# Total Optimization of Energy Networks in a Smart City by Multi-Population Global-Best Modified Brain Storm Optimization with Migration

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## Abstract

**:**

_{2}emission. The energy and environmental problem of smart city can be formulated as a mixed integer nonlinear programming (MINLP) problem. Therefore, evolutionary computation methods including variations of recently developed Brain Storm Optimization (BSO) such as Global-best BSO (GBSO), Modified BSO (MBSO), and Global-best Modified BSO (GMBSO) have been adopted to the problem. However, there is still room for improvement of quality of solution. Evolutionary computation methods with multi-population have been applied to various problems and verified to improve quality of solution. Therefore, the approach can be expected to improve quality of solution. The proposed MS-GMBSO utilizes only migration for multi-population models instead of abest which is the best individual among all sub-populations so far and both migration and abest. Various multi-population models, migration policies, the number of sub-populations, and migration topologies are also investigated. It is verified that the proposed MP-GMBSO based method with migration using ring topology, the W-B policy, and 320 individuals is the most effective among all of multi-population parameters.

## 1. Introduction

_{2}emission [1,2]. Smart city is an eco-city which can realize low carbon emission and energy consumption by Internet of Things (IoT), renewable energies, storage batteries, and so on. For example, in Japan, after the Great East Japan Earthquake, introduction of the smart city was investigated, especially in Tohoku area [3].

_{2}emission reduction in the actual smart city. Therefore, a smart city model including various sector models should be utilized for the evaluation. Static models which can treat various energy balances and dynamic models which can treat dynamic behaviors have been developed separately in each sector [4,5,6,7]. However, a smart city model, which can calculate various environmental and energy loads among all sectors considering environmental and energy flow among various sectors, had not been developed. Considering these backgrounds, a smart city model was developed in order to evaluate energy costs and CO

_{2}emission of the whole smart city considering environmental and energy flow among various sectors in Japan [8,9,10].

_{2}emission by Particle Swarm Optimization (PSO) [11], Differential Evolution (DE) [12], Differential Evolutionary PSO (DEEPSO) [13], BSO [14], MBSO [15], GBSO [16], and GMBSO [17]. In addition, considering energy facility characteristics, energy load characteristics, cost characteristics, and continuity of weekday operation of various energy facilities, the authors have proposed reduction methods of search space in order to solve the problem effectively [11,12]. Therefore, GMBSO considering the reduction of search space can obtain the highest quality solution so far [17]. However, there is still room for improvement of quality of solution.

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- A proposal of a new evolutionary computation method, namely MP-GMBSO with migration, in order to realize improvement of quality of solution,
- -
- An application of MP-GMBSO with migration-based method to total optimization of energy networks in a smart city,
- -
- Verification of efficacy of the conventional GMBSO based method for total optimization of smart city by comparing with the conventional DEEPSO, BSO, MBSO, and GBSO based methods,
- -
- Verification of efficacy of the proposed MP-GMBSO based method with migration for total optimization of smart city by comparing with the original GMBSO (GMBSO with one population) based method, and the MP-GMBSO based methods with various interaction model (only using migration, only using abest, and using both migration and abest), various policies, various topologies, various numbers of individuals, and various numbers of sub-populations,
- -
- It is verified that quality of solution is the most improved by the proposed MP-GMBSO with migration-based method using the ring topology with 16 sub-populations and 320 individuals, and the W-B policy (the worst individual of a sub-population is substituted by the best individual of other sub-populations) among all of multi-population parameters.

## 2. Smart City Model

#### 2.1. A Summary of the Whole Smart City Model

_{2}emission or energy costs of whole smart city can be quantitatively calculated by the model. Various sectors are included in the model (see Figure 1). The model deals with energy supply-side and demand-side groups. Natural gas and electric power utilities, and drinking water and wastewater treatment plant sectors are categorized into the energy supply-side group. The other sectors are categorized into the demand-side group [8,27].

#### 2.2. Sector Models in the Supply-Side Group

#### 2.3. Sector Models in the Demand-Side Group

## 3. Problem Formulation of Total Optimization of Energy Networks in a Smart City

_{2}emission. Purchase costs of various fuels by electric power and gas companies from resource companies are not included in the problem.

#### 3.1. Decision Variables

#### 3.2. Objective Function

_{2}emission in the whole city as the third term. The functions are shown in the following Equation (1):

_{2}emission, $EC$ is a coefficient between purchased electric power and CO

_{2}emission, and ${w}_{1}$, ${w}_{2}$, and ${w}_{3}$ are weighting coefficients (${w}_{1}+{w}_{2}+{w}_{3}=1$).

#### 3.3. Constraints

- (1)
- Energy balances: Electric power, hot and cold heat, and steam energy balances are considered. These energy balances are expressed using the following equation:$${g}_{nr}\left({y}_{i},{z}_{i}\right)=0,\left(n=1,\dots ,NumSec,\text{}r=1,\dots ,Num{E}_{n},i=1,\dots ,NumDim\right)$$
- (2)
- Facility characteristics: Efficiency functions of facilities, and upper and lower bounds of various facilities in each sector can be expressed using the following equation:$${h}_{nq}\left({y}_{i},{z}_{i}\right)\le 0,\hspace{1em}\left(n=1,\dots ,NumSec,\text{}q=1,\dots ,Num{F}_{n},i=1,\dots ,NumDim\right)$$Efficiency of facility should be sometimes expressed with nonlinear functions. Hence, the problem is considered as one of mixed-integer nonlinear optimization problems (MINLPs) and evolutionary computation methods should be utilized in order to treat the problem.

