# Optimal Location and Sizing of Distributed Generators in DC Networks Using a Hybrid Method Based on Parallel PBIL and PSO

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## Abstract

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## 1. Introduction

#### 1.1. General Context

#### 1.2. Motivation

#### 1.3. Literature Review

#### 1.4. Contributions and Scope

- The implementation of a mathematical formulation that considers the characteristics of DC grids, including the set of constraints associated with the representation of DC electrical networks with distributed generation, which is needed to perform an optimal integration of the distributed generators for the minimization of power loss.
- A methodology for the optimal integration of DGs in DC grids that uses a parallel master-slave strategy to obtain the satisfactory results in terms of the quality of the solution and the processing time.
- The adoption of parallel processing tools for the planning of DC grids in order to improve the performance of the methodologies for sizing and location.
- The adoption of sequential programming in order to enable the use of free programming languages for the optimal sizing and location of DGs in DC grids and avoid the use of specialized and costly software.

#### 1.5. Organization of the Document

## 2. Mathematical Formulation

## 3. Proposed Hybrid Method

#### 3.1. Codification of the Solution

#### 3.2. Master Stage: Optimal Location of DGs by Using PPBIL

Algorithm 1: Pseudo-code proposed for the PPBIL algorithm |

- Step 1. Assign initial conditions:In this step, four main parameters are selected to start the iterative process of the PBIL. First, the population size ($PS$) is assigned to it. In order to ensure that the evaluation of the complete population is done in one cycle of the parallel units, the $PS$ selected is kept equal to the number of cores of the processor. The second parameter is the initial probability of each option in the probability matrix. For an adequate exploration of the solution space, the PBIL starts the iterative process with the same probability for all the possible options. The initial probability for this application is equal to $0.5$ due to which each bus has two options: the option of locating (50%) or not locating (50%) a DG on the bus. The third parameter includes the type of learning rate (LR), for example, linear, exponential, sigmoidal or bell shaped. The LR bounds are within the range of 0 to 1 [49]. In regard to the location of DGs, it was found in a heuristic form that the best type of LR is sigmoidal and the best values for the LR bounds are 0.25 and 0.50, respectively. The fourth parameter is the stopping criterion, which corresponds to the entropy tolerance, and it is selected as 0.1 in accordance with the work reported in [32].
- Step 2. Initialization of the probability matrix (PM):The PM is formed by (n) columns, which represent the elements to be considered while formulating the solution of the problem, and (m) rows, which represent the possible solutions for each element. For the location of DGs, n is equal to ($\left|\mathcal{N}\right|-1$), thereby excluding the slack bus, and $m=2$ since only two options are available for each node. Figure 2 shows an example of the probability matrix.At the start of the iterative process, all the components of the PM start with the same probability of $1/m=0.5$. Moreover, the PM must satisfy the constraint given in (9) where $P\left(j,h\right)$ represents the probability of the option j to be selected on element h. Such a restriction ensures that the accumulative probability of the options for a single element is 100%.$$\sum _{j=1}^{m}P\left(j,h\right)=1\phantom{\rule{1.em}{0ex}}\forall h=1,2,\dots ,n$$
- Step 3. Generate the population:The population is generated by using the PM data. The objective is to create individuals by using the information of each option with each element of the PM to avoid identical individuals and guarantee an adequate exploration of the solution space. Therefore, when an individual is repeated, it is replaced by a randomly generated one.
