Integration of Stand-Alone Controlled Active Power Filters in Harmonic Power Flow of Radial Distribution Networks

Abstract

Utilization of active power filters (APFs) is the most efficient method to reduce harmonic pollution in distribution networks. Previous approaches utilized APFs in integrated control schemes based on broad data-gathering systems. Since a broad data-gathering system is not available in most practical distribution networks, previously proposed approaches may not readily be implemented. This paper presents the utilization of stand-alone controlled APFs (SACAPFs) in radial distribution networks. Utilizing APFs with a stand-alone control system decreases implementation costs and complexity by making them autonomous and independent of integrated control systems, which are complicated and expensive in practical applications. In this paper, a single SACAPF is modeled as a dependent current source where its injection current is equal in amplitude but opposite in phase compared to the harmonic content of the current passing through the point of common coupling (reference current). Due to the presence of both linear and nonlinear loads in the distribution network, the reference current changes after injection by SACAPF, so it is necessary to modify the injection current until reaching a constant value in the reference current. This is considered via an iterative procedure in the modeling scheme. Operation of multiple SACAPFs is handled using a backward procedure based on a priority list. Simulation results on an IEEE 18-bus test system show the proper operation of the stand-alone control systems for both single and multiple SACAPF implementation. Furthermore, optimal allocation of the proposed SACAPFs is performed in an IEEE 33-bus test network and a 9-bus test network, and the results are discussed and compared with the allocation of integrated control system APFs.

1. Introduction

Harmonic pollution is one of the main concerns in power quality (PQ) problems in modern power systems. Harmonic pollution can lead to different problems, such as the maloperation of protective relays, excessive heating, and extra losses [1]. The major source of harmonic pollution is the presence of nonlinear loads, which are directly or indirectly connected to distribution networks. Due to the growing use of power electronic devices [2,3] in recent years, there has been an increasing concern about this subject. Hence, it is necessary to design and implement proper methods to reduce harmonic pollution.

Filtering harmonic currents by passive or active power filters (APFs) is one of the best ways to reduce harmonic pollution in distribution networks [4]. Passive filters deal with harmonic currents by directing them into a bypass path, while active filters eliminate harmonic currents by injecting a current that is equal in amplitude but in the opposite phase with respect to the harmonic content of the nonlinear loads [5,6,7,8,9].

In distribution networks, passive filters cannot compensate for harmonic currents effectively because of some problems, such as time-varying harmonics and the widespread utilization of small power electronic-based devices [2,5]. In contrast to passive filters, APFs can compensate for harmonic currents in distribution networks more significantly and efficiently thanks to the flexibility of their power electronic elements [10,11].

A schematic diagram of an APF, which is used to protect the source from the harmonic current of a nonlinear load, is shown in Figure 1. In this system, the control system is working to determine proper switching patterns of the power electronic switches incorporated in the filter structure in order to compensate harmonic content of the nonlinear loads’ currents adequately and make the source current (the current flowing through the upstream branch at the point of common coupling (PCC)) as pure sinusoidal as possible.

The application of APFs in distribution networks was recently studied by many power system researchers. Grady et al. [12] developed an APF to minimize the voltage harmonics of all of the network buses. Grady et al. [13] optimized the injected current of an APF regarding different objective functions, such as voltage distortion, telephone interference factor, and motor load losses. Chang et al. [14] optimized the allocation (sitting and sizing) problem of an APF. The allocation of multiple APFs was explained in the work of Chang et al. [15] for the first time. More recently, the focus on APF research in distribution networks has shifted to optimization methodology. Some papers [16,17,18] employed analytical methods. However, due to the main disadvantage of analytical methods (i.e., sensitivity to local optima), evolutionary algorithms were applied to the APF allocation problem [5,19,20,21,22,23]. A decision tree-based algorithm for the allocation of APFs was proposed in [24], and software based on a brute force algorithm, tested on a low voltage network, was developed in [25]. Zhao et al. [4] used a chance-constrained algorithm to consider uncertainty in the APF allocation problem. The application of multiobjective approaches was demonstrated by Shivaie et al. [6] and Carpinelli et al. [26]. Furthermore, Carpillei et al. [27] took the uncertainty into account for multiobjective programming procedures. A detailed review of the applied methods for the allocation of APFs was discussed in [28].