## 4. The Proposed Multi-Population Global-Best Modified Brain Storm Optimization with Migration

#### 4.1. Overview of BSO

- Step 1
**Initialization**: Randomly generate $NumInd$ individuals and calculate the objective function values of $NI$ individuals.- Step 2
**Clustering**: The k-means method is applied to divide $NumInd$ individuals into $K$ clusters.- Step 3
**Generation of New individual**: One or two clusters are randomly selected, and new individuals are generated.- Step 4
**Selection of individuals**: Individuals which are newly generated are compared with the current individuals which have the same individual indices of the newly generated individuals. Keep the better ones and the individuals are stored as the current individuals.- Step 5
**Evaluation of individuals**: The newly stored $NumInd$ individuals are evaluated.- Step 6
- The procedure can be stopped and go to Step 7 when the iteration number reaches the maximum iteration number which is pre-determined. Otherwise, go to Step 2 and repeat the procedures.
- Step 7
- The objective function value and the finally obtained variables are output as a set of the final solution.

#### 4.1.1. Clustering of BSO

- Step 2-1
- Clustering: The k-means algorithm is applied to divide $NumInd$ individuals into $K$ clusters.
- Step 2-2
- A value ${r}_{clustering}$ is randomly generated in random (1,0).
- Step 2-3
- Individuals are ranked in each cluster.If ${r}_{clustering}\ge {p}_{clustering}$ (a pre-determined probability),the best individual is set as the cluster center in each cluster,Otherwise,One individual is randomly selected in each cluster, and the selected individual is set as the cluster center.

#### 4.1.2. Generation of New Individual of BSO

#### 4.2. Overview of MBSO

#### 4.2.1. Clustering of MBSO

- Step 1
- $K$ different individuals are randomly selected from the current generation as group centers of $K$ groups.
- Step 2
- Calculate distances between the individuals and each group center. Distances to all group centers are compared. The individuals are assigned to the closest group.
- Step 3
- Individuals are ranked in each cluster.If ${r}_{clustering}\ge {p}_{clustering}$(a pre-determined probability),the best individual is set as the cluster center in each cluster,Otherwise,One individual is randomly selected in each cluster, and the selected individual is set as the cluster center.

#### 4.2.2. Generation of New Individuals in MBSO

#### 4.3. Overview of GBSO

#### 4.3.1. Clustering

- Step 1
- Individuals are ranked using calculated values of the objective function.
- Step 2
- $NumInd$ individuals are divided into $K$ groups using (8).$$g\left(i\right)=\left(r\left(i\right)-1\right)\%K+1\left(i=1,..,NumInd\right))$$

#### 4.3.2. Generation of New Individuals of GBSO

#### 4.4. Overview of GMBSO

- Step 1
**Initialization**: $NumInd$ individuals are randomly generated and evaluated.- Step 2
**Clustering**: $NumInd$ individuals are divided into $K$ clusters by Fitness-based grouping explained in 4.3.- Step 3
**Generation of new individuals**: Randomly select one cluster or two clusters. When the condition (9) is satisfied, information of “gbest” is applied to ${x}_{ij}^{old}$ using (11). Then, new individuals are generated using Equation (7) explained in 4.2.- Step 4
**Selection**: The individuals which are newly generated are compared with the current individuals with the same individual indices. The better one is kept and stored as the current individual.- Step 5
**Evaluation**: The $NumInd$ individuals are evaluated.- Step 6
- The procedure can be stopped and go to Step 7 if the number of current iteration reaches the maximum number of iteration which is pre-determined. Otherwise, go to Step 2 and repeat the procedures.
- Step 7
- The objective function value and the finally obtained variables are output as a final solution.

#### 4.5. Overview of the Proposed MP-GMBSO

- -
**The number of sub-populations**: the number of sub-populations which performs GMBSO independently.- -
**Migration topology**: topological structures of sub-populations. Ring topology with 2, 4, 8, and 16 sub-populations (see Figure 6a–d), trigonal pyramid topology with four sub-populations, a cube topology with eight sub-populations, or hyper-cube topology with 16 sub-populations (see Figure 7a–c) can be utilized.- -
**Migration interval**: how often searching individuals migrate.- -
**Migration policy**: the way to select searching individuals for replacement in the receiving sub-population and the way to select searching individuals for migration in the sending sub-population. The worst individual of the receiving sub-population is replaced by the best individual of the sending sub-populations (W-B) (see Figure 8) , a randomly selected individual of the receiving sub-population is replaced by the best individual of the sending sub-populations (R-B) , the best individual of the receiving sub-population is replaced by the best individual of the sending sub-populations (B-B), the worst individual of the receiving sub-population is replaced by a randomly selected individual of the sending sub-populations (W-R), a randomly selected individual of the receiving sub-population is replaced by a randomly selected individual of the sending sub-populations (R-R), the best individual of the receiving sub-population is replaced by a randomly selected individual of the sending sub-populations (B-R), the worst individual of the receiving sub-population is replaced by the worst individual of the sending sub-populations (W-W), a randomly selected individual of the receiving sub-population is replaced by the worst individual of the sending sub-populations (R-W), or the best individual of the receiving sub-population is replaced by the worst individual of the sending sub-populations (B-W), can be utilized .

#### 4.6. Update Equations of The Proposed MP-GMBSO with Migration

- -
- the proposed MP-GMBSO with only migration model (see Figure 3):$${x}_{ijs}^{old}={x}_{ijs}^{old}+rand\left(1,0\right)\times C\times \left({x}_{js}^{gbest}-{x}_{ijs}^{old}\right)\left(i=1,\dots ,NumInd,j=1,\dots ,NumDim,s=1,\dots ,NumSubPop\right)$$
- -
- only abest model (see Figure 4), and a model using both migration and abest (see Figure 5):$${x}_{ijs}^{old}={x}_{ijs}^{old}+rand\left(1,0\right)\times C\times \left({x}_{j}^{abest}-{x}_{ijs}^{old}\right)\left(i=1,\dots ,NumI,j=1,\dots ,NumDim,s=1,\dots ,NumSubPop\right)$$

## 5. Total Optimization of Energy Networks in a Smart City by Multi-Population Global-Best Modified Brain Storm Optimization with Migration

#### 5.1. Cutout Transformation Function

#### 5.2. Reduction of Search Space

#### 5.3. The Proposed Total Optimization Algorithm of Smart City by MP-GMBSO with Migration

- Step 1
**Initialization**: Divide all individuals into $NumSubPop$ sub-populations. Generate initial individuals at each sub-population considering the reduced search space for a smart city.- Step 2
- Calculate the objective function at all individuals in sub-populations.
- Step 3
**Clustering**:- Step 3-1
- Generate clusters using $FbG$ in all sub-populations.
- Step 3-2
- Calculate objective functions of all individuals in each cluster in all sub-populations.
- Step 3-3
- Rank individuals ascending order.
- Step 3-4
- The highest rank individuals at each cluster of all sub-populations are set as cluster centers. If ${p}_{clustering}>rand\left(1,0\right),$ randomly generate a new individual and replace a cluster center with the newly generated individual.