- Step 4. Evaluate the Fitness Function ($FF$):The evaluation of the $FF$ of each individual is the step that takes up more time in the PBIL process. Therefore, the solution proposed in this work uses the parallel processing structure reported in [32], which enables the evaluation of the $FF$ of all the individuals at the same time. If the population size is not equal to the number of cores in the processor (W), the time required by the paralleling process ($PPT$) is equal to $PPT=CEIL(PS/W)\xb7MTRP$ where $MTRP$ is the maximum time required to evaluate the $FF$ of an individual and $CEIL\left(\xb7\right)$ returns the integer that is higher than or equal to a real number. In this paper, the evaluation of the $FF$ is performed in the slave stage, which optimizes the size of the distributed generators for each individual of the PPBIL population.
- Step 5. Select the best individual:The individual with the best $FF$ value is selected. In this case, the individual with the minimum power loss fulfills the set of constraints discussed in Section 2.
- Steps 6 and 7. Update the probability matrix and the learning rate:The values of the probability matrix are updated by using Equation (10) where $P{(i,j)}_{Old}$ is the non-updated probability of positions $(i,j)$ and $P{(i,j)}_{Act}$ is the updated probability.$${P\left(i,j\right)}_{Act}={P\left(i,j\right)}_{Old}+\left(1-{P\left(i,j\right)}_{Old}\right)\xb7LR\phantom{\rule{4pt}{0ex}}$$Then, the index k is assigned to the option with the highest fitness value of the element j, and the update of the probability matrix is performed by using Equation (11); the probability of the option with the highest fitness value is increased while the probability of the other options (different to k) are reduced. In such an equation, $P{(i,j)}_{New}$ denotes the actualized value of the probability at positions $(i,j)$. Finally, all the elements must satisfy the restriction (9) after updating the probability matrix.$$\begin{array}{c}\hfill P{\left(i,j\right)}_{\mathrm{New}}=\left\{\begin{array}{cc}P{\left(i,j\right)}_{\mathrm{Act}}& \mathrm{if}\phantom{\rule{0.277778em}{0ex}}i=k\\ \left(1-{P\left(i,j\right)}_{\mathrm{Act}}\right)\xb7\frac{{P\left(i,j\right)}_{\mathrm{Old}}}{1-{P\left(i,j\right)}_{\mathrm{Old}}}& \mathrm{if}\phantom{\rule{0.277778em}{0ex}}i\ne k\end{array}\right.\end{array}$$Algorithm 1 reports that the learning rate must also be updated. The value of LR is updated by using Equation (12) where $L{R}_{min}$ and $L{R}_{max}$ are the bounds assigned in Step 1. Moreover, ${E}_{n}$ is the entropy of the $PM$ in the first iteration of the algorithm ${E}_{n}=1$; therefore, the probability matrix exhibits a high dispersion of the data, as all the options have the same probability in each element [45,49].$$LR=L{R}_{max}-\frac{L{R}_{max}-L{R}_{min}}{1+{e}^{-10\xb7\left({E}_{n}-0.5\right)}}$$
- Steps 8 and 9. Calculate the entropy and evaluate the stopping criteria:The entropy ${E}_{n}$ enables the quantification of the distribution of the probabilities of the solution. In that way, the probabilities in $PM$ are completely dispersed when ${E}_{n}=1$, and they converge into a single solution when ${E}_{n}=0$. However, a particular tolerance ${E}_{TOL}$ is used as an approximation to ${E}_{n}=0$ in order to allow converging into a solution in a finite time. Equation (13) describes the mathematical formulation of the entropy, which is calculated by adding all the probabilities in $PM$ and using the total number of elements n to normalize the entropy into $0\le {E}_{n}\le 1$. The stopping criterion for the PPBIL algorithm is ${E}_{n}<{E}_{TOL}$ with ${E}_{TOL}=0.1$. The selection of the ${E}_{TOL}$ value has been discussed in [32].$${E}_{n}=\frac{-{\displaystyle \sum _{i=1}^{m}}{\displaystyle \sum _{j=1}^{n-1}}{P}_{\left(i,j\right)}\xb7log\left[{P}_{\left(i,j\right)}\right]}{n}$$
- Steps 10 and 11. Extract the best individual from the probability matrix and evaluate the $FF$:When the PPBIL algorithm reaches the stopping criterion, the best solution is extracted from $PM$ by selecting the options on each element with the highest probability. Finally, the $FF$ of the best configuration is calculated, which enables the evaluation of the impact associated with the best solution on the DC network.