The common strategy of many published papers in the field of APF application in distribution networks is the utilization of an integrated control system for APFs. Figure 2a shows a schematic diagram of such an integrated control system. In this system, all of the network information, including network structure and the condition of linear and nonlinear loads, are available, and the current to be injected by the APFs is determined based on data processing in an integrated control system.

Since there is no available broad infrastructure in most practical distribution networks, none of the previous approaches are readily applicable. Hence, this paper proposes the utilization of standalone controlled APFs (SACAPFs) in distribution networks. In an SACAPF, the current injected by the filter is determined based on local information. In other words, the flowing current to the downstream branches at the PCC is utilized to determine the SACAPF injection current (as explained).

The operational approach diagram of the SACAPF is shown in Figure 2b where it is clear that the current injected by the SACAPF is determined only based on local information. The main advantage of stand-alone control systems over integrated control systems is that there is no need to utilize a broad data transmission infrastructure, so there is a considerable decrease in implementation costs. It is worth mentioning that due to the harmonic content of transmitted information, the required transmission infrastructure would be more expensive than that required for a fundamental current. Furthermore, the utilization of stand-alone control systems is simpler and more practical compared to integrated control systems.

The implementation of a single SACAPF was performed based on the behavior of the SACAPF in the harmonic power flow (HPF) procedure presented in Section 2.2 and was developed for multiple SACAPFs in Section 2.3. The implementation procedure presented in this paper is explained in three stages. In the first stage, HPF for networks with no SACAPF is discussed. In the second stage, the modeling of stand-alone control systems in the HPF procedure is presented for networks with only one SACAPF. In the third stage, the HPF procedure is extended to multiple SACAPFs. The results of this paper would be completely applicable in an analysis of SACAPF operation as well as the allocation of SACAPFs in electric power distribution networks.

Detailed explanations about the proposed HPF are presented in Section 2. Section 3 discusses the allocation of SACAPFs in distribution networks. Two simulation studies are presented in Section 4. In the first study, the proposed HPF procedure was tested on a modified 18-bus IEEE test network, and in the second study, the allocation of SACAPFs was performed on a 33-bus IEEE test network and a 9-bus test network. Finally, some concluding remarks are given in Section 5.

2. Modeling of APFs with Stand-Alone Control Systems (SACAPFs)

According to the SACAPF’s injected current, an SACAPF can be modeled as a dependent current source at different harmonic orders. Therefore, the operation of an SACAPF in distribution networks affects the operation of the other installed SACAPFs in that network. Hence, in this paper, a backward procedure is proposed to handle the operation of multiple SACAPFs in HPF. The operation procedure of SACAPFs in HPF is explicitly presented in the following sections.

2.1. Harmonic Power Flow for Networks with No SACAPF

HPF is a tool for calculating harmonic voltages and harmonic currents in different nodes and branches of power networks. HPF methods are basically classified into two main groups, including time-domain (TDHPF) and frequency-domain (FDHPF) methods. The TDHPF methods solve differential equations of the network using numerical techniques, while the FDHPF methods are an extension or reformulation of the conventional power flow (CPF) to include both fundamental and harmonic frequencies. FDHPF methods are used in most harmonic problems due to the high calculation burden required in TDHPF methods for determining the steady-state network currents and voltages [29]. There are several methods published in the literature which focus on FDHPF. Some of these solve HPF for all harmonics in a coupled manner, so harmonic voltages for each harmonic are dependent on the voltages of other harmonics (for example, iterative harmonic penetration, simplified harmonic load flow, and complete harmonic load flow). Other FDHPF methods neglect correlation between harmonics (actually, these methods suppose no interaction between network and nonlinear loads) and calculate harmonic voltages for each harmonic independently (for example, the harmonic penetration method) [29]. In addition to complexity and time consumption, coupled methods need exact information about nonlinear loads, including their dependency on the harmonic content of the supplied voltage. In some cases, these methods cannot be practically employed due to a large number of installed nonlinear loads throughout the network. Although decoupled methods are less accurate compared to coupled ones, the simplicity and promptitude of these methods have led them to be used in many research works [30].

