# A Kriging Model Based Optimization of Active Distribution Networks Considering Loss Reduction and Voltage Profile Improvement

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## Abstract

**:**

## 1. Introduction

- (1)
- The optimal operation and schedule model of ADN is proposed considering multiple controllable resources such as battery storage, DGs, etc. The objectives include reducing the power losses and improving the voltage profile.
- (2)
- The Kriging model is used to approximate the complex active distribution network, speeding up the solving process.
- (3)
- The Kriging model based optimization method named ISO-MI is proposed to solve the optimization problem, which improves the solving efficiency.

## 2. Problem Formulation

#### 2.1. Objective Function

#### 2.2. Constraints

## 3. Model Solution

#### 3.1. Kriging Model

#### 3.2. Kriging Model Based Optimization

#### 3.3. Improved Surrogate Optimization-Mixed-Integer Algorithm

**Step 1—Construct Initial Kriging model.**In this step, it is easy to construct the Kriging model with high accuracy using the measured history operation data of distribution system. ${S}_{n}^{\mathrm{ini}}\left(x\right)$ denotes the initial surrogate model built by the set of sampled points ${\mathit{B}}_{n}^{\mathrm{ini}}=\left\{{x}_{1},\dots ,{x}_{n}\right\}$;

**Step 2—Initialize optimization parameters.**In this step, first, a series of parameters need to be set for the beginning of the optimization program, such as maximum evolution number M of expensive simulation, initial coordinate perturbation range ${r}_{0}$ and the minimum range ${r}_{\mathrm{min}}$, etc. Second, the LHD method is used to generate initial feasible design points. To construct a new Kriging model, it is necessary to do the expensive simulations and get the responses from them. Finally, find the best point ${x}_{k}^{\ast}$, which represents the control variables of the distribution network, and the corresponding objective function value $f\left({x}_{k}^{\ast}\right)$ in the current design points.

**Step 3—Iterate until the evolution number, m > M, maximum evolution number.**In this step, first, create four groups of candidate points by randomly perturbing coordinates around ${x}_{k}^{\ast}$. The purpose is to improve the efficiency of local search by adding the coordinate perturbation. The generation of the four groups in the global range are shown as follows:

- (1)
- Group 1: Uniformly and randomly perturb the continuous coordinates of ${x}_{k}^{\ast}$ at the range of ${r}_{k}$, ${x}_{\mathrm{G}1,i}={x}_{k}^{\ast}+\alpha \cdot {r}_{k}$, where i is the index of points and $\alpha ~N\left(0,1\right)$;
- (2)
- Group 2: Uniformly and randomly perturb the discrete coordinates of ${x}_{k}^{\ast}$ at the range of ${r}_{k}$, ${x}_{\mathrm{G}2,i}={x}_{k}^{\ast}+round\left(\alpha \cdot {r}_{k}\right)$;
- (3)
- Group 3: Uniformly and randomly perturb all coordinates of ${x}_{k}^{\ast}$ at the range of ${r}_{k}$, ${x}_{\mathrm{G}3,i}={x}_{k}^{\ast}+\alpha \cdot {r}_{k}$ and round the discrete coordinates to the closet integers;
- (4)
- Group 4: Select candidate points ${x}_{\mathrm{G}4}$ in the whole design space using LHS.

- (1)
- Calculate the objective function using the initial Kriging model ${S}_{n}^{\mathrm{ini}}\left(x\right)$ and current new Kriging model ${S}_{n,k}\left(x\right)$ of candidate points in four groups and compute the objective function score ${s}_{1}\left({x}_{i}\right)$ and ${s}_{2}\left({x}_{i}\right)$ of all points, where ${s}_{1}$ and ${s}_{2}$ are the normalized objective functions. Their value can be calculated by Euclidean distance in n dimensional space.
- (2)
- Compute the distance score of all design points.
- (3)
- Compute the weighted score $S\left({x}_{i}\right)={\omega}_{1}{s}_{1}\left({x}_{i}\right)+{\omega}_{2}{s}_{2}\left({x}_{i}\right)$, where $S$ denotes the objective function F in this paper, and select the point with minimum score $S\left({x}_{i}\right)$ to add it into design points set ${B}_{n,k+1}$ and do the expensive function evaluation, again.

**Step 4—Output the value of best solution**${x}^{\ast}$

**and the corresponding objective functions found so far.**

## 4. Simulation and Case Studies

#### 4.1. Test System Specification

#### 4.2. Accuracy of Kriging Model in Distribution System

- (1)
- Root Mean Square Error (RMSE):$$RMSE=\sqrt{\frac{1}{N}{\displaystyle \sum _{i=1}^{N}{\left({y}_{i}-\hat{{y}_{i}}\right)}^{2}}}$$
- (2)
- Relative Maximum Absolute Error (RMAX):$$RMAX=\mathrm{max}\left|\left({y}_{i}-\hat{{y}_{i}}\right)/{y}_{i}\right|$$
- (3)
- Relative Average Absolute Error (RAAE):$$RAAE=\frac{1}{N}{\displaystyle \sum _{i=1}^{N}\left|\left({y}_{i}-\hat{{y}_{i}}\right)/{y}_{i}\right|}$$
- (4)
- R-Square:$${R}^{2}=1-{\displaystyle \sum _{i=1}^{N}{\left({y}_{i}-\hat{{y}_{i}}\right)}^{2}}/{\displaystyle \sum _{i=1}^{N}{\left({y}_{i}-\stackrel{-}{y}\right)}^{2}}=1-\frac{MSE}{Variance}$$