- Step 4
**Generation of new individual**: When condition (9) is satisfied, information of “gbest” is applied to ${x}_{ijs}^{old}$ using (13) or (14). New individuals are generated considering several conditions explained in 4.1.2 using Equation (15).- Step 5
**Selection**:- Step 5-1
- The objective function values are calculated for all individuals.
- Step 5-2
- The new individual is compared with the current individual with the same individual index. Keep the better one and the individual is stored as the current individual in all sub-populations.

- Step 6
**Evaluation**: Calculate objective function values of individuals in all sub-populations. The best individual is updated when the objective function value of the individual is better than the current best individual.- Step 7
- Individuals are migrated when the current iteration number reaches the migration interval which is pre-determined.
- Step 8
- the whole procedure is stopped and go to Step 9 when the current number of iterations reaches the maximum number of iterations which is pre-determined. Otherwise, go to Step 3 and repeat the procedures.
- Step 9
- The finally obtained objective function value and the best operational values are output as a final solution.

## 6. Simulations

#### 6.1. Simulation Conditions

- Case 1:
- a goal of a general smart city considering all three terms of the objective function equally,${w}_{1}:0.333,{w}_{2}:0.333,{w}_{3}:0.333$
- Case 2:
- a goal of an industrial park which usually concentrates only minimization of total energy cost${w}_{1}:1,{w}_{2}:0,{w}_{3}:0$
- Case 3:
- a goal of local government of a city which usually concentrates only minimization of CO
_{2}emission.${w}_{1}:0,{w}_{2}:0,{w}_{3}:1$

- $\tau $: 0.2, ${\tau}^{\prime}$: 0.006, $p$: 0.75, the initial weight coefficients (A, B, and C): 0.5, the number of clones: 1.

- ${p}_{clustering}:0.5$, ${p}_{generation}:0.5$, ${p}_{OneCluster}:0.2$, ${p}_{TwoCluster}:0.2$, $pr:$ 0.2 (for MBSO, GMBSO), ${c}_{max}:0.7$, ${c}_{min}:0.2$ (for GBSO and GMBSO).

- -
- The initial weight coefficients of each term (D) is set to 0.5,
- -
- The number of sub-populations (NumSubPop): 2, 4, 8, and 16,
- -
- The total number of individuals (NumInd): 1280 (640 individuals/sub-population for 2 sub-populations, 320 individuals/sub-population for 4 sub-populations, 160 individuals/sub-population for 8 sub-populations, and 80 individuals/sub-population for 16 sub-populations), 640 (320 individuals/sub-population for 2 sub-populations, 160 individuals/sub-population for 4 sub-populations, 80 individuals/sub-population for 8 sub-populations, and 40 individuals/sub-population for 16 sub-populations), 320 (160 individuals/sub-population for 2 sub-populations, 80 individuals/sub-population for 4 sub-populations, 40 individuals/sub-population for 8 sub-populations, and 20 individuals/sub-population for 16 sub-populations), 160 (80 individuals/sub-population for 2 sub-populations, 40 individuals/sub-population for 4 sub-populations, 20 individuals/sub-population for 8 sub-populations, and 10 individuals/sub-population for 16 sub-populations), and 80 (40 individuals/sub-population for 2 sub-populations, 20 individuals/sub-population for 4 sub-populations, 10 individuals/sub-population for 8 sub-populations).
- -
- -
- Migration interval: 10 to 100 in 10 increments,
- -
- Migration policy: W-B, R-B, B-B, W-R, R-R, B-R, W-W, R-W, B-W.

- -
- The number of trials: 50
- -
- The maximum iteration number for BSO, GBSO, MBSO, and GMBSO based methods: 2000
- -
- The maximum iteration number for DEEPSO based method is set to 1000

#### 6.2. Simulation Results

**bold numbers**). It can be considered that the GMBSO based method can focus on intensification more than the other methods including the multi-swarm DEEPSO based method with a model using both migration and abest [26]. Therefore, the only migration model is the best model especially for the proposed MP-GMBSO based model in order to balance diversification and intensification. In addition, the mean rank by the only migration model with 16 sub-populations is the best among all parameters in the table.

**bold numbers**), and the average rank by the “W-B” policy with 16 sub-populations and the ring topology is the best among all parameters in the table.

**(bold numbers)**. It is also studied that there are significant differences at 0.05 significance level among all parameters in Table 6.

**bold numbers**). Namely, the proposed method with such conditions can balance diversification and intensification the most effectively for the problem. In addition, the average rank is the best when the number of individuals is set to 320 with 16 sub-populations using ring topology among all parameters in Table 8.

**bold numbers**). It is also verified that there are significant differences at 0.05 significance level among all methods.

**bold numbers**). Table 11 shows comparison of the best facility operation of an industrial model for Case 3 using only migration model, and W-B migration policy with ring topology when the number of individuals is set to 320 among various numbers of sub-populations. Electric power which is purchased from electric power utility should be reduced and electric power which is output of a GTG should be increased a whole day for reduction of CO

_{2}emission in the model. It was verified that electric power which is purchased from electric power utility can be reduced and electric power output of a GTG can be increased at most a whole day by the proposed method with 16 (

**bold numbers**).

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Configuration of a smart city model (© 2018 IEEE [27]).