In the harmonic penetration method, network voltages at the fundamental frequency are calculated using CPF considering nonlinear loads as PQ buses (constant power). The harmonic current of nonlinear loads for each harmonic order is then calculated using (1) [31,32,33,34]:

$$i_{nb}^h=c\left(h\right)·i_{nb}^1,$$

(1)

where $c\left(h\right)$ is the current injection factor and can be estimated by field test and Fourier analysis for all customers along the feeder [30,32,33,34,35,36], and $i_{nb}^1$ is the current of nonlinear loads at the fundamental frequency calculated by (2):

$$i_{nb}^1={\left(\frac{P_{nb}+i·Q_{nb}}{V_b^1}\right)}^{*}.$$

(2)

For each harmonic order, nonlinear loads are assumed as current sources, and harmonic voltages are calculated by the node voltage method using the admittance matrix. To build the admittance matrix for each harmonic, the harmonic model of linear components is needed, and if the skin effect is ignored, the admittances of linear loads, shunt capacitors, and line feeders are expressed by:

$$y_{lb}^h=\frac{P_{lb}}{{\left|V_b^1\right|}^2}-i\frac{Q_{lb}}{h·{\left|V_b^1\right|}^2},$$

(3)

$$y_{cb}^h=h·y_{cb}^1,$$

(4)

$$y_{b,b+1}^h=\frac{1}{r_{b,b+1}+i·h·x_{b,b+1}}.$$

(5)

Therefore, the admittance matrix for each harmonic is built as follows:

$$Y_{a,b}^h=\left\{\begin{array}{c}-y_{a,b}^h if a\ne b\\ y_{a-1,b}+y_{a+1,b}+y_{cb}^h+y_{lb}^h if a=b\end{array}\right..$$

(6)

Note that $y_{a-1,b}$ and $y_{a+1,b}$ are the symbols of the lines connected to bus B (the number of connected lines can be more or less than two depending on the network structure).

By means of the node voltage method, harmonic voltages for each harmonic are given as:

$$V^h={\left(Y^h\right)}^{-1}·I^h.$$

(7)

Finally, after calculating the network voltages for all harmonic orders, the total harmonic distortion ($TH{D}_b$) and individual harmonic distortion ($IH{D}_b^h$) factors for network voltages can be examined by (8) and (9), respectively:

$$TH{D}_b=\left[\frac{\sqrt{{{\displaystyle \sum }}_{h=2}^H{\left|v_b^h\right|}^2}}{\left|v_b^1\right|}\right],$$

(8)

$$IH{D}_b^h=\frac{\left|v_b^h\right|}{\left|v_b^1\right|}.$$

(9)

2.2. Harmonic Power Flow for Networks Equipped with a Single SACAPF

SACAPFs are a combination of power electronic switches and energy storage devices. The stand-alone control system of an SACAPF acts as the brain of the filter and determines the pattern of switching based on the current flowing through the downstream branch at the PCC to improve the current of the upstream branch into pure sinusoidal. It should be noted that the rating of filters is limited to the size of energy storage and switching elements incorporated in the structure of the filters. If the RMS value of the harmonic current flowing through the downstream branch at the PCC is more than the SACAPF rating, the filter injects current at the PCC at its full rating. Therefore, using this simple but novel assumption, modeling of an SACAPF in the HPF procedure can be achieved. Moreover, the current injected by the filter should be equal to the whole or part of the harmonic current flowing through the PCC, which is given as:

$$i^{\mathrm{SACAPF}}\left(t\right)=-{{\displaystyle \sum }}_{h\ne 1}\sqrt{2}·\lambda ·\left|i^{\mathrm{PCC},h}\right|·\sin \left(h\omega t+{\phi }^{\mathrm{PCC},h}\right),$$

(10)

where $\lambda$ is the compensation rate and determines the amount of current reduction as follows:

$$\lambda =\frac{1}{\max \left(1,\frac{i^{\mathrm{PCC},\mathrm{rms}}}{S^{\mathrm{SACAPF}}}\right)},$$

(11)

where $S^{\mathrm{SACAPF}}$ is the rating of the filter dependent on the size of the energy storage element incorporated in the filter structure. The energy storage element, usually a capacitor, should be capable of supplying the variation of injection current; hence, the size of the SACAPF is directly dependent on the size of this element, and the injection current of the filter should be limited below its rating.

It should be noted that the focus of this paper is on the application of SACAPFs in distribution networks. The control strategy of the filters is assumed ideal and, thus, out of the scope of the paper. Related works can be found in references [7,8,9].

