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This paper proposes a novel hybrid algorithm based on a combination of the simple quadratic interpolation and the symbiosis organisms search algorithm (SQI-SOS) for finding the optimal location and size of capacitors in radial distribution networks. The objective of the problem is to minimize the system operating cost so that the net yearly savings of the system are increased. The effectiveness of the SQI-SOS has been tested on 33-, 69-, and 119-bus radial distribution networks with different load models. The obtained results from the test system by the proposed SQI-SOS are compared with those from the conventional SOS and other mature optimization methods in the literature. The result comparison has shown that the proposed SQI-SOS algorithm can provide a better solution than the other methods. Accordingly, the proposed SQI-SOS can be a very effective and efficient method for dealing with the optimal capacitor placement problem in distribution networks.

The reactive power compensation plays an important role in the operation of power distribution networks since it brings many technical benefits such as decreasing power losses, enhancing voltage profiles, correcting power factor, and releasing system capacity. To acquire such benefits, shunt capacitors are widely employed in distribution networks to inject the reactive power. However, the inappropriate placement of capacitors may result in reducing the benefits of the system and even risking the whole system operation. Therefore, it is very important to determine the optimal location and size of capacitors to be installed in distribution networks so that the maximum benefits of the system can be acquired while satisfying all the operating constraints of the system.

In the recent past decades, the optimal capacitor placement (OCP) problem has been formulated with many objective functions such as power loss reduction, voltage profile improvement, capacitor installation cost minimization, network stability maximization, and reduction of burden on existing lines [

In recent years, artificial intelligence- (AI-) based methods have been developed and effectively implemented for dealing with the OCP problem. A survey of studies from the literature has shown that the OCP problem has been successfully solved by the genetic algorithm (GA) [

Symbiotic organisms search (SOS) is new a population-based algorithm proposed by Cheng and Prayogo [

The current paper contains the main contributions as follows:

Firstly, we propose a novel hybrid method via the combination of SQI and SOS (SQI-SOS) to estimate the optimal siting and sizing of capacitors for the first time, in which the overall cost objective function is minimized.

Secondly, we introduce a new initialization process for treating discrete capacitor sizing variables. Based on this initialization method, only a few adjustments are required in the original solution method.

Thirdly, this is the first time in this study where both continuous and discrete capacitor sizes have been investigated for a result comparison.

Fourthly, the actual voltage-dependent nonlinear load models have been utilized for investigations.

Finally, the proposed SQI-SOS method offers the solutions with a better quality than those acquired by the previously reported methods in the comparative cases.

The remaining of this paper is organized as follows: Section

In this study, the optimization problem of the OCP is formulated with an objective of minimizing the system operating cost which is described by the following equation [

In order to analyze the OCP problem with an actual voltage-dependent load model, the original objective function can be modified as follows (MC):

Real and reactive power balance constraint:

Voltage limits at buses:

Maximum real power flow constraint:

Reactive power compensation limits:

Total reactive power compensation limits:

Overall system power factor limits:

In the previous load flow studies, the load models with constant active and reactive powers, i.e., constant loads, were commonly utilized. However, power demands of practical loads highly depend on the network voltage. So, these loads can be modeled as voltage-dependent loads which include residential, industrial, and commercial loads. Mathematically, the voltage-dependent load models can be formulated as [

Constant load:

Industrial load:

Residential load:

Commercial load:

Mixed or practical load: the aggregation of the different load types is implemented by

Load types and exponent values.

Load type | Constant load | Industrial load | Residential load | Commercial load |
---|---|---|---|---|

Exponents |

To examine the standard radial distribution systems with a practical mixed load model, we assume that these systems only include industrial, residential, and commercial loads as in [

The SOS proposed by Cheng and Prayogo in 2014 [

Mutualism is a symbiotic relationship where both organisms get benefits from each other. In the SOS, an organism _{j} is randomly selected from the ecosystem, which is used to interact with an organism _{i}. The organism _{i} is the _{i} and _{j} are generated based on the mutualistic symbiosis relationship between organisms _{i} and _{j} by the following equations [_{best} represents the best organism in an ecosystem; MV denotes a mutual vector that represents the mutualistic symbiosis relationship between organisms _{i} and _{j}; _{1} and _{2} are the benefit factors which describe the level of benefit to each organism. These factors are stochastically selected as either 1 or 2 (1 is for partial benefit while 2 is for full benefit).

The new organisms are accepted only if they give a better fitness value compared to the previous organisms.

Commensalism is a symbiotic relationship where one organism is benefited and the other is neither harmed nor benefited. In this phase, an organism _{j} is randomly selected from the ecosystem to interact with the organism _{i}. As a result from the interaction, organism _{i} benefits while the organism _{j} is neither harmed nor benefited. The new organism of _{i} produced by this interaction is calculated as follows [

According to the rules, the new organism is only updated if it gives a better fitness value compared to the prior organism.

Parasitism is a symbiotic relationship between two different organisms where one benefits and the other is harmed. In this phase, organism _{i} is offered a role player of a parasite through a vector called “_{i} is duplicated and modified itself by using a random number to create a _{j} is randomly chosen and serves as a host to the _{j} and replace it in the ecosystem. Otherwise, organism _{j} will have immunity from the parasite, and the

In 2016, Nama et al. [

Considering two organism _{j} and _{i} is updated by the three-point SQI. The _{i}, _{j}, and _{k} are the fitness values of the

The new organism is set to the ecosystem if its fitness value is better than that of the corresponding organism in the ecosystem.

When a new organism is created, it is further checked for boundary violation. If any organism violates the boundary, that organism will be repaired by the following strategy:_{i} and UB_{i} are the lower and upper bounds of the

The pseudocode of the SQI-SOS algorithm is depicted in Algorithm

Set control parameters (dimension of problem

Initialize the population of organisms randomly

Set

Identify the best organism _{best} in an ecosystem;

Randomly select one organism _{j}, where _{j} _{i};

Modify organisms _{i} and _{j} using equations (

Check the new organisms for boundary violation and repair according to equation (

Calculate fitness value of the new organisms;

Update the new organisms with better fitness;

Randomly select one organism _{j}, where _{j} _{i};

Modify organism _{i} with the help of organism _{j} using equation (

Check the new organism for boundary violation and repair according to equation (

Calculate fitness value of the new organism;

Update the new organism with better fitness;

Randomly select one organism _{j}, where _{j} _{i};

Modify organism _{i} according to Section

Check the new organism for boundary violation and repair according to equation (

Calculate fitness value of the new organism;

Update the new organism with better fitness;

Randomly select two organisms _{j} and _{k}, where _{j} _{k} _{i};

Modify organism _{i} according to equation (

Check the new organism for boundary violation and repair according to equation (

Calculate the fitness value of the new organism;

Update the new organism with better fitness;

A population of organisms is created by a matrix with

In the SQI-SOS, each organism of the population is randomly initialized. The solution for the number of buses and sizes of capacitors for each load level is initialized as follows:

For the size of capacitors, there are two different ways to initialize before starting the optimization process depending on the study cases. In the case with continuous capacitor size, the capacitor sizes corresponding to each load level are initialized using equation (

Assuming that the actual capacitor size has a range of _{Cmin} to Ind_{Cmax} corresponding to the values from 1 to _{max}. Obviously, each discrete size value will be identified by an index Ind_{C}. Also, a randomization in the initial discrete size initialization process can be fulfilled thanks to the manipulation of equation (_{1} is a uniformly distributed random number in [0, 1] for each population of organisms; Ind_{C,ij} is an index representing the _{Cmin,ij} and Ind_{Cmax,ij} are the lower and upper limits of the index that deputize the discrete size value of the

After initialization, each organism needs to be evaluated by calculating its fitness function. The fitness function is formulated based on two components of the objective function and dependent variables. These dependent variables are the inclusion of bus voltages, line flows, power factor, and maximum allowable reactive power. The fitness function is calculated as follows:_{f}, _{p}, and _{q} are the penalty factors for bus voltages, line flows, power factor, and maximum allowable reactive power, respectively. In this study, the used penalty factors are set to 100000.