**Figure 2.**A configuration of an industrial model (© 2018 IEEE [27]).

**Figure 3.**A concept of a multi-population model using only migration (© 2018 IEEE [27]).

**Figure 4.**A concept of a multi-population model using only abest (© 2018 IEEE [27]).

**Figure 5.**A concept of a multi-population model using both migration and abest (© 2018 IEEE [27]).

**Figure 6.**Examples of ring topologies using various numbers of sub-populations (© 2018 IEEE [27]).

**Figure 7.**Topologies with three and more connections using various numbers of sub-populations (© 2018 IEEE [27]).

**Figure 9.**A conventional search space (inside solid lines) and a reduced search space (shaded area) of a heat storage tank (© 2018 IEEE [27]).

Sector | Decision Variables |
---|---|

Industrial sector | Output of electric power of a gas turbine generator (GTG), Heat output of turbo refrigerators (TRs), Heat output of stream refrigerators (SRs), Charged or discharged electric power of a storage battery (SB) |

Building sector | Output of electric power of a GTG, Heat output of TRs, Heat output of SRs |

Residential sector | Heat output of SRs, Output of electric power of a fuel cell, Heat output of a heat pump water heater, Charged or discharged electric power of a SB |

Railroad sector | The number of passengers/h, Average of journey distance by one passenger/h, The number of operated trains/h, The numbers of passenger cars/set, Average of journey distance by one train/h, Average of speed/h, The number of passengers/car |

Drinking water treatment plant sector | Inflow from river, Inflow of water into a service reservoir, Output of electric power output of a co-generator (CoGen), Charged or discharged electric power of a SB |

Wastewater treatment plant sector | Input of Pumped wastewater, Output of electric power of a CoGen, Charged or discharged electric power of a SB |

**Table 2.**Comparison of the mean, the minimum, the maximum, and the standard deviation of the objective function value among conventional DEEPSO, BSO, MBSO, GBSO, and the proposed GMBSO based methods with 80 individuals.

^{1}

Case | Mean | Min. | Max. | Std. | |
---|---|---|---|---|---|

1 | DEEPSO | 100.00 | 98.75 | 101.63 | 0.57 |

BSO | 97.13 | 96.46 | 97.96 | 0.30 | |

GBSO | 95.94 | 95.55 | 97.03 | 0.26 | |

MBSO | 97.20 | 96.75 | 97.66 | 0.20 | |

GMBSO | 95.06 | 94.90 | 95.29 | 0.09 | |

2 | DEEPSO | 100.00 | 99.53 | 100.58 | 0.20 |

BSO | 99.28 | 98.98 | 99.60 | 0.14 | |

GBSO | 98.29 | 98.22 | 98.42 | 0.04 | |

MBSO | 99.38 | 99.15 | 99.50 | 0.06 | |

GMBSO | 98.26 | 98.17 | 98.36 | 0.04 | |

3 | DEEPSO | 100.00 | 99.44 | 100.88 | 0.32 |

BSO | 99.64 | 99.38 | 99.87 | 0.09 | |

GBSO | 99.36 | 99.12 | 99.53 | 0.10 | |

MBSO | 98.37 | 98.30 | 98.46 | 0.04 | |

GMBSO | 98.10 | 98.05 | 98.16 | 0.03 |

^{1}All values are calculated when the mean of the objective function value of the proposed method with a single population is set to 100%.

**Table 3.**Results of average ranks and Friedman Test through 50 trials among the conventional DEEPSO, BSO, GBSO, MBSO, and the proposed GMBSO based methods.

DEEPSO | BSO | GBSO | MBSO | GMBSO | p-value | |
---|---|---|---|---|---|---|

Case 1 | 5 | 3.34 | 2 | 3.66 | 1 | $2.26\times {10}^{-36}$ |

Case 2 | 5 | 3.22 | 1.6 | 3.78 | 1.4 | $9.75\times {10}^{-37}$ |

Case 3 | 4.92 | 4.08 | 3 | 2 | 1 | $2.26\times {10}^{-39}$ |

**Table 4.**Comparison of the mean, the minimum, the maximum, the standard deviation, and average rank of the objective function value of the optimal objective function values among various numbers of sub-populations, and topologies using both migration with the W-B policy and abest, only migration with the W-B policy, and only abest with 640 individuals through 50 trials for Case 1 and a p-value by Friedman test.

^{1}

Model | Mig. Policy | # of sub-pop. | Ring | Trigonal Pyramid/Cube/Hyper-Cube | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Min. | Max. | Std. | Ave. Rank | Mean | Min. | Max. | Std. | Ave. Rank | |||

- | - | 1 | 100.00 | 99.90 | 100.17 | 0.06 | 18.82 | - | - | - | - | - |

Abest | - | 2 | 99.94 | 99.82 | 100.06 | 0.05 | 17.44 | - | - | - | - | - |

4 | 99.89 | 99.79 | 99.99 | 0.04 | 16.22 | - | - | - | - | - | ||

8 | 99.80 | 99.73 | 99.89 | 0.03 | 11.66 | - | - | - | - | - | ||

16 | 99.71 | 99.59 | 99.83 | 0.04 | 5.64 | - | - | - | - | - | ||

Abest & Mig. | W-B | 2 | 99.85 | 99.73 | 99.92 | 0.05 | 14.28 | - | - | - | - | - |

4 | 99.79 | 99.67 | 99.87 | 0.05 | 10.52 | 99.80 | 99.71 | 99.91 | 0.05 | 11.02 | ||

8 | 99.73 | 99.62 | 99.86 | 0.05 | 6.76 | 99.75 | 99.53 | 99.86 | 0.06 | 8.3 | ||

16 | 99.67 | 99.5 | 99.83 | 0.08 | 4.52 | 99.73 | 99.53 | 99.87 | 0.08 | 7.04 | ||

Mig. | W-B | 2 | 99.86 | 99.78 | 99.97 | 0.04 | 15.46 | - | - | - | - | - |

4 | 99.78 | 99.69 | 99.89 | 0.05 | 9.94 | 99.79 | 99.71 | 99.87 | 0.04 | 11.18 | ||

8 | 99.72 | 99.59 | 99.82 | 0.06 | 6.18 | 99.73 | 99.6 | 99.89 | 0.06 | 6.8 | ||

16 | 99.63 | 99.45 | 99.82 | 0.09 | 3.14 | 99.68 | 99.52 | 99.91 | 0.08 | 5.08 | ||

p-value | $4.89\times {10}^{-120}$ |

^{1}All values are calculated when the mean of the objective function value of the proposed method with a single population is set to 100%.