As it is inferred from the equation, in the overload condition of the SACAPF, the injected current in all harmonic orders is downgraded equally, aimed at keeping the injected current in the maximum allowable value. This procedure could be readily customized by the SACAPF designer. Hence, some harmonic orders can be specified to have higher priority than others in the downgrading procedure.

It is clear that the harmonic current flowing to the downstream branch of the PCC (reference current) is a combination of linear and nonlinear load harmonic current. Due to the decoupled assumption on different harmonic orders, the operation of an SACAPF in the harmonic penetration (HP) method does not affect the currents and voltage condition of the system at the fundamental frequency. Hence, from (1) and (2), the harmonic currents of nonlinear loads remain constant, whereas the harmonic currents of linear loads change. This is due to the fact that the harmonic current of linear loads depends on the harmonic content of the supplied voltage at the connection point (injection by the SACAPF changes $I^h$ results in variation of $V^h$). In other words, linear loads are modeled as constant impedances in different harmonic orders [32]. Hence, linear load currents are affected by the injection currents anywhere in the network. Actually, the harmonic content of the reference current changes after the current injection by the SACAPF. Therefore, it is required to update the switching pattern (current injection) until reaching a stable condition in which there is no change in the harmonic content of the reference current in two consequent iterations of the SACAPF operation.

2.3. Harmonic Power Flow for Networks Equipped with Multiple SACAPFs

Herein, a backward procedure is proposed to calculate the current injected by the SACAPFs in radial distribution networks equipped with multiple SACPAFs. The proposed procedure is based on a priority list that sorts the SACAPFs by their distance from the main substation. To form the priority list, first, it is needed to sort the network layers, and then the highest priority is assigned to the SACAPF installed in the layer farthest from the main substation. As the location layer of the filter is increased, the priority of the corresponding filter is also increased. Figure 3 illustrates an example of a layered network with a priority list of SACAPFs.

Now, beginning with the priority list, the current injected by each SACAPF is calculated based on the explanations in Section 2.2, while the currents injected by other SACAPFs are assumed to be constant (in the first iteration, the currents injected by other filters are zero). This process continues until there is no change in the harmonic currents at all PCCs in two consecutive iterations. Figure 4 shows the flowchart of the proposed procedure.

3. Allocation of SACAPFs

Allocation of SACAPFs is the procedure of determining the optimal sizes and locations of SACAPFs to yield desired network harmonic conditions. Each SACAPF helps improve power quality by compensating for harmonic currents, which are the main source of harmonic pollution.

The objective function of the allocation problem is the summation of the cost of APFs as follows:

$$\min C_T={{\displaystyle \sum }}_{i=1}^{NA}C\left(S_i^{\mathrm{SACAPF}}\right),$$

(12)

where $C\left(S_i^{\mathrm{SACAPF}}\right)$ is the cost of the $i_{th}$ SACAPF and is calculated using (13).

$$C\left(S_i^{\mathrm{SACAPF}}\right)=C^{\mathrm{fix}}+C^{\mathrm{var}}·S_i^{\mathrm{SACAPF}},$$

(13)

where $C^{\mathrm{fix}}$ and $C^{\mathrm{var}}$ are factors of fixed and variable costs of SACAPFs, respectively.

Constraints of the allocation problem related to the size of the SACAPFs and the condition of the network are stated as follows:

$$i_i^{\mathrm{rms},\mathrm{SACAPF}}\le S_i^{\mathrm{SACAPF}}, \mathrm{for} i=1,2,3,\dots ,NA,$$

(14)

$$S_i^{\mathrm{SACAPF}}\in \mathrm{\Omega }, \mathrm{for} i=1,2,3,\dots ,NA,$$

(15)

$$TH{D}_b\le TH{D}^{\max }, \mathrm{for}b=1,2,3,\dots ,NB,$$

(16)

$$IH{D}_b\le IH{D}^{\max }, \mathrm{for}b=1,2,3,\dots ,NB,$$

(17)

Equations (14) and (15) are related to the SACAPFs where (14) enforces the total current of each SACAPF below the rating of the corresponding SACAPF, and (15) reveals that the rating of the SACAPFs should be selected from standard values. Equations (16) and (17) are related to the harmonic condition of the network where both THD and IHD of the network voltages should be less than the maximum allowable values.