The limit values of the dependent variables in equation (_{k}, PF_{overall}, and _{TC} and ^{lim} represents the limits of _{k}, PF, and _{TC}.

This study uses the maximum number of iterations (

The flowchart of the proposed SQI-SOS method for solving the OCP problem is given in Figure

Flowchart of the SQI-SOS for solving the OCP problem.

To validate the effectiveness of the proposed SQI-SOS algorithm, it is tested on several test systems including 33-bus, 69-bus, and 119-bus radial distribution networks to find the optimal locations and sizes of capacitors for minimizing the objective function as formulated in Section _{c} and SQI-SOS_{c}.

The implementation of the SQI-SOS to the OCP problem is coded in the Matlab R2016a platform, and 50 independent trials are run on a computer with Intel core i5-3337U CPU of 1.80 GHz speed and 8 GB RAM. The Matpower 6.0 toolbox [

In order to examine variable load conditions in Scenario 1, it is assumed that the networks are operated at three load levels: 0.5 (light), 0.75 (medium), and 1.0 (full) for time percentages of 25, 35, and 40%, respectively. To calculate the total cost, the rates stated in Table

Rates for energy, purchase, operation, and maintenance costs.

Item | Rate |
---|---|

Average energy cost (KP) | 0.06k$/kWh |

Purchase cost (KC) | 3k$/kVAr |

Installation cost (Kci) | 1000l$/location |

Operating cost (Kco) | 300y$/year/location |

Hours per year (T) | 8760.0000 |

There are two control parameters of the SQI-SOS (i.e., ecosystem size

Parameter setting for the SQI-SOS algorithm and inequality constraint setting.

Item | 33-bus test system | 69-bus test system | 119-bus test system |
---|---|---|---|

Population size | 90 | 90 | 150 |

Maximum number of iterations | 100 | 100 | 200 |

Maximum power flow—MPF (kW) | 3925.99 (#_{1-2}) | 4027.1 (#_{1-2}) | 10677.9 (#_{1-2}) |

Maximum limit of power flow—MLPF (kW) | 4000 | 4200 | 11000 |

Bus voltage constraint | |||

Power factor constraint | |||

Number of capacitors to be installed | 3 | 2 | 8 |

Allowable capacitor range | 0–1500 kVAr for continuous size variables; 0–1500 kVAr with the fixed step of 50 kVAr for discrete size variables |

The first test system is a 33-bus radial distribution network with the line and load data from [

In the first scenario, the OCP problem was examined with a CP load model to make a performance comparison with previously reported methods.

Table _{c} and SOS_{c} as well as the proposed SQI-SOS and SOS at different load levels. Fixed and switched capacitors on the load levels by the proposed SQI-SOS and SOS are provided in Table

Optimal siting and sizing of capacitors for the 33-bus system with the CP load model at different load levels.

Load level | Method | Optimal siting and sizing in kVAr | Total kVAr | Total kVAr demand |
---|---|---|---|---|

Light 50% | SOSc | (13, 191.51); (25, 200.72); (30, 516.28) | 908.51 | 1150 |

SQI-SOSc | (13, 191.48); (25, 200.85); (30, 516.33) | 908.67 | ||

SOS | (14, 200); (25, 200); (30, 500) | 900 | ||

SQI-SOS | (14, 200); (25, 200); (30, 500) | 900 | ||

Medium 75% | SOSc | (13, 295.44); (25, 200.72); (30, 789.82) | 1285.99 | 1725 |

SQI-SOSc | (13, 295.42); (25, 200.85); (30, 789.84) | 1286.11 | ||

SOS | (14, 300); (25, 200); (30, 800) | 1300 | ||

SQI-SOS | (14, 300); (25, 200); (30, 800) | 1300 | ||

Full 100% | SOSc | (13, 325.37); (25, 200.72); (30, 900.73) | 1426.84 | 2300 |

SQI-SOSc | (13, 325.37); (25, 200.85); (30, 900.75) | 1426.97 | ||

SOS | (14, 300); (25, 200); (30, 900) | 1400 | ||

SQI-SOS | (14, 300); (25, 200); (30, 900) | 1400 | ||

Final optimal ratings (location and size) by proposed approach | SOS | Fixed: (14, 200); (25, 200); (30, 500) | ||

SQI-SOS | Fixed: (14, 200); (25, 200); (30, 500) |

Table _{1-2}) is decreased from 3925.99 kW to 3856.54 kW due to optimal compensation by the proposed SQI-SOS, and this value is much lower than the maximum limit of power flow (MLPF) of 4000 kW. Furthermore, the connection of capacitors to the system leads to the reduction of power flow in most lines. The detail of power loss comparison before and after compensation for this system corresponding to the full load level using the SQI-SOS method in each line is given in Table

Comparison results of the 33-bus system before and after compensation with the CP load model at different load levels.

Load level | Item | Uncompensated | Compensated | |||
---|---|---|---|---|---|---|

SOS_{c} | SQI-SOS_{c} | SOS | SQI-SOS | |||

Light 50% | 0.9540 | 0.9667 | 0.9667 | 0.9678 | 0.9678 | |

P_{loss}, kW | 48.7870 | 32.8848 | 32.8848 | 32.916 | 32.916 | |

PF_{overall} | 0.8497 | 0.9904 | 0.9904 | 0.9897 | 0.9897 | |

MPF, kW | 1906.28 | 1890.38 | 1890.38 | 1890.41 | 1890.41 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Medium 75% | 0.9295 | 0.9495 | 0.9495 | 0.9512 | 0.9512 | |

P_{loss}, kW | 113.9869 | 76.1249 | 76.1243 | 76.1849 | 76.1849 | |

PF_{overall} | 0.8494 | 0.9856 | 0.9856 | 0.9864 | 0.9864 | |

MPF, kW | 2900.23 | 2862.37 | 2862.37 | 2862.43 | 2862.43 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Full 100% | 0.9038 | 0.9271 | 0.9271 | 0.9275 | 0.9275 | |

P_{loss}, kW | 210.9875 | 141.1855 | 141.1840 | 141.5439 | 141.5439 | |

PF_{overall} | 0.8490 | 0.9698 | 0.9698 | 0.9682 | 0.9682 | |

MPF, kW | 3925.99 | 3856.18 | 3856.18 | 3856.54 | 3856.54 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Total annual cost ($/year) | 71,737.7 | 56,188.36 | 56,188.36 | 56,198.33 | 56,198.33 | |

Annual net savings ($/year) | — | 15,549.33 | 15,549.33 | 15,539.36 | 15,539.36 | |

Worst cost ($/year) | — | 56,449.50 | 56,449.50 | 56,507.02 | 56,458.73 | |

Best cost ($/year) | — | 56,188.36 | 56,188.36 | 56,198.33 | 56,198.33 | |

Mean cost ($/year) | — | 56,252.59 | 56,228.78 | 56,265.06 | 56,227.01 | |

Standard deviation ( | — | 97.37 | 78.96 | 107.41 | 62.58 |

Convergence curves of the total cost of the 33-bus system under the CP load model with 100% loading.