**Table 5.**The best values of the mean, the minimum, the maximum, the standard deviation, and the average rank of the objective function value of the optimal objective function values of each migration model with 640 individuals through 50 trials for Case 1 from Table 4 and a p-value by Friedman test.

^{1}

Model | Mig. Policy | # of sub-pop. | Ring | ||||
---|---|---|---|---|---|---|---|

Mean | Min. | Max. | Std. | Ave. Rank | |||

- | - | 1 | 100.00 | 99.90 | 100.17 | 0.06 | 4 |

Abest | - | 16 | 99.71 | 99.59 | 99.83 | 0.04 | 2.4 |

Abest & Mig. | W-B | 16 | 99.67 | 99.5 | 99.83 | 0.08 | 2.02 |

Mig. | W-B | 16 | 99.63 | 99.45 | 99.82 | 0.09 | 1.58 |

p-value | $1.48\times {10}^{-21}$ |

^{1}All values are calculated when the mean of the objective function value of the proposed method with a single population is set to 100%.

**Table 6.**Comparison of the mean, the minimum, the maximum, the standard deviation values, and the average rank of the optimal objective function values using only migration model among various migration policies, various topologies, and various numbers of sub-populations with 640 individuals through 50 trials for Case 1 and a p-value by Friedman test.

^{1}

Policy | NSP | Ring | Ave. Rank | Cube/Trigonal Pyramid/Hypercube | Ave. Rank | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Min. | Max. | Std. | Mean | Min. | Max. | Std. | ||||

- | 1 | 100 | 99.9 | 100.17 | 0.06 | 38.78 | - | - | - | - | - |

B-B | 2 | 99.93 | 99.82 | 100.08 | 0.05 | 33.48 | - | - | - | - | - |

4 | 99.89 | 99.75 | 100.05 | 0.05 | 28.48 | 99.89 | 99.8 | 100 | 0.05 | 27.94 | |

8 | 99.82 | 99.72 | 99.88 | 0.04 | 19.14 | 99.81 | 99.72 | 99.95 | 0.05 | 18.78 | |

16 | 99.71 | 99.61 | 99.85 | 0.06 | 8.2 | 99.71 | 99.61 | 99.79 | 0.04 | 7.88 | |

W-W | 2 | $6.54\times {10}^{10}$ | $1.17\times {10}^{9}$ | $1.89\times {10}^{11}$ | $5.06\times {10}^{10}$ | 48.12 | - | - | - | - | - |

4 | $7.35\times {10}^{10}$ | $4.41\times {10}^{8}$ | $3.25\times {10}^{11}$ | $5.42\times {10}^{10}$ | 48.82 | $8.56\times {10}^{10}$ | $1.62\times {10}^{9}$ | $2.28\times {10}^{11}$ | $5.49\times {10}^{10}$ | 49.46 | |

8 | $1.13\times {10}^{11}$ | $8.82\times {10}^{8}$ | $3.29\times {10}^{11}$ | $7.45\times {10}^{10}$ | 51.2 | $9.55\times {10}^{10}$ | $1.11\times {10}^{10}$ | $2.48\times {10}^{11}$ | $6.09\times {10}^{10}$ | 50.32 | |

16 | $1.27\times {10}^{11}$ | 101.55 | $2.80\times {10}^{11}$ | $8.34\times {10}^{10}$ | 52.24 | $1.36\times {10}^{11}$ | 102.12 | $4.13\times {10}^{11}$ | $8.45\times {10}^{10}$ | 52.52 | |

W-B | 2 | 99.86 | 99.78 | 99.97 | 0.04 | 25.6 | - | - | - | - | |

4 | 99.78 | 99.69 | 99.89 | 0.05 | 14.96 | 99.79 | 99.71 | 99.87 | 0.04 | 16.62 | |

8 | 99.72 | 99.59 | 99.82 | 0.06 | 8.98 | 99.73 | 99.6 | 99.89 | 0.06 | 10.42 | |

16 | 99.63 | 99.45 | 99.82 | 0.09 | 4.54 | 99.68 | 99.52 | 99.91 | 0.08 | 7.54 | |

B-W | 2 | $1.59\times {10}^{11}$ | $4.60\times {10}^{10}$ | $3.63\times {10}^{11}$ | $7.17\times {10}^{10}$ | 54.6 | - | - | - | - | - |

4 | $1.95\times {10}^{11}$ | $3.75\times {10}^{10}$ | $3.39\times {10}^{11}$ | $7.09\times {10}^{10}$ | 56.8 | $2.00\times {10}^{11}$ | $3.67\times {10}^{10}$ | $3.74\times {10}^{11}$ | $8.06\times {10}^{10}$ | 56.98 | |

8 | $2.64\times {10}^{11}$ | $5.38\times {10}^{10}$ | $4.89\times {10}^{11}$ | $9.61\times {10}^{10}$ | 59.36 | $2.12\times {10}^{11}$ | $4.96\times {10}^{10}$ | $4.06\times {10}^{11}$ | $8.03\times {10}^{10}$ | 57.22 | |

16 | $3.58\times {10}^{11}$ | $6.11\times {10}^{10}$ | $6.25\times {10}^{11}$ | $1.49\times {10}^{11}$ | 61.16 | $3.18\times {10}^{11}$ | $2.26\times {10}^{10}$ | $5.83\times {10}^{11}$ | $1.21\times {10}^{11}$ | 60.44 | |