#### 3.3. Slave Problem: Optimal Sizing of DGs by Using PSO

Algorithm 2: Pseudo-code for the PSO algorithm |

## 4. Test Scenarios and Considerations

#### 4.1. Test Systems

#### 4.1.1. 10 Bus Test System

#### 4.1.2. The 21 Bus Test System

#### 4.1.3. 69 Bus Test System

#### 4.2. General Considerations

- All nodes except the slack node are candidates for locating DGs.
- The maximum power level that can be supplied by each generator is 1.2 p.u for the 10 bus system, 1.5 p.u for the 21 bus system and 12 p.u for the 69 bus system. The minimum power level of each generator is 0 p.u. for all the cases [15].
- The current ratings of the DC lines for all the test systems were calculated by using a power flow analysis without taking the integration of DGs into consideration. The calibers and maximum current levels were defined based on those values of the branch currents and by using Table 310.16 of the National Electrical Code (NEC) for operation at ambient temperatures of 75 and 30 °C, respectively. For the 10 and 21 bus test systems, the caliber assigned was 900 kcmil with a current limit of 520 A (5.2 p.u. considering the base values of both voltage and power), while the caliber assigned for the 69 bus test system was 400 kcmil with a current limit of 335 A (42.11 p.u.). The high p.u. value for the current rating of the 69 bus is caused by the base value of the current (7.89 p.u.), which is obtained from the base values of both the voltage and the power. Finally, the test systems consider the installation of the same caliber for all the branches forming the electrical network.
- In accordance with the recommendation given in [58], all test systems were assigned a maximum distributed power penetration level equal to 40% of the total power supplied by the slack bus without considering the integration of DGs (base case). Such a constraint was considered in order to account for the restriction of the distributed generation present in some national electrical regulations; hence, the proposed mathematical formulation provides a general solution because such a limit could be set to zero when it is not needed. Moreover, this consideration enables the analysis of the impact of the penetration level of the DG on the electrical system.

#### 4.3. Fitness Function

#### 4.4. Comparison Methods and Parameters

## 5. Simulation Results

#### 5.1. 10 Bus Test System Results

#### 5.2. 21 Bus Test System Results

#### 5.3. 69 Bus Test System Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Codification strategy for the location and sizing of distributed generators (DGs) in direct current (DC) networks.

**Figure 3.**Proposed method based on the Population-Based Incremental Learning (PPBIL) and the Particle Swarm Optimization (PSO) algorithms.

**Figure 4.**Electrical configuration for the 10-bus test feeder [11].

**Figure 7.**Results obtained with the 10 bus test system. (

**a**) Reduction of power losses (%); (

**b**) Processing Time (s); (

**c**) Reduction of square error voltage (%); (

**d**) Worst Voltage profile (p.u).

**Figure 9.**Results obtained with the 21 bus test system. (

**a**) Reduction of power losses (%); (

**b**) Processing Time (s); (

**c**) Reduction of square error voltage (%); (

**d**) Worst Voltage profile (p.u).

**Figure 11.**Results obtained with the 69 bus test system. (

**a**) Reduction of power losses (%); (

**b**) Processing Time (s); (

**c**) Reduction of square error voltage (%); (

**d**) Worst Voltage profile (p.u).

From | To | R (pu) | Type of Node | P (pu) − R (pu) |
---|---|---|---|---|

1 (slack) | 2 | 0.0050 | Step-node | - |

2 | 3 | 0.0015 | P | −0.8 |

2 | 4 | 0.0020 | P | −1.3 |

4 | 5 | 0.0018 | P | −0.5 |

2 | 6 | 0.0023 | R | 2.0 |

6 | 7 | 0.0017 | Step-node | 0 |

7 | 8 | 0.0021 | P | −0.3 |

7 | 9 | 0.0013 | P | −0.7 |

3 | 10 | 0.0015 | R | 1.25 |

From | To | R (pu) | P (pu) | From | To | R (pu) | P (pu) |
---|---|---|---|---|---|---|---|

1(slack) | 2 | 0.0053 | −0.70 | 11 | 12 | 0.0079 | −0.68 |

1 | 3 | 0.0054 | 0.00 | 11 | 13 | 0.0078 | −0.10 |

3 | 4 | 0.0054 | −0.36 | 10 | 14 | 0.0083 | 0.00 |

4 | 5 | 0.0063 | −0.04 | 14 | 15 | 0.0065 | −0.22 |

4 | 6 | 0.0051 | −0.36 | 15 | 16 | 0.0064 | −0.23 |

3 | 7 | 0.0037 | 0.00 | 16 | 17 | 0.0074 | −0.43 |

7 | 8 | 0.0079 | −0.32 | 16 | 18 | 0.0081 | −0.34 |

7 | 9 | 0.0072 | −0.80 | 14 | 19 | 0.0078 | −0.09 |

3 | 10 | 0.0053 | 0.00 | 19 | 20 | 0.0084 | −0.21 |

10 | 11 | 0.0038 | −0.45 | 19 | 21 | 0.0082 | −0.21 |

**Table 3.**Parameters of the location techniques [32].

Method | GA | PMC | PPBIL |
---|---|---|---|

Population size | 12 | 12 | 12 |

Selection method | Tournament | Repeated random sampling | Initial probability: 0.5 |

Learning rate | Cross over: simple | - - - | Sigmoidal LRmin: 0.25 LRmax: 0.50 |

Mutation | Binary simple | - - - | Random population |

Stopping criterion | Maximum generational cycles: 40 | Maximum iterations: 10 | Entropy: 0.1 |