Voltage profile of the 33-bus system before and after compensation with the CP load model at 100% loading.

Comparison of lines power flow of the 33-bus system before and after compensation with the CP load model at 100% loading.

Results of real power loss in all lines of the 33-bus system before and after the installation of capacitors.

Line | Real power loss (kW) | |
---|---|---|

Uncompensated | Compensated by SQI-SOS | |

1-2 | 12.300 | 9.127 |

2-3 | 52.077 | 37.285 |

3-4 | 20.053 | 13.143 |

4-5 | 18.850 | 12.187 |

5-6 | 38.566 | 24.693 |

6-7 | 1.946 | 1.588 |

7-8 | 11.873 | 9.488 |

8-9 | 4.266 | 3.366 |

9-10 | 3.620 | 2.826 |

10-11 | 0.565 | 0.437 |

11-12 | 0.899 | 0.710 |

12-13 | 2.721 | 2.234 |

13-14 | 0.744 | 0.658 |

14-15 | 0.364 | 0.346 |

15-16 | 0.287 | 0.273 |

16-17 | 0.257 | 0.244 |

17-18 | 0.054 | 0.051 |

2–19 | 0.161 | 0.161 |

19-20 | 0.832 | 0.831 |

20-21 | 0.101 | 0.101 |

21-22 | 0.044 | 0.044 |

3–23 | 3.182 | 2.740 |

23-24 | 5.144 | 4.389 |

24-25 | 1.288 | 1.038 |

6–26 | 2.602 | 1.199 |

26-27 | 3.330 | 1.466 |

27-28 | 11.306 | 4.728 |

28-29 | 7.837 | 3.077 |

29-30 | 3.897 | 1.392 |

30-31 | 1.594 | 1.509 |

31-32 | 0.213 | 0.202 |

32-33 | 0.013 | 0.012 |

The test results of the 33-bus system acquired by the applied SOS and proposed SQI-SOS methods with different voltage-dependent load models are represented in Table _{1-2} is reduced to 3753.63 kW, which is lower than a maximum limit of power flow (MLPF) of 4000 kW. Moreover, the power flow in most lines for the compensated case is lower than that for the uncompensated case. Based on the experiment results, it can be concluded that the proposed SQI-SOS is capable of dealing with the practical voltage-dependent load model scenarios for this system.

Comparative results of the 33-bus system after compensation with different load models.

Load type | Item | Uncompensated | Compensated | |
---|---|---|---|---|

SOS | SQI-SOS | |||

Industrial load | 0.9152 (18) | 0.9305 (18) | 0.9305 (18) | |

P_{loss}, kW | 167.7916 | 135.9869 | 135.9869 | |

PF_{overall} | 0.9042 | 0.9809 | 0.9809 | |

Total cost, $/year | 88,191.24 | 80,024.69 | 80,024.69 | |

Annual savings ($/year) | — | 8,166.55 | 8,166.55 | |

MPF, kW | 3,851.37 | 3,825.86 | 3,825.86 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Residential load | 0.9160 (18) | 0.9314 (18) | 0.9314 (18) | |

P_{loss}, kW | 164.5408 | 127.1548 | 127.1548 | |

PF_{overall} | 0.8822 | 0.9766 | 0.9766 | |

Total cost, $/year | 86,482.65 | 75,232.54 | 75,232.54 | |

Annual savings ($/year) | — | 11,250.10 | 11,250.10 | |

MPF, kW | 3,723.14 | 3,718.49 | 3,718.49 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Commercial load | 0.9175 (18) | 0.9236 (18) | 0.9236 (18) | |

P_{loss}, kW | 159.5012 | 127.1272 | 127.1272 | |

PF_{overall} | 0.8707 | 0.9580 | 0.9580 | |

Total cost, $/year | 83,833.85 | 74,168.07 | 74,168.07 | |

Annual savings ($/year) | — | 9,665.77 | 9,665.77 | |

MPF, kW | 3,625.88 | 3,625.81 | 3,625.81 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Mixed load | 0.9159 (18) | 0.9307 (18) | 0.9307 (18) | |

P_{loss}, kW | 164.9165 | 130.0845 | 130.0845 | |

PF_{overall} | 0.8908 | 0.9776 | 0.9776 | |

Total cost, $/year | 86,680.09 | 76,772.43 | 76,772.43 | |

Annual savings ($/year) | — | 9,907.66 | 9,907.66 | |

MPF, kW | 3,765.53 | 3,753.63 | 3,753.63 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) |

Optimal locations and kVArs of capacitors for the 33-bus system with different load models.

Load type | Location and injected kVAr | Total kVAr | Method |
---|---|---|---|

Industrial load | (13, 350); (25, 250); (30, 950) | 1550 | SOS |

(13, 350); (25, 250); (30, 950) | 1550 | SQI-SOS | |

Residential load | (13, 300); (25, 250); (30, 950) | 1500 | SOS |

(13, 300); (25, 250); (30, 950) | 1500 | SQI-SOS | |

Commercial load | (5, 350); (25, 100); (30, 700) | 1150 | SOS |

(5, 350); (25, 100); (30, 700) | 1150 | SQI-SOS | |

Mixed load | (13, 300); (25, 250); (30, 950) | 1500 | SOS |

(13, 300); (25, 250); (30, 950) | 1500 | SQI-SOS |

Annual cost of the 33-bus system before and after compensation with different load models.

Convergence curves of the total cost of the 33-bus system with the mixed load model.

Voltage profile of the 33-bus system before and after compensation with the mixed load model.

Comparison of lines power flow of the 33-bus system before and after compensation with the mixed load model.

The second test system is a 69-bus radial distribution network with the total load demand of 3.8 MW and 2.69 MVAr. The data for branch and load of this system are taken from [

In this scenario, the siting and sizing of capacitors obtained by the proposed SQI-SOS, applied SOS, and other methods are shown in Table

Optimal siting and sizing of capacitors for the 69-bus system with the CP load model at different load levels.