As the ratings of SACAPFs are discrete values chosen from a set of standard values, the determination of optimal sizes (ratings) is a mixed integer problem, and due to the nonlinear behavior of power systems, the allocation problem of SACAPFs is a mixed integer nonlinear problem (MINLP). A PSO algorithm was hired as the optimization tool for the allocation of the proposed SACAPFs. The structure of the utilized particles is illustrated in Figure 5, where each particle is a vector including a number of cells (equal to the number of candidate locations); each cell represents the rating of the SACAPF that should be installed on the corresponding candidate location.

The location of the particles in each iteration of the optimization procedure is updated based on the following equations:

$$V_i^{k+1}=\omega V_i^k+c_1·r_1·\left({\mathrm{Pbest}}_i^k-X_i^k\right)+c_2·r_2·\left({\mathrm{Gbest}}^k-X_i^k\right),$$

(18)

$$X_i^{k+1}=X_i^k+V_i^{k+1},$$

(19)

where $k$ is the iteration number; ω$, c_1$, and $c_2$ are weight parameters; and $r_1$ and $r_2$ are random values ranging from 0 to 1. Particles in each iteration are evaluated based on the following fitness function:

$$F=C_t+P_1\left[{{\displaystyle \sum }}_{b=1}^{NB}\max \left(0, TH{D}_b-TH{D}^{\max }\right)\right]+P_2\left[{{\displaystyle \sum }}_{b=1}^{NB}{{\displaystyle \sum }}_{h=3, 5,\dots }\max \left(0, IH{D}_b^h-IH{D}^{\max }\right)\right].$$

(20)

The optimization procedure performs the following steps to estimate the optimum solution:

• Evaluate all particles;
• Update $\mathrm{Pbests}$ and $\mathrm{Gbest}$;
• If the convergence criteria are satisfied, report Gbest as the optimum solution, else go to 4;
• Update the location of particles (18) and (19);
• Go to 1.

In this paper, the number of iterations is large enough to guarantee convergence. The number of iterations and basic parameters of the PSO algorithm are designed using sensitivity analysis and are reported for each study in the corresponding section.

4. Simulation Results and Comparison

The simulation results were tested in two studies. In the first study, the proposed algorithm for modeling SACAPFs in the HPF procedure was tested, and the optimal allocation of SACAPFs was determined in the second study.

4.1. Modeling of SACAPFs in HPF Procedure

The algorithm proposed for modeling stand-alone control systems in HPF was tested on a modified 18-bus IEEE test network simulated in MATLAB software. Detailed data of the network branches and linear loads of the network are presented in [2]. Two sets of simulations were performed. In the first set, the operation of a single SACAPF was evaluated, and in the second set, the results were illustrated for multiple SACAPFs.

All nonlinear loads are six-pulse converters, and the harmonic spectra are presented in Figure 6.

Table 1. Data of Nonlinear Load Power.

Name of Nonlinear Load	Power
	P (MW)	Q (MVAr)
6-pulse 1	3	2.26
6-pulse 2	1	0.45
6-pulse 3	0.45	0.28
6-pulse 4	0.31	0.23

shows the data of the nonlinear load power.

4.1.1. Single SACAPF

In order to verify the operation of SACAPFs in distribution networks, two different evaluations were performed for single and multiple SACAPFs implemented in the network. This section focuses on the results of a single SACAPF implementation in the network.

The single-line diagram of the studied network is illustrated in Figure 7. The HPF results of the network are presented in

Table 2. Network voltages when a 3 MW 2.25 MVAr six-pulse converter is implemented.

Bus	V1 (pu)	THD (%)	Bus	V1 (pu)	THD (%)
1	0.97	5.19	9	0.96	6.27
2	0.96	6.28	20	0.95	4.91
3	0.95	7.85	21	0.94	4.78
4	0.95	8.7	22	0.93	4.78
5	0.93	11.23	23	0.92	4.66
6	0.93	11.2	24	0.91	4.66
7	0.92	11.17	25	0.91	4.64
8	0.92	11.15	26	0.91	4.64

where it is evident that the maximum THD value of the network voltages is 11.23% at bus 5.