Method | CGA | BH | PSO |
---|---|---|---|

Number of particles | 30 | 30 | 30 |

Selection method | Tournament | Event horizon radius | Cognitive and social component: 1.4 |

Update population method | Cross over: promedium | Cognitive and social component | Speed (max–min): (0.1–0.1) Inertia (max–min): (0.7–0.001) |

Mutation | Random population | Random population | R1 = R2: Random |

Stopping criterion | Maximum iterations: 200 Iteration without improving: 50 | Maximum iterations: 200 Iteration without improving: 50 | Maximum iterations: 200 Iteration without improving: 50 |

Method: Location/ Sizing | DG Location/ Size (kW) | ${\mathit{P}}_{\mathit{loss}}$ (kW)/ %${\mathit{P}}_{\mathit{loss}}$ Reduction | ${\mathit{V}}_{\mathit{error}}$ (p.u)/ %${\mathit{V}}_{\mathit{error}}$ Reduction | Worst Voltage Profile (p.u)/ Bus | Maximum Current (p.u) | Time (s) |
---|---|---|---|---|---|---|

Without DGs | - - - | 14.3628 | 0.0075 | 0.9690/ 9 | 5.2 | - - - |

PPBIL/ PSO | 5/67.12 9/82.51 10/49.10 | 4.8526/ 66.21 | 0.0024/ 67.85 | 0.9829/ 8 | 2.91 | 38.30 |

PPBIL/ CGA | 5/88.90 9/74.85 10/34.95 | 4.8777/ 66.03 | 0.0024/ 67.59 | 0.9826/ 8 | 2.91 | 91.07 |

PPBIL/ BH | 4/85.84 5/59.24 9/48.38 | 5.2283/ 63.59 | 0.0026/ 64.54 | 0.9812/ 8 | 2.97 | 85.35 |

PMC/ PSO | 3/66.56 5/63.53 7/68.73 | 5.0169/ 65.06 | 0.0025/ 66.25 | 0.9820/ 9 | 2.91 | 45.79 |

PMC/ CGA | 4/79.85 9/79.18 10/39.75 | 4.8899/ 65.95 | 0.0024/ 67.11 | 0.9824/ 5 | 2.91 | 111.37 |

PMC/ BH | 2/63.56 4/29.79 8/99.17 | 5.6770/ 60.47 | 0.0025/ 65.61 | 0.9810/ 5 | 2.98 | 25.38 |

GA/ PSO | 4/75.42 9/71.95 10/51.45 | 4.8869/ 65.97 | 0.0024/ 66.75 | 0.9823/ 5 | 2.91 | 108.93 |

GA/ CGA | 4/67.68 9/84.71 10/46.18 | 4.9024/ 65.86 | 0.0024/ 67.35 | 0.9821/ 5 | 4.97 | 228.25 |

GA/ BH | 5/86.42 6/65.21 9/33.90 | 5.5162/ 61.59 | 0.0027/ 63.82 | 0.9811/ 10 | 1.59 | 56.47 |

Method: Location/ Sizing | DG Location/ Size (kW) | ${\mathit{P}}_{\mathit{loss}}$ (kW)/ %${\mathit{P}}_{\mathit{loss}}$ Reduction | ${\mathit{V}}_{\mathit{error}}$ (p.u)/ %${\mathit{V}}_{\mathit{error}}$ Reduction | Worst Voltage Profile (p.u)/ Bus | Maximum Current (p.u) | Time (s) |
---|---|---|---|---|---|---|