Load level | Method | Optimal siting and sizing in kVAr | Total kVAr | Total kVAr demand |
---|---|---|---|---|

Light 50% | CSA [ | (21, 0); (62, 600) | 600 | 1347.3 |

IHA [ | (21, 0); (61, 550) | 550 | ||

FPA [ | (17, 0); (61, 550) | 550 | ||

SOSc | (18, 178.14); (61, 627.28) | 805.42 | ||

SQI-SOSc | (18, 178.09); (61, 627.43) | 805.52 | ||

SOS | (18, 200); (61, 600) | 800 | ||

SQI-SOS | (18, 200); (61, 600) | 800 | ||

Medium 75% | CSA [ | (21, 0); (62, 950) | 950 | 2020.95 |

IHA [ | (21, 0); (61, 900) | 900 | ||

FPA | (17, 0); (61, 900) | 900 | ||

SOSc | (18, 242.09); (61, 952.60) | 1194.69 | ||

SQI-SOSc | (18, 242.01); (61, 952.60) | 1194.62 | ||

SOS | (18, 250); (61, 950) | 1200 | ||

SQI-SOS | (18, 250); (61, 950) | 1200 | ||

Full 100% | CSA [ | (21, 250); (62, 1200) | 1450 | 2694.6 |

IHA [ | (21, 350); (61, 1350) | 1700 | ||

FPA [ | (17, 300); (61, 1250) | 1550 | ||

SOSc | (18, 242.09); (61, 1113.35) | 1355.44 | ||

SQI-SOSc | (18, 242.01); (61, 1113.41) | 1355.43 | ||

SOS | (18, 250); (61, 1100) | 1350 | ||

SQI-SOS | (18, 250); (61, 1100) | 1350 | ||

Final optimal ratings (location and size) by proposed approach | SOS | Fixed: (18, 200); (61, 600) | ||

Switched: (18, 50); (61, 500) | ||||

SQI-SOS | Fixed: (18, 200); (61, 600) | |||

Switched: (18, 50); (61, 500) |

Results of real power loss in all lines of the 69-bus system before and after the installation of capacitors.

Line | Real power loss (kW) | |
---|---|---|

Uncompensated | Compensated by SQI-SOS | |

1-2 | 0.075 | 0.055 |

2-3 | 0.075 | 0.055 |

3-4 | 0.195 | 0.140 |

4-5 | 1.937 | 1.306 |

5-6 | 28.244 | 19.036 |

6-7 | 29.352 | 19.780 |

7-8 | 6.895 | 4.633 |

8-9 | 3.375 | 2.247 |

9-10 | 4.778 | 3.621 |

10-11 | 1.015 | 0.763 |

11-12 | 2.193 | 1.559 |

12-13 | 1.287 | 0.885 |

13-14 | 1.247 | 0.858 |

14-15 | 1.206 | 0.833 |

15-16 | 0.224 | 0.155 |

16-17 | 0.321 | 0.228 |

17-18 | 0.003 | 0.002 |

18-19 | 0.104 | 0.103 |

19-20 | 0.067 | 0.066 |

20-21 | 0.108 | 0.106 |

21-22 | 0.001 | 0.001 |

22-23 | 0.005 | 0.005 |

23-24 | 0.011 | 0.011 |

24-25 | 0.006 | 0.006 |

25-26 | 0.002 | 0.002 |

26-27 | 3.50 | 3.44 |

3–28 | 3.47 | 3.47 |

28-29 | 0.003 | 0.003 |

29-30 | 0.006 | 0.006 |

30-31 | 0.001 | 0.001 |

31-32 | 0.005 | 0.005 |

32-33 | 0.012 | 0.012 |

33-34 | 0.010 | 0.010 |

34-35 | 4.79 | 4.79 |

3–36 | 0.001 | 0.001 |

36-37 | 0.015 | 0.015 |

37-38 | 0.017 | 0.017 |

38-39 | 0.005 | 0.005 |

39-40 | 1.98 | 1.98 |

40-41 | 0.049 | 0.049 |

41-42 | 0.020 | 0.020 |

42-43 | 0.003 | 0.003 |

43-44 | 0.001 | 0.001 |

44-45 | 0.006 | 0.006 |

45-46 | 1.26 | 1.26 |

4–47 | 0.023 | 0.023 |

47-48 | 0.583 | 0.583 |

48-49 | 1.634 | 1.633 |

49-50 | 0.116 | 0.116 |

8–51 | 0.002 | 0.002 |

51-52 | 4.38 | 4.34 |

9–53 | 5.781 | 3.701 |

53-54 | 6.711 | 4.296 |

54-55 | 9.125 | 5.830 |

55-56 | 8.790 | 5.607 |

56-57 | 49.685 | 31.694 |

57-58 | 24.489 | 15.622 |

58-59 | 9.506 | 6.064 |

59-60 | 10.671 | 6.787 |

60-61 | 14.026 | 8.921 |

61-62 | 0.112 | 0.108 |

62-63 | 0.135 | 0.129 |

63-64 | 0.661 | 0.634 |

64-65 | 0.041 | 0.040 |

11–66 | 0.003 | 0.003 |

66-67 | 1.53 | 1.52 |

12–68 | 0.023 | 0.023 |

68-69 | 3.71 | 3.66 |

To verify the efficacy of the proposed SQI-SOS, the resulting solutions acquired by the SQI-SOS are compared with those from the other well-established methods such as SOS, SQI-SOS_{c}, SOS_{c}, CSA [_{1-2} is reduced to 3,950.52 kW by the proposed SQI-SOS, and it is still within permissible limit. In addition, the power flow on most of the lines is diminished due to reactive power compensation. Generally, the proposed SQI-SOS method outperforms the previously published methods regarding the gain of the lowest total annual cost for the test system in this scenario. On the other hand, the SQI-SOS approach introduces a fairly new solution to the OCP problem for the 69-bus system with the CP load model. Therefore, the SQI-SOS method exposes the ability not only to improve the convergence speed, but also to enhance the robustness of the original SOS algorithm when solving the problem under the CP load model.

Comparison results of the 69-bus system before and after compensation with the CP load model at different load levels.

Load level | Item | Uncompensated | Compensated | ||||||
---|---|---|---|---|---|---|---|---|---|

CSA [ | IHA [ | FPA [ | SOS_{c} | SQI-SOS_{c} | SOS | SQI-SOS | |||

Light 50% | 0.9567 | 0.966 | 0.9652 | 0.965 | 0.9666 | 0.9666 | 0.9663 | 0.9663 | |

P_{loss}, kW | 51.6064 | 35.89 | 35.9451 | 36.5235 | 34.3536 | 34.3536 | 34.3938 | 34.3938 | |

PF_{overall} | 0.8184 | 0.93 | 0.9218 | 0.9218 | 0.9608 | 0.9608 | 0.9601 | 0.9601 | |

MPF, kW | 1952.65 | — | — | — | 1935.40 | 1935.40 | 1935.44 | 1935.44 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | |||||

Medium 75% | 0.9335 | 0.9486 | 0.949 | 0.9506 | 0.9491 | 0.9491 | 0.9491 | 0.9491 | |

P_{loss}, kW | 121.0301 | 83.19 | 82.57 | 82.5706 | 79.7548 | 79.755 | 79.7399 | 79.7399 | |

PF_{overall} | 0.8198 | 0.94 | 0.93 | 0.9299 | 0.9592 | 0.9592 | 0.9597 | 0.9597 | |

MPF, kW | 2972.60 | — | — | — | 2931.32 | 2931.32 | 2931.31 | 2931.31 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | |||||

Full 100% | 0.9092 | 0.930 | 0.937 | 0.95 | 0.9282 | 0.9282 | 0.9281 | 0.9281 | |