An ideal SACAPF (without any limitation on SACAPF rating) was installed at bus 5. It is worth mentioning that the installation of an optimum rated SACAPF in an optimum location of the network affects its operation effectiveness in terms of harmonic compensation and cost-effective perspective. However, this paper is focused on proposing a locally controlled APF, and the presented numerical results in the studied networks aimed at a performance evaluation of the proposed SACAPF control method. The subject of the optimum rating and location of an SACAPF is under study and will be published in the near future.

Figure 8 shows the RMS value of the SACAPF current, the harmonic content of the nonlinear load current, the harmonic content of the upstream branch current (branch 4–5), and the harmonic content of the downstream branch current (branch 5–6) in 5 iterations of sampling and injection. Figure 8 shows that the SACAPF has absolutely compensated harmonic content of the nonlinear load, and no harmonic current flows through the system in a steady state (the harmonic content of the upstream and downstream branches is zero). In this condition, no harmonic voltage exists in the network, and the supplied current is purely sinusoidal.

Different constraints were applied to the rating of the SACAPF as follows:

Scenario 1: There is no SACAPF installed in the network.

Scenario 2: The installed SACAPF rating is limited to 25% of the harmonic content of the nonlinear load.

Scenario 3: The installed SACAPF rating is limited to 50% of the harmonic content of the nonlinear load.

Scenario 4: The installed SACAPF rating is equal to the harmonic content of the nonlinear load.

These scenarios are explained in

Table 3. Ratio of the rating of the SACAPF to the harmonic content of the nonlinear load.

	Scenario 0	Scenario 1	Scenario 2	Scenario 3
$S^{\mathrm{SACAPF}}/i_5^{\mathrm{hrms}}$	-	25%	50%	100%

where $S^{\mathrm{SACAPF}}$ and $i_5^{\mathrm{hrms}}$ represent the SACAPF rating and the RMS value of the harmonic content of the nonlinear load, respectively.

Figure 9 shows the THD value of network voltages in steady-state conditions while the SACAPF rating is constrained. It is notable that, in Figure 9, as the SACAPF rating increases, the network voltages yield to sinusoidal condition, and finally, in the third scenario, the network voltages are purely sinusoidal.

In order to provide a comparison between the SACAPF output current and nonlinear load current, the resultant waveforms from different scenarios are plotted in the same graph (Figure 10). As it is shown in the figure, the main logic of the current injection (as it is explained previously in the Introduction section) is always realized. However, it is possible that the injection current magnitude becomes smaller than that of the harmonic content of the nonlinear load due to the limited SACAPF rating. For instance, this happened in Scenario 1 and Scenario 2.

In order to evaluate the SACAPF performance installed in different locations of the network, the following study was performed when the location of the SACAPF was moved to bus 4 of the network, and Figure 11 shows the THD value of the network voltage in a steady-state condition. The scenarios are the same in Table 3. As it is demonstrated in Figure 11, a greater SACAPF results in a more sinusoidal condition of the network similar to the case of bus 5. Hence, when the SACAPF is located at bus 4, and the rating of the SACAPF is adequate, all upstream buses are free from harmonics, so the proper operation of the SACAPF is confirmed. The other point in Figure 11 is the presence of harmonics in buses 5, 6, 7, and 8 (buses downstream to bus 4) in all scenarios, which shows the importance of the installation location.

4.1.2. Multiple SACAPFs

In order to evaluate the performance of the proposed method for multiple stand-alone controlled SACAPFs, two different conditions of the predefined case study network were utilized: one containing a single nonlinear load and the other containing multiple nonlinear loads.

Network with a Single Nonlinear Load

The test network is the same as in Figure 7, which contains one nonlinear load installed at bus 5. Two SACAPFs are installed in the network at buses 4 and 5. The considered SACAPFs ratings are presented in

Table 4. Ratings of SACAPFs in comparison to harmonic content of nonlinear load.

Name of SACAPF	Scenario 1	Scenario 2	Scenario 3	Scenario 0
A 1 (Bus 5)	25%	50%	100%	No installed SACAPF
A 2 (Bus 4)	100%	100%	100%

, and Figure 12 shows the voltage THD value of the network buses.

Figure 13 shows the RMS value of the SACAPF output currents (A 1 and A 2), the harmonic content of the nonlinear load (N), the harmonic content of branch 4–5 (B 4–5), and the harmonic content of branch 5-6 (B 5–6). As shown in the graph, it is obvious that as the rating of A 1 increases, the compensation current of A 2 decreases, and finally, in Scenario 3, there is no need for compensation by A 2.

