATOM devices in the system. The simulation results showed the positive effect of using DSTATOM if optimally located in the system. From the view of the different objectives, optimizing the voltage deviation target will increase the total cost of the system. As a result, the necessity of using a sufficient multi-objective framework is evident. Also, it was seen that considering uncertainty will affect the optimal values of the objective functions without which the reliability of the analysis is reduced. The simulation results show the high ability of the proposed method for solving the problem.

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