P_{loss}, kW | 225.0006 | 147.95 | 145.3236 | 145.14 | 148.323 | 148.323 | 148.4248 | 148.4248 | |

PF_{overall} | 0.8213 | 0.95 | 0.9656 | 0.9559 | 0.9419 | 0.9419 | 0.9415 | 0.9415 | |

MPF, kW | 4027.10 | — | — | — | 3950.42 | 3950.42 | 3950.52 | 3950.52 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | |||||

Total annual cost ($/year) | 76,349.9 | 58,074.59 | 58,165.60 | 57,753.11 | 57,035.52 | 57,035.52 | 57,043.14 | 57,043.14 | |

Annual net savings ($/year) | — | 18,275.31 | 18,184.30 | 18,596.79 | 19,314.37 | 19,314.37 | 19,306.75 | 19,306.75 | |

Worst cost ($/year) | — | — | — | — | 57,035.64 | 57,035.52 | 57,043.61 | 57,043.14 | |

Best cost ($/year) | — | — | — | — | 57,035.52 | 57,035.52 | 57,043.14 | 57,043.14 | |

Mean cost ($/year) | — | — | — | — | 57,035.53 | 57035.52 | 57,043.15 | 57,043.14 | |

Standard deviation ( | — | — | — | — | 0.0167 | 0.0 | 0.0657 |

Convergence curves of the total cost of the 69-bus system with the CP load model at 100% loading.

Voltage profile of the 69-bus system before and after compensation with the CP load model at 100% loading.

Comparison of lines power flow of the 69-bus system before and after compensation with the CP load model at 100% loading.

The OCP results for the 69-bus system with voltage-dependent load models are tabulated in Table _{1-2} is decreased from 3,863.60 kW (in the uncompensated case) to 3,844.38 kW (in the compensated case). Besides, the power flow in most lines is improved after compensation. From the simulated results, it can be realized that the proposed SQI-SOS method can effectively deal with the problem in the voltage-dependent load models.

Comparative results of the 69-bus system after compensation with different load models.

Load type | Item | Uncompensated | Compensated | |
---|---|---|---|---|

SOS | SQI-SOS | |||

Industrial load | 0.9187 (65) | 0.9304 (65) | 0.9304 (65) | |

P_{loss}, kW | 175.0872 | 143.121 | 143.121 | |

PF_{overall} | 0.8752 | 0.9538 | 0.9538 | |

Total cost, $/year | 92,025.82 | 82,174.39 | 82,174.39 | |

Annual savings ($/year) | — | 9,851.42 | 9,851.42 | |

MPF, kW | 3,946.63 | 3,919.18 | 3,919.18 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Residential load | 0.9203 (65) | 0.9335 (65) | 0.9335 (65) | |

P_{loss}, kW | 170.8264 | 131.1693 | 131.1693 | |

PF_{overall} | 0.8516 | 0.9497 | 0.9497 | |

Total cost, $/year | 89,786.35 | 75,892.56 | 75,892.56 | |

Annual savings ($/year) | — | 13,893.79 | 13,893.79 | |

MPF, kW | 3,823.35 | 3,808.35 | 3,808.35 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Commercial load | 0.9222 (65) | 0.9351 (65) | 0.9351 (65) | |

P_{loss}, kW | 165.0464 | 123.2511 | 123.2511 | |

PF_{overall} | 0.8393 | 0.9442 | 0.9442 | |

Total cost, $/year | 86,748.37 | 71,580.75 | 71,580.75 | |

Annual savings ($/year) | — | 15,167.61 | 15,167.61 | |

MPF, kW | 3,731.56 | 3,728.19 | 3,728.19 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Mixed load | 0.9199 (65) | 0.9325 (65) | 0.9325 (65) | |

P_{loss}, kW | 171.3912 | 134.9734 | 134.9734 | |

PF_{overall} | 0.8606 | 0.9512 | 0.9512 | |

Total cost, $/year | 90,083.21 | 77,892.02 | 77,892.02 | |

Annual savings ($/year) | — | 12,191.18 | 12,191.18 | |

MPF, kW | 3,863.60 | 3,844.38 | 3,844.38 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) |

Optimal locations and sizes of capacitors for the 69-bus system with different load models.

Load type | Locations and injected kVArs | Total kVAr | Method |
---|---|---|---|

Industrial load | (18, 300); (61, 1150) | 1450 | SOS |

(18, 300); (61, 1150) | 1450 | SQI-SOS | |

Residential load | (18, 300); (61, 1150) | 1450 | SOS |

(18, 300); (61, 1150) | 1450 | SQI-SOS | |

Commercial load | (18, 300); (61, 1100) | 1400 | SOS |

(18, 300); (61, 1100) | 1400 | SQI-SOS | |

Mixed load | (18, 300); (61, 1150) | 1450 | SOS |

(18, 300); (61, 1150) | 1450 | SQI-SOS |

Annual cost of the 69-bus system before and after compensation with different load models.

Convergence curves of the total cost of the 69-bus system with the mixed load model.

Voltage profile of the 69-bus system before and after compensation with the mixed load model.

Comparison of lines power flow of the 69-bus system before and after compensation with the mixed load model.

The performance of the proposed SQI-SOS is finally tested on a large-scale radial distribution network with 119 buses. The total load demand of this system is 22.709 MW and 17.041 MVAr with a rated voltage of 11 kV. The system data are referred from [

Table _{c}, SOS_{c}, CSA [

Optimal siting and sizing of capacitors for the 119-bus system with the CP load model at different load levels.

Item | Load level—light 50% | ||||||||
---|---|---|---|---|---|---|---|---|---|

Method | CSA [ | IHA [ | FPA [ | ABC [ | MGABC [ | SOS_{c} | SQI-SOS_{c} | SOS | SQI-SOS |

Locations (kVArs) | 32 (0) | 39 (1200) | 32 (0) | 32 (500) | 33 (300) | 96 (561.52) | 50 (977.53) | 42 (400) | 32 (850) |

39 (1100) | 43 (0) | 40 (1100) | 35 (0) | 35 (750) | 107 (636.82) | 96 (548.25) | 111 (750) | 80 (800) | |

40 (0) | 70 (900) | 70 (750) | 40 (0) | 45 (300) | 80 (807.44) | 32 (899.34) | 80 (800) | 74 (750) | |

70 (350) | 74 (0) | 74 (1100) | 50 (0) | 46 (150) | 50 (982.00) | 80 (811.47) | 74 (750) | 111 (800) | |

74 (500) | 86 (0) | 89 (500) | 70 (0) | 49 (900) | 74 (753.84) | 107 (626.67) | 96 (550) | 96 (550) | |

86 (450) | 91 (600) | 104 (0) | 73 (750) | 54 (450) | 42 (393.18) | 42 (392.52) | 107 (600) | 107 (550) | |

108 (0) | 107 (0) | 109 (0) | 79 (500) | 71 (750) | 32 (904.33) | 111 (733.85) | 50 (950) | 50 (1000) | |

118 (1000) | 109 (0) | 112 (0) | 105 (300) | 76 (150) | 111 (753.84) | 74 (751.84) | 33 (750) | 42 (400) | |