Network with Multiple Nonlinear Loads

In this section, three nonlinear loads (six-pulse converters) are considered in the case study network. Figure 14 shows the single-line diagram of the test network. The SACAPFs are ideal, and there is no limitation on the compensation rate. In order to provide a comprehensive study that encompasses different installed SACAPFs throughout the network, three different scenarios were devised. The first scenario represents the basic form of the case study network in which no SACAPF was installed. The second and third scenarios represent the installation of three and two SACAPFs in the network, respectively.

Table 5. Location of installed SACAPFs in the network with multiple nonlinear loads.

Location of Ideal APFs	Scenario 0	Scenario 1	Scenario 2
Bus	-	5, 24, 25	5, 23

shows the bus locations where the SACAPFs were installed.

The results for the THD values of voltages in a steady state are presented in Figure 15. As can be seen, the utilization of three SACAPFs (Scenario 1) results in an almost ideal operation of the network in terms of harmonic emission. In Scenario 2, there are yet small voltage harmonics at buses 24, 25, and 26 due to the electrical distances between the SACAPF locations and the nonlinear loads that do not appear in Scenario 1.

Figure 16 shows the RMS value of the harmonic content of the nonlinear loads in comparison to the RMS value of the injected currents by the SACAPFs. The total RMS value of the injected currents by the SACAPFs in Scenario 1 is 6.38% and 6.35% in Scenario 2. These values are very close to each other, and the small difference is due to the nonlinear load locations in Scenario 2.

4.2. Optimal Allocation of SACAPFs

In these sets of simulations, the optimal allocation of the proposed SACAPFs were compared to the optimal allocation of conventional APFs (APFs with integrated control systems). The tests were performed on a 33-bus IEEE test network and a 9-bus test network.

4.2.1. 33-Bus IEEE Test Network

The single-line diagram of the 33-bus IEEE test network is shown in Figure 17. This network includes 33 buses in 12.6 kV, and the base power is 10 MVA. Detailed data of the line sections are reported in [37]. In this network, a part of all loads is assumed nonlinear, and detailed data of the loads are reported in

Table 6. Load data of modified 33-bus IEEE test network.

Bus	Linear kW	Linear kVAr	Nonlinear kW	Nonlinear kVAr	THD (%)	Bus	Linear kW	Linear kVAr	Nonlinear kW	Nonlinear kVAr	THD (%)
1	0	0	0	0	3.14	18	54	24	36	16	9.26
2	60	36	40	24	3.28	19	54	24	36	16	9.32
3	54	24	36	16	3.95	20	54	24	36	16	3.58
4	72	48	48	32	4.31	21	54	24	36	16	3.64
5	36	18	24	12	4.68	22	54	24	36	16	3.7
6	36	12	24	8	5.98	23	54	24	36	16	4.17
7	120	60	80	40	6.46	24	252	120	168	80	4.62
8	120	60	80	40	7.32	25	252	120	168	80	4.85
9	36	12	24	8	7.72	26	36	15	24	10	6.9
10	36	12	24	8	8.09	27	36	15	24	10	6.23
11	27	18	18	12	8.13	28	36	12	24	8	7.13
12	36	21	24	14	8.18	29	72	42	48	28	7.8
13	36	21	24	14	8.61	30	120	360	80	240	8.02
14	72	48	48	32	8.83	31	90	42	60	28	8.35
15	36	6	24	4	8.93	32	126	60	84	40	8.42
16	36	12	24	8	9.02	33	36	24	24	16	8.45
17	36	12	24	8	9.21

. The type of nonlinear load is the same six-pulse converters where their harmonic spectra are illustrated in Figure 6.

The population size was set to 100 for the allocation of the proposed SACAPFs and set to 250 for the allocation of the conventional APFs. The number of iterations was set to 80 for the allocation of the proposed SACAPFs and set to 100 for the allocation of the conventional APFs. The size of the SACAPFs was a discrete multiple of 0.01 pu limited to 0.1 pu. $C^{\mathrm{var}}$ and $C^{\mathrm{fix}}$ were set to $72,000 and $90,000 [4]. The maximum allowable values for the voltage THD and IHD were based on IEEE 1574 and were set to 5% and 8%, respectively.