#### 4.3. Solving Efficiency of ISO-MI

_{1}and ω

_{2}, are both 0.5. The simulation results are displayed as follows. The comparisons among ISO-MI, GA and PSO are shown in Figure 8, Figure 9 and Figure 10. The optimization results of tap positions of Reg1–Reg4 are displayed in Figure 11 and the charge power and SOC of BAT1 are shown in Figure 12.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 8.**Convergence characteristic by Improved Surrogate Optimization-Mixed-Integer (ISO-MI), genetic algorithm (GA) and particle swarm optimization (PSO).

**Figure 10.**Different optimization results of voltage profiles of all nodes during 24 h between ISO-MI, GA and PSO.

Name | Installed Location | Phases | Tap Range | Voltage Regulation Range | Maximum Operating Times |
---|---|---|---|---|---|

Reg1 | 150–149 | A-B-C | [−16, +16] | [0.95, 1.05] | 10 |

Reg1 | 25–26 | A-C | [−16, +16] | [0.95, 1.05] | 10 |

Reg1 | 9–14 | A | [−16, +16] | [0.95, 1.05] | 10 |

Reg1 | 160–67 | A-B-C | [−16, +16] | [0.95, 1.05] | 10 |

Name | Installed Location | Installed Capacity (kVar) | Maximum Operating Times | ||
---|---|---|---|---|---|

Phase A | Phase B | Phase C | |||

Cap1 | 83 | 100 | 100 | 100 | 10 |

Cap2 | 88 | 50 | 0 | 0 | 10 |

Cap3 | 90 | 0 | 50 | 0 | 10 |

Cap4 | 92 | 0 | 0 | 50 | 10 |

Name | Installed Location | Type | Rated Power (kW) | Power Factor |
---|---|---|---|---|

DG1 | 66 | WT | 150 | 0.9 |

DG1 | 51 | PV | 100 | 0.9 |

DG1 | 30 | MT | 150 | 0.9 |

DG1 | 18 | MT | 200 | 0.9–1.0 |

DG1 | 60 | MT | 150 | 0.9–1.0 |

DG1 | 108 | MT | 200 | 0.9–1.0 |

DG1 | 77 | MT | 150 | 0.9–1.0 |

**Table 4.**ZIP coefficients of the loads [46].

ZIP Coefficients | Z | I | P |
---|---|---|---|

Active load | 0.418 | 0.135 | 0.447 |

Reactive load | 0.515 | 0.023 | 0.462 |

Name | Installed Location | Power (kW) | Capacity (kWh) | Efficiency | |
---|---|---|---|---|---|

Charging | Discharging | ||||

BAT1 | 86 | [–150, 150] | 750 | 0.9 | 0.9 |

N | 50 | 100 | 200 | |
---|---|---|---|---|

Index | ||||

Voltage fluctuation | RMSE | 9.4 × 10^{−5} | 9.0 × 10^{−5} | 9.1 × 10^{−5} |

RMAE | 7.5 × 10^{−4} | 7.2 × 10^{−4} | 6.2 × 10^{−4} | |

RAAE | 6.3 × 10^{−5} | 5.9 × 10^{−5} | 5.2 × 10^{−5} | |

R^{2} | 1.0000 | 1.0000 | 1.0000 | |

Power loss | RMSE | 2.3 × 10^{−3} | 1.9 × 10^{−3} | 1.3 × 10^{−3} |

RMAE | 3.4 × 10^{−3} | 2.5 × 10^{−3} | 1.9 × 10^{−3} | |

RAAE | 1.1 × 10^{−3} | 0.9 × 10^{−3} | 0.7 × 10^{−3} | |

R^{2} | 0.9954 | 0.9973 | 0.9988 |

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## Share and Cite

**MDPI and ACS Style**

Wang, D.; Hu, Q.; Tang, J.; Jia, H.; Li, Y.; Gao, S.; Fan, M.
A Kriging Model Based Optimization of Active Distribution Networks Considering Loss Reduction and Voltage Profile Improvement. *Energies* **2017**, *10*, 2162.
https://doi.org/10.3390/en10122162

**AMA Style**

Wang D, Hu Q, Tang J, Jia H, Li Y, Gao S, Fan M.
A Kriging Model Based Optimization of Active Distribution Networks Considering Loss Reduction and Voltage Profile Improvement. *Energies*. 2017; 10(12):2162.
https://doi.org/10.3390/en10122162

**Chicago/Turabian Style**

Wang, Dan, Qing’e Hu, Jia Tang, Hongjie Jia, Yun Li, Shuang Gao, and Menghua Fan.
2017. "A Kriging Model Based Optimization of Active Distribution Networks Considering Loss Reduction and Voltage Profile Improvement" *Energies* 10, no. 12: 2162.
https://doi.org/10.3390/en10122162