B-R | 2 | 99.92 | 99.81 | 99.99 | 0.04 | 31.46 | - | - | - | - | - |

4 | 100.14 | 100.02 | 100.25 | 0.05 | 42.56 | 100.11 | 100.02 | 100.22 | 0.05 | 42.26 | |

8 | 99.95 | 99.84 | 100.05 | 0.06 | 34.72 | 99.95 | 99.83 | 100.06 | 0.05 | 35.32 | |

16 | 99.92 | 99.81 | 99.99 | 0.04 | 31.46 | 99.88 | 99.79 | 99.96 | 0.04 | 27.84 | |

R-B | 2 | 99.96 | 99.83 | 100.05 | 0.06 | 35.46 | - | - | - | - | - |

4 | 99.89 | 99.76 | 99.98 | 0.05 | 28.2 | 99.91 | 99.8 | 100.05 | 0.05 | 30.84 | |

8 | 99.82 | 99.72 | 99.92 | 0.05 | 19.16 | 99.81 | 99.69 | 99.9 | 0.04 | 18.88 | |

16 | 99.7 | 99.58 | 99.84 | 0.06 | 7.34 | 99.7 | 99.58 | 99.89 | 0.06 | 7.3 | |

W-R | 2 | 99.99 | 99.87 | 100.09 | 0.06 | 38 | - | - | - | - | - |

4 | 99.91 | 99.83 | 100 | 0.04 | 31.04 | 99.9 | 99.75 | 99.96 | 0.04 | 29.84 | |

8 | 99.82 | 99.7 | 99.89 | 0.04 | 19.24 | 99.76 | 99.65 | 99.84 | 0.04 | 12.58 | |

16 | 99.69 | 99.61 | 99.77 | 0.03 | 6.14 | 99.64 | 99.55 | 99.77 | 0.05 | 3.74 | |

R-W | 2 | $8.58\times {10}^{10}$ | $9.25\times {10}^{8}$ | $3.48\times {10}^{11}$ | $7.48\times {10}^{10}$ | 49.42 | - | - | - | - | - |

4 | $1.17\times {10}^{11}$ | $8.86\times {10}^{8}$ | $4.16\times {10}^{11}$ | $9.66\times {10}^{10}$ | 51.16 | $1.11\times {10}^{11}$ | $7.72\times {10}^{8}$ | $3.55\times {10}^{11}$ | $8.10\times {10}^{10}$ | 51 | |

8 | $1.61\times {10}^{11}$ | $1.62\times {10}^{10}$ | $7.86\times {10}^{11}$ | $1.49\times {10}^{11}$ | 52.92 | $1.42\times {10}^{11}$ | $7.88\times {10}^{9}$ | $5.17\times {10}^{11}$ | $1.13\times {10}^{11}$ | 52.6 | |

16 | $3.07\times {10}^{11}$ | $5.37\times {10}^{10}$ | $1.19\times {10}^{12}$ | $2.25\times {10}^{11}$ | 58.84 | $2.85\times {10}^{11}$ | $4.49\times {10}^{10}$ | $1.02\times {10}^{12}$ | $1.85\times {10}^{11}$ | 58.78 | |

R-R | 2 | 99.93 | 99.85 | 100.05 | 0.04 | 33.42 | - | - | - | - | 26.56 |

4 | 99.87 | 99.79 | 99.99 | 0.04 | 26.54 | 99.94 | 99.85 | 100.07 | 0.04 | 18.48 | |

8 | 99.81 | 99.74 | 99.89 | 0.03 | 18.32 | 99.81 | 99.66 | 99.93 | 0.05 | 9.4 | |

16 | 99.7 | 99.62 | 99.84 | 0.05 | 7.56 | 99.71 | 99.6 | 99.87 | 0.06 | 26.56 | |

p-value | 0 |

^{1}All values are calculated when the mean of the objective function value of the proposed method with a single population is set to 100%. NSP stands for number of sub-populations.

**Table 7.**The best values of the mean, the minimum, the maximum, the standard deviation values, and the average rank of the optimal objective function values using only migration model of each migration policy with 640 individuals through 50 trials for Case 1 from Table 6 and a p-value by Friedman test.

^{1}

Policy | NSP | Ring | Ave. Rank | |||
---|---|---|---|---|---|---|

Mean | Min. | Max. | Std. | |||

- | 1 | 100 | 99.9 | 100.17 | 0.06 | 6.9 |

B-B | 16 | 99.71 | 99.61 | 99.85 | 0.06 | 3.42 |

W-W | 16 | $1.27\times {10}^{10}$ | 101.55 | $2.80\times {10}^{11}$ | $8.34\times {10}^{10}$ | 8.32 |

W-B | 16 | 99.63 | 99.45 | 99.82 | 0.09 | 2.16 |

B-W | 2 | $1.59\times {10}^{11}$ | $4.60\times {10}^{10}$ | $3.63\times {10}^{11}$ | $7.17\times {10}^{10}$ | 9.58 |

B-R | 16 | 99.92 | 99.81 | 99.99 | 0.04 | 6.04 |

R-B | 16 | 99.7 | 99.58 | 99.84 | 0.06 | 3.16 |

W-R | 16 | 99.69 | 99.61 | 99.77 | 0.03 | 3 |

R-W | 16 | $3.07\times {10}^{11}$ | $5.37\times {10}^{10}$ | $1.19\times {10}^{12}$ | $2.25\times {10}^{11}$ | 9.1 |

R-R | 16 | 99.7 | 99.62 | 99.84 | 0.05 | 3.32 |

p-value | $9.02\times {10}^{-79}$ |

^{1}All values are calculated when the mean of the objective function value of the proposed method with a single population is set to 100%.

**Table 8.**Comparison of the mean, the minimum, the maximum, the standard deviation, and average rank of the objective function value of the optimal objective function values among various migration topologies with W-B policy, and various numbers of sub-populations using only migration with various number of individuals through 50 trials for Case 1 and a p-value by Friedman test.