The results for the allocation of both SACAPFs and APFs in a 33-bus IEEE test network are reported in

Table 7. Results of allocation of active filters in 33-bus IEEE network.

Control Method	Proposed Approach (SACAPF)	Integrated Approaches (APF)
Number of Filters	2	1
Rating of Filter(s) (pu)	0.02 & 0.01	0.03
Location of Filter(s) (Bus)	6 & 11	7
Cost ($)	396,000	306,000
Max THD @ Bus	4.96 @ 33	5 @ 18
Max IHD (harmonic) @ Bus	8 (5) @ 33	8 (5) @ 18

. The proposed approach needs at least two SACAPFs for reducing harmonic pollution within acceptable ranges, while using only one integrated controlled APF results in an acceptable harmonic condition. The cost of filters for the conventional method is less than the proposed approach. Note that this advantage is only achievable in the presence of a wide communication infrastructure that is not yet implemented in most distribution networks. Hence, using the proposed approach can result in more economic solutions due to the independency of the communication networks.

Voltage THDs are shown in Figure 18 where it can be seen that both methods could reach acceptable ranges (THD for all buses is less than 5%), while the maximum THD location is different in methods that are the result of the difference between the control methods. In conventional methods, the control system can have a vision throughout the network and adjusts injection currents thanks to the presence of a communication network. In the proposed approach, the control system can only record and see what is going on through the installation location, and the injection current adjusted for current compensation results in a sinusoidal branch current. The current of branch 5-6 is shown in Figure 19 where it is clear that the proposed approach has reached sinusoidal condition, while the branch current in the conventional method is still polluted because current pollution is not considered in conventional methods.

4.2.2. Nine-Bus Test Network

The single-line diagram of the nine-bus test network is shown in Figure 20. Line data are reported in [32], and load data are presented in

Table 8. Load data of the 9-bus test network.

Bus	Linear kW	Linear kVAr	Nonlinear kW	Nonlinear kVAr	THD (%)
1	1012	253	0	0	0.83
2	539	187	0	0	2.06
3	985	245	0	0	4.6
4	0	0	1598	1840	5.94
5	0	0	1610	600	7.32
6	429	61	0	0	7.23
7	633	33	0	0	7.14
8	539	72	0	0	7.03
9	902	110	0	0	7.01

. Two nonlinear loads are placed on bus 4 and bus 5, and the type of the nonlinear loads is the same 6-pulse converters, and their harmonic spectra are shown in Figure 6. The objective of optimization in this network is to compensate for all harmonic currents and yield sinusoidal voltages throughout the network. To achieve this goal, the maximum allowable value of voltage THD ($TH{D}^{\max }$) and the maximum allowable value of voltage IHD ($IH{D}^{\max }$) were set to zero.

Both the proposed and the conventional methods resulted in the utilization of 2 filters located on bus 4 and bus 5. Detailed results are reported in

Table 9. Results of allocation of filters in 9-bus test network.

Item	Proposed Method	Conventional Method
Size of the filter located on bus 4 (%)	8	8
Total injection current (%)	7.28	7.36
Size of the filter located on bus 5 (%)	6	6
Total injection current (%)	5.24	5.29
Maximum THD of voltages (%)	0.00096	0.00951

. Although there is a small difference between the injection currents, the final solution of both methods (size and location of filters) is the same as expected.

5. Conclusions

In this paper, a stand-alone control procedure for APFs was proposed, which is capable of handling the operation of multiple SACAPFs using a novel backward procedure. The proposed control procedure uses local feedback from the flowing current in the SACAPF installation location and determines the injection current that has the same magnitude as the reference current but is an opposite phase angle. In addition, a downgrading procedure was proposed to manage SACAPF operation in a probable overload condition. The operation of the proposed stand-alone SACAPF was evaluated in different scenarios. Single and multiple SACAPFs were installed in different scenarios in an IEEE 18-bus distribution network with various nonlinear loads. The results confirmed the effective operation of SACAPFs in harmonic compensation of practical distribution networks where data transfer infrastructure is not readily available and employing APFs with integrated control systems is not applicable. The proposed approach can be used as an analysis tool in the allocation of SACAPFs in distribution networks.