118 (1000) | 118 (1000) | 106 (0) | 86 (750) | ||||||

109 (0) | 94 (300) | ||||||||

110 (950) | 101 (750) | ||||||||

110 (600) | |||||||||

111 (450) | |||||||||

114 (150) | |||||||||

115 (150) | |||||||||

Total kVAr | 3400 | 3700 | 3350 | 3900 | 6900 | 5777.11 | 5741.51 | 5550 | 5700 |

Total kVAr demand | 8520.55 | ||||||||

Load level—medium 75% | |||||||||

Locations (kVArs) | 32 (900) | 39 (1500) | 32 (0) | 32 (0) | 33 (600) | 96 (804.70) | 50 (1499.95) | 42 (550) | 32 (1050) |

39 (1500) | 43 (0) | 40 (1500) | 35 (800) | 35 (1200) | 107 (903.13) | 96 (820.33) | 111 (1100) | 80 (1100) | |

40 (0) | 70 (900) | 70 (750) | 40 (1200) | 45 (300) | 80 (1107.43) | 32 (1137.43) | 80 (1100) | 74 (1150) | |

70 (600) | 74 (600) | 74 (600) | 50 (450) | 46 (150) | 50 (1497.64) | 80 (1087.39) | 74 (1150) | 111 (1150) | |

74 (750) | 86 (0) | 89 (1500) | 70 (550) | 49 (1350) | 74 (1146.86) | 107 (904.62) | 96 (800) | 96 (800) | |

86 (700) | 91 (1500) | 104 (0) | 73 (750) | 54 (750) | 42 (553.04) | 42 (556.16) | 107 (950) | 107 (900) | |

108 (750) | 107 (0) | 109 (0) | 79 (850) | 71 (1050) | 32 (1122.45) | 111 (1132.41) | 50 (1500) | 50 (1500) | |

118 (1100) | 109 (0) | 112 (0) | 105 (0) | 76 (300) | 111 (1133.50) | 74 (1145.73) | 33 (1000) | 42 (600) | |

118 (1200) | 118 (1000) | 106 (0) | 86 (1050) | ||||||

109 (800) | 94 (450) | ||||||||

110 (1000) | 101 (1200) | ||||||||

110 (900) | |||||||||

111 (600) | |||||||||

114 (150) | |||||||||

115 (150) | |||||||||

Total kVAr | 6300 | 5700 | 5350 | 6400 | 10200 | 8268.79 | 8284.06 | 8150 | 8250 |

Total kVAr demand | 12780.83 | ||||||||

Load level—full 100% | |||||||||

Locations (kVArs) | 32 (1500) | 39 (1500) | 32 (1500) | 32 (850) | 33 (900) | 96 (804.85) | 50 (1499.95) | 42 (550) | 32 (1050) |

39 (1500) | 43 (1000) | 40 (1500) | 35 (1050) | 35 (1500) | 107 (903.63) | 96 (820.46) | 111 (1400) | 80 (1100) | |

40 (550) | 70 (1000) | 70 (850) | 40 (1300) | 45 (450) | 80 (1107.66) | 32 (1137.43) | 80 (1100) | 74 (1400) | |

70 (950) | 74 (1000) | 74 (1100) | 50 (800) | 46 (450) | 50 (1499.53) | 80 (1087.45) | 74 (1400) | 111 (1450) | |

74 (750) | 86 (900) | 89 (1500) | 70 (550) | 49 (1500) | 74 (1395.53) | 107 (904.64) | 96 (800) | 96 (800) | |

86 (1050) | 91 (1500) | 104 (500) | 73 (1300) | 54 (1050) | 42 (553.16) | 42 (556.20) | 107 (950) | 107 (900) | |

108 (1500) | 107 (850) | 109 (900) | 79 (1200) | 71 (1200) | 32 (1122.81) | 111 (1435.61) | 50 (1500) | 50 (1500) | |

118 (1200) | 109 (850) | 112 (250) | 105 (700) | 76 (450) | 111 (1436.75) | 74 (1408.13) | 33 (1000) | 42 (600) | |

118 (1200) | 118 (1150) | 106 (250) | 86 (1350) | ||||||

109 (800) | 94 (750) | ||||||||

110 (1200) | 101 (1500) | ||||||||

110 (1350) | |||||||||

111 (750) | |||||||||

114 (150) | |||||||||

115 (150) | |||||||||

Total kVAr | 9000 | 9800 | 9250 | 10000 | 13500 | 8823.94 | 8849.91 | 8700 | 8800 |

Total kVAr demand | 17041.1 | ||||||||

Fixed capacitors | SOS | (42, 400); (111, 750); (80, 800); (74, 750); (96, 550); (107, 600); (50, 950); (33, 750) | |||||||

SQI-SOS | (32, 850); (80, 800); (74, 750); (111, 800); (96, 550); (107, 550); (50, 1000); (42, 400) | ||||||||

Switched capacitors | SOS | (42, 150); (111, 650); (80, 300); (74, 650); (96, 250); (107, 350); (50, 550); (33, 250) | |||||||

SQI-SOS | (32, 200); (80, 300); (74, 650); (111, 650); (96, 250); (107, 350); (50, 500); (42, 200) |

The result comparison between the proposed SQI-SOS and other methods is represented in Table _{1-2}. Furthermore, the power flow on most lines after compensation is lower than that before compensation. The detail of power loss comparison before and after compensation corresponding to the full load level using the SQI-SOS method in each line is given in Table

Comparison results of the 119-bus system before and after compensation with the CP load model at different load levels.

Load level | Item | Uncomp. | Compensated | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

CSA [ | IHA [ | FPA [ | ABC [ | MGABC | SOS_{c} | SQI-SOS_{c} | SOS | SQI-SOS | |||

Light 50% | 0.9385 | 0.955 | 0.9552 | — | 0.9539 | 0.9566 | 0.9567 | 0.9567 | 0.9566 | 0.9566 | |

P_{loss}, kW | 297.15 | 208.96 | 206.8402 | 209.0868 | 207.52 | 206.65 | 194.8344 | 194.8289 | 195.4589 | 194.8911 | |

PF_{overall} | 0.7998 | 0.87 | 0.9188 | 0.914 | 0.8915 | 0.9883 | 0.9701 | 0.9693 | 0.9654 | 0.9685 | |

MPF, kW | 5233.93 | — | — | — | — | — | 5202.31 | 5202.31 | 5202.93 | 5202.32 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | |||||||

Medium 75% | 0.9049 | 0.933 | 0.92 | — | 0.9313 | 0.9336 | 0.9335 | 0.9335 | 0.9335 | 0.9335 | |

P_{loss}, kW | 697.33 | 471.01 | 473.1 | 488.83 | 471.78 | 482.01 | 451.2657 | 451.2566 | 452.5665 | 451.4294 | |

PF_{overall} | 0.7998 | 0.91 | 0.9207 | 0.9419 | 0.9068 | 0.9860 | 0.9636 | 0.9638 | 0.9618 | 0.9633 | |

MPF, kW | 7927.17 | — | — | — | — | — | 7852.32 | 7852.28 | 7853.62 | 7852.45 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | |||||||

Full 100% | 0.8688 | 0.906 | 0.902 | — | 0.90886 | 0.9073 | 0.9047 | 0.9049 | 0.9047 | 0.9047 | |