Without DGs | - - - | 27.6034 | 0.0567 | 0.9211/ 17 | 5.2 | - - - |

PPBIL/ PSO | 12/73.79 16/118.34 20/40.50 | 5.9697/ 78.37 | 0.0070/ 87.61 | 0.9759/ 9 | 2.57 | 68.74 |

PPBIL/ CGA | 12/81.11 16/110.27 21/40.46 | 6.0040/ 78.24 | 0.0073/ 86.95 | 0.9759/ 9 | 2.57 | 176.93 |

PPBIL/ BH | 12/86.84 16/91.90 19/50.46 | 6.1891/ 77.57 | 0.0082/ 85.49 | 0.9738/ 17 | 2.6 | 85.66 |

PMC/ PSO | 8/32.38 14/111.37 17/88.88 | 7.2063/ 73.89 | 0.0081/ 85.65 | 0.9704/ 12 | 2.58 | 74.99 |

PMC/ CGA | 7/52.63 11/147.70 13/32.09 | 11.5531/ 58.14 | 0.0209/ 63.13 | 0.9456/ 17 | 2.62 | 210.30 |

PMC/ BH | 4/19.28 9/118.47 12/76.08 | 13.1477/ 52.39 | 0.0276/ 51.22 | 0.9386/ 17 | 2.83 | 41.53 |

GA/ PSO | 3/31.61 8/55.46 17/145.56 | 8.6873/ 68.52 | 0.0103/ 81.82 | 0.9673/ 12 | 2.59 | 238.74 |

GA/ CGA | 9/59.30 11/134.55 13/38.77 | 11.1495/ 59.60 | 0.0209/ 62.97 | 0.9452/ 17 | 2.62 | 535.23 |

GA/ BH | 12/25.62 14/78.41 18/64.83 | 9.7973/ 64.50 | 0.0151/ 73.21 | 0.9633/ 17 | 3.24 | 110.06 |

Method: Location/ Sizing | DG Location/ Size (kW) | ${\mathit{P}}_{\mathit{loss}}$ (kW)/ %${\mathit{P}}_{\mathit{loss}}$ Reduction | ${\mathit{V}}_{\mathit{error}}$ (p.u)/ %${\mathit{V}}_{\mathit{error}}$ Reduction | Worst Voltage Profile (p.u)/ Bus | Maximum Current (p.u) | Time (s) |
---|---|---|---|---|---|---|

Without DGs | - - - | 153.8476 | 0.0769 | 0.9274/ 69 | 42.11 | - - - |

PPBIL/ PSO | 23/169.58 61/1200 67/247.65 | 13.8469/ 90.99 | 0.0062/ 91.91 | 0.9848/ 64 | 22.85 | 222.17 |

PPBIL/ CGA | 27/148.99 62/1167.96 65/294.86 | 14.8686/ 90.33 | 0.0062/ 91.84 | 0.9843/ 21 | 22.92 | 542.57 |

PPBIL/ BH | 60/448.52 62/395.63 65/296.11 | 36.1161/ 76.52 | 0.0185/ 75.88 | 0.9719/ 64 | 27.85 | 133.77 |

PMC/ PSO | 10/417.23 32/$1.02\times {10}^{-6}$ 63/1200 | 23.9597/ 84.42 | 0.0117/ 84.74 | 0.9764/ 69 | 22.95 | 225.80 |

PMC/ CGA | 14/942.14 37/222.53 46/157.29 | 126.45/ 17.80 | 0.0505/ 34.27 | 0.9333/ 69 | 27.07 | 675.88 |

PMC/ BH | 2/189.23 10/1042.66 33/51.06 | 122.71/ 20.23 | 0.0548/ 28.68 | 0.9339/ 69 | 29.18 | 135.00 |

GA/ PSO | 14/179.33 58/237.90 62/1200 | 17.4946/ 88.62 | 0.0084/ 89.04 | 0.9804/ 69 | 22.89 | 839.67 |

GA/ CGA | 59/446.07 63/1170.76 | 19.0251/ 87.63 | 0.0090/ 88.25 | 0.9779/ 27 | 22.91 | 1611.72 |

GA/ BH | 8/60.85 14/406.04 67/673.68 | 55.8518/ 63.69 | 0.0210/ 72.65 | 0.9596/ 61 | 28.04 | 305.31 |

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## Share and Cite

**MDPI and ACS Style**

Grisales-Noreña, L.F.; Montoya, O.D.; Ramos-Paja, C.A.; Hernandez-Escobedo, Q.; Perea-Moreno, A.-J. Optimal Location and Sizing of Distributed Generators in DC Networks Using a Hybrid Method Based on Parallel PBIL and PSO. *Electronics* **2020**, *9*, 1808.
https://doi.org/10.3390/electronics9111808

**AMA Style**

Grisales-Noreña LF, Montoya OD, Ramos-Paja CA, Hernandez-Escobedo Q, Perea-Moreno A-J. Optimal Location and Sizing of Distributed Generators in DC Networks Using a Hybrid Method Based on Parallel PBIL and PSO. *Electronics*. 2020; 9(11):1808.
https://doi.org/10.3390/electronics9111808

**Chicago/Turabian Style**

Grisales-Noreña, Luis Fernando, Oscar Danilo Montoya, Carlos Andrés Ramos-Paja, Quetzalcoatl Hernandez-Escobedo, and Alberto-Jesus Perea-Moreno. 2020. "Optimal Location and Sizing of Distributed Generators in DC Networks Using a Hybrid Method Based on Parallel PBIL and PSO" *Electronics* 9, no. 11: 1808.
https://doi.org/10.3390/electronics9111808