^{1}

NI | NSP | Ring | Ave.Rank | Cube/Trigonal Pyramid/Hypercube | Ave.Rank | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Mean | Min. | Max. | Std. | Mean | Min. | Max. | Std. | ||||

40 | 1 | 99.85 | 99.73 | 100.06 | 0.06 | 38.8 | - | - | - | - | - |

2 | 99.93 | 99.73 | 100.21 | 0.11 | 34.1 | - | - | - | - | - | |

4 | $3.71\times {10}^{7}$ | 99.78 | $5.32\times {10}^{18}$ | $1.26\times {10}^{8}$ | 39.42 | 100.03 | 99.79 | 100.33 | 0.12 | 38.8 | |

8 | $5.71\times {10}^{7}$ | 100.08 | $6.02\times {10}^{10}$ | $1.06\times {10}^{10}$ | 45.64 | $5.72\times {10}^{8}$ | 99.96 | $6.81\times {10}^{9}$ | $1.34\times {10}^{9}$ | 44.24 | |

80 | 1 | 99.91 | 99.81 | 100.02 | 0.04 | 34.32 | - | - | - | - | - |

2 | 99.79 | 99.64 | 100.03 | 0.08 | 22.84 | - | - | - | - | - | |

4 | 99.81 | 99.63 | 100.01 | 0.09 | 24.02 | 99.86 | 99.65 | 100.09 | 0.10 | 12.1 | |

8 | 99.87 | 99.64 | 100.18 | 0.12 | 29.06 | 99.86 | 99.71 | 100.11 | 0.09 | 19.08 | |

16 | 99.72 | 99.54 | 99.89 | 0.08 | 14.72 | 99.91 | 99.70 | 100.12 | 0.10 | 31.56 | |

160 | 1 | 99.95 | 99.82 | 100.14 | 0.07 | 36.64 | - | - | - | - | - |

2 | 99.77 | 99.68 | 99.90 | 0.05 | 31.34 | - | - | - | - | - | |

4 | 99.77 | 99.58 | 99.99 | 0.08 | 16.42 | 99.77 | 99.61 | 100.03 | 0.08 | 15.78 | |

8 | 99.73 | 99.56 | 99.89 | 0.08 | 7 | 99.73 | 99.52 | 99.98 | 0.11 | 8.48 | |

16 | 99.57 | 99.42 | 99.86 | 0.09 | 3.64 | 99.67 | 99.53 | 99.92 | 0.09 | 7.66 | |

320 | 1 | 99.97 | 99.83 | 100.10 | 0.06 | 38.38 | - | - | - | - | - |

2 | 99.81 | 99.68 | 99.97 | 0.05 | 36.56 | - | - | - | - | - | |

4 | 99.77 | 99.64 | 99.88 | 0.05 | 27.64 | 99.77 | 99.64 | 99.87 | 0.05 | 27.52 | |

8 | 99.71 | 99.56 | 99.81 | 0.06 | 13.74 | 99.69 | 99.54 | 99.82 | 0.07 | 10.02 | |

16 | 99.56 | 99.41 | 99.77 | 0.09 | 2 | 99.66 | 99.51 | 99.93 | 0.09 | 4.38 | |

640 | 1 | 100.00 | 99.90 | 100.17 | 0.06 | 39.64 | - | - | - | - | - |

2 | 99.86 | 99.78 | 99.97 | 0.04 | 31.02 | - | - | - | - | - | |

4 | 99.78 | 99.69 | 99.89 | 0.05 | 21.46 | 99.79 | 99.71 | 99.87 | 0.04 | 23.24 | |

8 | 99.72 | 99.59 | 99.82 | 0.06 | 14.88 | 99.73 | 99.60 | 99.89 | 0.06 | 16.12 | |

16 | 99.63 | 99.45 | 99.82 | 0.09 | 8.58 | 99.68 | 99.52 | 99.91 | 0.08 | 12.16 | |

1280 | 1 | 100.75 | 100.59 | 100.91 | 0.07 | 44.66 | - | - | - | - | - |

2 | 99.88 | 99.79 | 99.96 | 0.04 | 32.34 | - | - | - | - | - | |

4 | 99.83 | 99.76 | 99.89 | 0.04 | 25.9 | 99.84 | 99.75 | 100.03 | 0.05 | 27.6 | |

8 | 99.77 | 99.66 | 99.84 | 0.04 | 18.86 | 99.77 | 99.68 | 99.88 | 0.04 | 19.7 | |

16 | 99.68 | 99.57 | 99.82 | 0.06 | 12.74 | 99.70 | 99.51 | 99.82 | 0.06 | 15.2 | |

p-value | 0 |

^{1}All values are calculated when the mean of the objective function value of the proposed method with a single population and

**640**individuals is set to 100%. NI stands for number of individuals.

**Table 9.**The best values of the mean, the minimum, the maximum, the standard deviation, and average rank of the objective function value of the optimal objective function values of each number of individuals with WB policy using only migration and ring topology through 50 trials for Case 1 from Table 8 and a p-value by Friedman test.

^{1}

NI | NSP | Ring | Ave. Rank | |||
---|---|---|---|---|---|---|

Mean | Min. | Max. | Std. | |||

40 | 1 | 99.85 | 99.73 | 100.06 | 0.06 | 6 |

80 | 16 | 99.72 | 99.54 | 99.89 | 0.08 | 1.42 |

160 | 16 | 99.57 | 99.42 | 99.86 | 0.09 | 3.08 |

320 | 16 | 99.56 | 99.41 | 99.77 | 0.09 | 2.2 |

640 | 16 | 99.63 | 99.45 | 99.82 | 0.09 | 4.06 |

1280 | 16 | 99.68 | 99.57 | 99.82 | 0.06 | 6 |

^{1}All values are calculated when the mean of the objective function value of the proposed method with a single population and 640 individuals is set to 100%.