P_{loss}, kW | 1298.09 | 858.89 | 843.1459 | 853.1543 | 854.39 | 874.03 | 846.6694 | 846.2974 | 847.3224 | 847.0243 | |

PF_{overall} | 0.7998 | 0.92 | 0.9488 | 0.9419 | 0.9295 | 0.9844 | 0.9360 | 0.9363 | 0.9344 | 0.9357 | |

MPF, kW | 10677.92 | — | — | — | — | — | 10547.83 | 10547.60 | 10548.47 | 10548.23 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | (#_{1-2}) | |||||||

Total annual cost ($/year) | 440,236.40 | 332,077.38 | 332,573.27 | 336,216.33 | 337,983.73 | 359,583.69 | 323,491.68 | 323,488.99 | 323,578.50 | ||

Annual net savings ($/year) | — | 108,159.02 | 107,663.13 | 104,020.07 | 102,252.67 | 80,652.70 | 116,744.71 | 116,747.40 | 116,657.89 | ||

Worst cost ($/year) | — | — | — | — | — | — | 327346.43 | 325,776.65 | 326,745.33 | ||

Best cost ($/year) | — | — | — | — | — | — | 323,491.68 | 323,488.99 | 323,578.50 | ||

Mean cost ($/year) | — | — | — | — | — | — | 324,441.73 | 324,248.97 | 324,445.95 | ||

Standard deviation ( | — | — | — | — | — | — | 786.63 | 629.26 | 644.11 |

The sign “

Convergence curves of the total cost of the 119-bus system with the CP load model at 100% loading.

Voltage profile of the 119-bus system before and after compensation with the CP load model at 100% loading.

Comparison of lines power flow of the 119-bus system before and after compensation with the CP load model at 100% loading.

Results of real power loss in all lines of the 119-bus system before and after the installation of capacitors.

Line | Real power loss (kW) | |
---|---|---|

Uncompensated | Compensated by SQI-SOS | |

1-2 | 54.693 | 40.757 |

2-3 | 1.07 | 1.07 |

2–4 | 43.480 | 30.111 |

4-5 | 0.032 | 0.032 |

5-6 | 0.021 | 0.021 |

6-7 | 0.008 | 0.008 |

7-8 | 0.003 | 0.003 |

8-9 | 0.002 | 0.002 |

2–10 | 9.016 | 9.009 |

10-11 | 5.060 | 5.056 |

11-12 | 0.242 | 0.242 |

12-13 | 0.155 | 0.155 |

13-14 | 0.141 | 0.140 |

14-15 | 0.018 | 0.018 |

15-16 | 0.010 | 0.010 |

16-17 | 0.002 | 0.002 |

11–18 | 5.692 | 5.687 |

18-19 | 2.999 | 2.997 |

19-20 | 3.288 | 3.286 |

20-21 | 0.829 | 0.829 |

21-22 | 0.439 | 0.439 |

22-23 | 3.596 | 3.593 |

23-24 | 0.516 | 0.516 |

24-25 | 0.080 | 0.080 |

25-26 | 0.007 | 0.007 |

26-27 | 0.002 | 0.002 |

4–28 | 13.066 | 8.854 |

28-29 | 8.888 | 5.880 |

29-30 | 37.661 | 22.342 |

30-31 | 40.934 | 22.299 |

31-32 | 17.805 | 9.099 |

32-33 | 18.523 | 10.325 |

33-34 | 16.343 | 8.905 |

34-35 | 12.331 | 6.343 |

30–36 | 2.083 | 2.046 |

36-37 | 0.348 | 0.342 |

29–38 | 11.434 | 8.464 |

38-39 | 9.456 | 6.855 |

39-40 | 3.689 | 2.654 |

40-41 | 4.037 | 2.855 |

41-42 | 7.621 | 6.211 |

42-43 | 0.263 | 0.257 |

43-44 | 0.064 | 0.063 |

44-45 | 0.028 | 0.028 |

45-46 | 0.005 | 0.005 |

35–47 | 15.022 | 7.490 |

47-48 | 8.107 | 3.970 |

48-49 | 9.464 | 4.546 |

49-50 | 8.580 | 3.956 |

50-51 | 2.016 | 1.887 |

51-52 | 0.525 | 0.492 |

52-53 | 1.340 | 1.254 |

53-54 | 1.086 | 1.017 |

29–55 | 4.162 | 4.140 |

55-56 | 3.834 | 3.815 |

56-57 | 2.957 | 2.941 |

57-58 | 4.074 | 4.053 |

58-59 | 0.329 | 0.327 |

59-60 | 0.275 | 0.274 |

60-61 | 0.081 | 0.081 |

61-62 | 0.014 | 0.014 |

1–63 | 21.514 | 14.882 |

63-64 | 78.020 | 53.567 |

64-65 | 76.194 | 52.927 |

65-66 | 26.431 | 18.290 |

66-67 | 37.674 | 25.135 |

67-68 | 48.439 | 32.331 |

68-69 | 37.607 | 24.970 |

69-70 | 84.702 | 55.975 |

70-71 | 9.090 | 6.320 |

71-72 | 8.740 | 6.211 |

72-73 | 7.149 | 5.446 |

73-74 | 4.499 | 3.497 |

74-75 | 0.313 | 0.289 |

75-76 | 0.613 | 0.565 |

76-77 | 0.009 | 0.008 |

64–78 | 37.404 | 24.564 |

78-79 | 9.678 | 6.266 |

79-80 | 3.949 | 2.301 |

80-81 | 1.822 | 1.773 |

81-82 | 0.461 | 0.449 |

82-83 | 0.152 | 0.148 |

83-84 | 0.019 | 0.018 |

84-85 | 0.005 | 0.005 |

79–86 | 0.419 | 0.409 |

86-87 | 0.011 | 0.011 |

87-88 | 0.001 | 0.001 |

65–89 | 20.677 | 14.298 |

89-90 | 7.558 | 5.200 |

90-91 | 7.180 | 4.964 |

91-92 | 0.291 | 0.283 |

92-93 | 0.103 | 0.100 |

93-94 | 0.022 | 0.021 |

94-95 | 0.022 | 0.021 |

91–96 | 2.974 | 2.224 |

96-97 | 0.917 | 0.890 |

97-98 | 0.012 | 0.012 |

98-99 | 0.002 | 0.002 |

1-100 | 24.074 | 15.852 |

100-101 | 45.611 | 28.730 |

101-102 | 33.412 | 20.393 |

102-103 | 43.442 | 25.825 |

103-104 | 69.359 | 38.948 |

104-105 | 22.986 | 12.902 |

105-106 | 43.600 | 24.262 |

106-107 | 19.223 | 10.705 |

107-108 | 31.590 | 18.174 |

108-109 | 11.153 | 6.153 |

109-110 | 15.047 | 7.897 |

110-111 | 4.069 | 2.678 |

110-112 | 0.409 | 0.388 |

112-113 | 0.011 | 0.010 |

100-114 | 2.160 | 2.157 |

114-115 | 0.210 | 0.210 |

115-116 | 0.248 | 0.248 |

116-117 | 0.031 | 0.031 |

117-118 | 0.006 | 0.006 |

To sum up, the proposed SQI-SOS reveals an outstanding performance when compared with the previous methods available in the literature in terms of the obtained minimum total annual cost. Moreover, it can be realized that the proposed SQI-SOS is more efficient and robust than the original SOS algorithm in solving the OCP problem in this scenario.

The simulation results for the 119-bus system after compensation with the nonlinear load models are given in Table _{1-2} is reduced to 10,344.84 kW as compared to 10,360.30 kW for the case without capacitors. Besides, the power flow on most lines is also improved after compensation. In summary, the proposed SQI-SOS again shows the ability to cope effectively with the nonlinear load models when applied to the large-scale distribution system.

Comparative results of the 119-bus system after compensation with the different load models.

Load type | Item | Uncompensated | Compensated | |
---|---|---|---|---|

SOS | SQI-SOS | |||

Industrial load | 0.8888 (77) | 0.9069 (77) | 0.9081 (77) | |

P_{loss}, kW | 996.4081 | 804.7756 | 802.4568 | |

PF_{overall} | 0.8627 | 0.9564 | 0.9581 | |

Total cost, $/year | 523,712.09 | 462,490.07 | 461,721.30 | |

Annual savings ($/year) | — | 61,222.01 | 61,990.79 | |

MPF, kW | 10,532.80 | 10,483.41 | 10,480.68 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Residential load | V_{min} (bus) | 0.8910 (77) | 0.9121 (77) | 0.9121 (77) |

P_{loss}, kW | 977.8818 | 755.8888 | 753.3145 | |

PF_{overall} | 0.8361 | 0.9393 | 0.9426 | |

Total cost, $/year | 513,974.67 | 433,645.15 | 433,042.09 | |

Annual savings ($/year) | — | 80,329.51 | 80,932.57 | |

MPF, kW | 10,273.71 | 10,273.42 | 10,272.36 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Commercial load | 0.8939 (77) | 0.9139 (77) | 0.9148 (77) | |

P_{loss}, kW | 948.3656 | 748.8654 | 738.5942 | |

PF_{overall} | 0.8222 | 0.9244 | 0.9356 | |

Total cost, $/year | 498,460.96 | 426,653.67 | 423,805.13 | |

Annual savings ($/year) | — | 71,807.28 | 74,655.83 | |

MPF, kW | 10,069.95 | 10,069.89 | 10,069.86 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) | ||

Mixed load | 0.8905 (77) | 0.9103 (77) | 0.9104 (77) | |

P_{loss}, kW | 979.2163 | 769.1398 | 765.9692 | |

PF_{overall} | 0.8464 | 0.9481 | 0.9522 | |

Total cost, $/year | 514,676.09 | 442,259.89 | 441,643.41 | |

Annual savings ($/year) | — | 72,416.20 | 73,032.68 | |

MPF, kW | 10,360.30 | 10,347.46 | 10,344.84 | |

(#_{1-2}) | (#_{1-2}) | (#_{1-2}) |

Optimal locations and kVArs of capacitors for the 119-bus system with different load models.

Load type | Locations and injected kVArs | Total kVAr | Method |
---|---|---|---|

Industrial load | (50, 1500); (41, 950); (107, 1050); (91, 950); (72, 1450); (35, 1050); (110, 1500); (80, 1250) | 9700 | SOS |

(80, 1250); (32, 1450); (50, 1500); (96, 900); (41, 850); (107, 1050); (111, 1400); (74, 1450) | 9850 | SQI-SOS | |

Residential load | (74, 1350); (111, 1100); (52, 650); (50, 1300); (96, 900); (41, 950); (109, 1150); (80, 1250) | 8650 | SOS |

(74, 1350); (32, 1000); (96, 1000); (50, 1400); (80, 1100); (42, 650); (111, 1400); (107, 1000) | 8900 | SQI-SOS | |

Commercial load | (111, 1450); (108, 900); (6, 1100); (81, 1050); (58, 500); (52, 700); (75, 1250); (96, 600) | 7550 | SOS |

(107, 1000); (74, 1300); (4, 1300); (96, 850); (51, 750); (111, 1400); (58, 600); (80, 1200) | 8400 | SQI-SOS | |

Mixed load | (74, 1350); (80, 1300); (50, 1500); (41, 950); (47, 850); (91, 800); (111, 1400); (107, 1050) | 9200 | SOS |

(74, 1350); (50, 1500); (111, 1050); (96, 900); (80, 1200); (32, 1400); (41, 900); (108, 1250) | 9550 | SQI-SOS |

Annual cost of the 119-bus system before and after compensation with different load models.

Convergence curves of the total cost of the 119-bus system with the mixed load model.

Voltage profile of the 119-bus system before and after compensation with the mixed load model.

Comparison of lines power flow of the 119-bus system before and after compensation with the mixed load model.

In this paper, the proposed SQI-SOS has been successfully implemented for solving the OCP problem with the objective of total operating cost reduction at different load models. The proposed SQI-SOS is an improvement of the SOS method to enhance its searchability in terms of the solution quality and convergence speed. The main advantage of the SQI-SOS is that it has a simple structure with only two controllable parameters; thus, it is easy to be implemented to optimization problems. The proposed SQI-SOS has been tested on different large-scale distribution systems with 33, 69, and 119 buses. The original SOS method has also been implemented to validate the exploitation capacity of the SQI-SOS. The result simulations have confirmed that the convergence speed of the proposed SQI-SOS is faster than that of the conventional SOS. In addition, the SQI-SOS method has also offered a better solution quality than other compared methods such as SOS and many other methods in terms of the total annual cost. It proves that the SQI-SOS has a good performance to compete with other optimization methods in terms of the solution quality and convergence rate for the OCP problem as well as for other optimization problems in power systems. Therefore, the proposed SQI-SOS can be a favorable method for solving the OCP problem in distribution systems.

Detailed results of real power loss in all lines of 33-, 69-, and 119-bus test systems are tabulated in Tables

_{P}:

Energy cost per each kWh

_{loss,i}:

Real power loss at any load level

_{i}:

Duration of load level

_{C}:

Purchase rate of the capacitor per kVAr

_{C,j}:

Size of the capacitor placed at the bus

_{ci}:

Installation cost

_{o}:

Operating cost

Number of load levels

Designed period

Number of capacitor locations

_{slack}:

Active power of the slack bus

_{slack}:

Reactive power of the slack bus

_{D,i}:

Active power demand at the

_{Do,i}:

Active power demand operating point at the

_{D,i}:

Reactive power demand at the

_{Do,i}:

Reactive power demand operating point at the

_{L,j}:

Real power loss at the

_{L,j}:

Reactive power loss at the

_{C,j}:

Amount of reactive power at the

Minimum reactive power limit of the compensated bus

Maximum reactive power limit of the compensated bus

_{TC}:

Total reactive power injected by capacitors

_{l}:

Number of load buses

_{br}:

Total number of branches

Number of buses where to install the capacitors

Minimum voltage level at bus

Maximum voltage level at bus

Voltage at the

Operating point voltage

_{b}:

Total number of buses of the network

_{k}:

Power flow through the

The maximum limit of power flow through the

_{overall}:

System overall power factor

_{min}:

Minimum system power factor limit at the slack bus

_{max}:

Maximum system power factor limit at the slack bus.

No data were used to support this study.

The authors declare that there are no conflicts of interest regarding the publication of this paper.