**Table 10.**Comparison of the best facility operation in an industrial model of Case 2 using migration model, and W-B migration policy with ring topology when the number of individuals is set to 320 among 1, 2, 4, 8, and 16 sub-populations.

^{1}Column A shows the amount of electric power output by a gas turbine generator, Column B shows the amount of purchased electric power, and total is the summation of each column from 8 to 22 h.

**Table 11.**Comparison of the best facility operation in an industrial model of Case 3 using migration model, and W-B migration policy with ring topology when the number of individuals is set to 320 among 1, 2, 4, 8, and 16 sub-populations.

^{1}

# of Pop. | 1 | 2 | 4 | 8 | 16 | |||||
---|---|---|---|---|---|---|---|---|---|---|

Hours | A | B | A | B | A | B | A | B | A | B |

1 | 6.23 | 0.90 | 0.00 | 7.14 | 6.64 | 0.36 | 6.00 | 1.27 | 7.04 | 0.21 |

2 | 6.28 | 0.97 | 0.00 | 7.00 | 6.56 | 0.81 | 6.32 | 0.89 | 6.56 | 0.75 |

3 | 6.65 | 0.64 | 6.72 | 0.47 | 6.00 | 1.20 | 7.04 | 0.23 | 6.28 | 1.02 |

4 | 6.00 | 1.18 | 0.00 | 7.30 | 6.14 | 1.19 | 6.30 | 0.94 | 7.06 | 0.26 |

5 | 7.52 | 1.82 | 8.69 | 0.57 | 7.84 | 1.47 | 9.00 | 0.22 | 6.00 | 3.23 |

6 | 6.77 | 2.49 | 6.00 | 3.11 | 6.68 | 2.45 | 6.73 | 2.17 | 7.99 | 1.24 |

7 | 7.10 | 2.02 | 7.38 | 1.52 | 6.77 | 2.13 | 8.10 | 1.04 | 8.55 | 0.61 |

8 | 7.37 | 1.73 | 7.13 | 2.13 | 7.66 | 1.35 | 8.86 | 0.31 | 7.40 | 1.71 |

9 | 10.16 | 0.91 | 9.12 | 1.91 | 9.65 | 1.38 | 9.86 | 1.25 | 10.32 | 0.87 |

10 | 13.65 | 1.33 | 13.52 | 1.28 | 13.22 | 1.74 | 13.89 | 1.21 | 14.94 | 0.11 |

11 | 16.67 | 2.03 | 15.53 | 3.25 | 17.07 | 1.79 | 18.31 | 0.64 | 17.02 | 2.06 |

12 | 20.00 | 4.75 | 17.79 | 7.22 | 17.73 | 7.24 | 15.46 | 9.41 | 19.39 | 5.45 |

13 | 16.65 | 0.95 | 17.09 | 0.67 | 16.82 | 0.97 | 17.70 | 0.15 | 16.79 | 1.11 |

14 | 19.29 | 2.84 | 18.92 | 3.15 | 18.81 | 3.38 | 19.53 | 2.56 | 19.49 | 2.72 |

15 | 20.00 | 3.18 | 20.00 | 3.42 | 18.98 | 4.24 | 19.30 | 3.86 | 19.34 | 3.81 |

16 | 15.72 | 5.43 | 19.24 | 1.94 | 18.69 | 2.67 | 18.07 | 3.25 | 19.47 | 1.83 |

17 | 20.00 | 3.05 | 19.24 | 3.78 | 19.92 | 2.87 | 19.56 | 3.32 | 18.31 | 4.65 |

18 | 20.00 | 2.05 | 16.25 | 5.79 | 18.79 | 3.39 | 18.73 | 3.40 | 20.00 | 2.21 |

19 | 18.79 | 4.24 | 18.98 | 4.08 | 16.80 | 6.41 | 19.87 | 3.21 | 19.98 | 3.23 |

20 | 12.76 | 8.43 | 20.00 | 1.13 | 20.00 | 1.00 | 20.00 | 1.23 | 19.51 | 1.57 |

21 | 16.91 | 0.41 | 15.54 | 1.63 | 16.52 | 0.64 | 17.33 | 0.03 | 15.71 | 1.36 |

22 | 9.20 | 3.04 | 10.55 | 1.88 | 11.39 | 0.90 | 11.75 | 0.53 | 10.96 | 1.22 |

23 | 11.35 | 1.56 | 11.83 | 1.22 | 11.45 | 1.51 | 12.51 | 0.34 | 12.32 | 0.63 |

24 | 8.00 | 2.40 | 7.27 | 2.90 | 10.01 | 0.24 | 9.00 | 1.34 | 9.27 | 1.18 |

Total | 237.18 | 44.39 | 238.90 | 43.27 | 242.06 | 39.96 | 248.23 | 34.35 | 248.64 | 33.92 |

^{1}Column A shows the amount of electric power output by a gas turbine generator, Column B shows the amount of purchased electric power, and Total is the summation of each column whole of a day.

© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Sato, M.; Fukuyama, Y.; Iizaka, T.; Matsui, T. Total Optimization of Energy Networks in a Smart City by Multi-Population Global-Best Modified Brain Storm Optimization with Migration. *Algorithms* **2019**, *12*, 15.
https://doi.org/10.3390/a12010015

**AMA Style**

Sato M, Fukuyama Y, Iizaka T, Matsui T. Total Optimization of Energy Networks in a Smart City by Multi-Population Global-Best Modified Brain Storm Optimization with Migration. *Algorithms*. 2019; 12(1):15.
https://doi.org/10.3390/a12010015

**Chicago/Turabian Style**

Sato, Mayuko, Yoshikazu Fukuyama, Tatsuya Iizaka, and Tetsuro Matsui. 2019. "Total Optimization of Energy Networks in a Smart City by Multi-Population Global-Best Modified Brain Storm Optimization with Migration" *Algorithms* 12, no. 1: 15.
https://doi.org/10.3390/a12010015