# A Fast Reactive Power Optimization in Distribution Network Based on Large Random Matrix Theory and Data Analysis

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## Abstract

**:**

## 1. Introduction

## 2. Random Matrix and Data Model in Reactive Power Optimization

#### 2.1. Random Matrix of Loads

#### 2.2. Lengths and Covariance of Random Matrix of Loads

#### 2.3. Data Model of Loads

#### 2.4. Load Grouping

## 3. Problem Formulation

#### 3.1. Overview

#### 3.2. Objective Function

**x**is expressed as follow:

#### 3.3. Constraints

#### 3.3.1. Equality Constraints

#### 3.3.2. Inequality Constraints

#### 3.3.3. Constraints on Equipment Operations Number

#### 3.4. Overall Formulation

## 4. The Proposed Method for Predicting the Reactive Power Adjustment

#### 4.1. Sensitivity Analysis

#### 4.2. Load Similarity

#### 4.3. Reactive Power Optimization Method Based on Big Data

#### 4.3.1. Data Preparation

#### 4.3.2. Load Similarity Matching

**Step 1**: Establish the forecasting load augmented matrix of the day ahead, based on the forecasting load. According to the date of the day ahead, determine the load grouping properties $\mathsf{\lambda}$ and $\mathsf{\mu}$.

**Step 2**: Based on grouping properties, select the group $t\in c(\mathsf{\lambda},\mathsf{\mu})$ and establish the corresponding historical load augmented matrices ${A}_{H}(t)$, where $t\in c(\mathsf{\lambda},\mathsf{\mu})$.

**Step3**: According to Equation (32), calculate the load similarity $s(t)$ of historical load augmented matrix ${A}_{H}(t)$ and forecasting load augmented matrix ${A}_{F}$, when $t\in c(\mathsf{\lambda},\mathsf{\mu})$.

**Step 4**: According to Equation (33), the best matching day $t={t}_{\text{max}}$ can be found when the load similarity becomes the maximum.

**Step 5**: Set the minimum load similarity margin ${s}_{\text{min}}$ based on experience.

**Step 6**: Compare the largest load similarity $s({t}_{\text{max}})$ with the minimum load similarity margin ${s}_{\text{min}}$.

**Step 7**: If $s({t}_{\text{max}})\ge {s}_{\text{min}}$, the historical load of the day with date $t={t}_{\text{max}}$ and forecasting load have high similarity. The reactive power control devices dispatching scheme can be obtained from the historical sequence of tap setting and sequence of compensation capacity.

**Step 8**: If $s({t}_{\text{max}})\le {s}_{\text{min}}$, the historical load of the day with date $t={t}_{\text{max}}$ and forecasting load have low similarity. The reactive power control devices dispatching scheme of the day ahead should be calculated with a fine adjustment method based on sensitivity analysis.

**Step 9**: Store the RPO data into database including the forecasting load and the reactive power control devices dispatching scheme.

#### 4.3.3. Fine Adjustment Method Based on Sensitivity Analysis

Algorithm 1. Control variable increment calculation rules | ||

if the sensitivity of active loss to control variable ${S}_{u}<0$ | ||

if ${u}^{h}>{u}^{h+1}$ or ${u}^{h}>{u}^{h-1}$ | ||

control variable increment $\Delta {u}^{h}=\text{max}({u}^{h+1},{u}^{h-1})-{u}^{h}$ | ||

end if | ||

else if the sensitivity of active loss to control variable ${S}_{u}>0$ | ||

if ${u}^{h}<{u}^{h+1}$ or ${u}^{h}<{u}^{h-1}$ | ||

control variable increment $\Delta {u}^{h}=\text{min}({u}^{h+1},{u}^{h-1})-{u}^{h}$ | ||

end if | ||

Else control variable increment $\Delta {u}^{h}=0$ | ||

end if |

**Step 1**: According to reactive power control devices dispatching scheme achieved by load similarity matching, initialize the control variable ${u}_{k}^{h}$, where $h=1,2,3,\mathrm{...},24$. Let the iteration number k = 1.

**Step 2**: Calculate the reactive power loss ${P}_{loss,k}$ when the control variable is ${u}_{k}^{h}$.

**Step 3**: Calculate the active power loss sensitivity ${S}_{u,k}$ to the control variable.

**Step 4**: According to the control variable increment calculation rules, calculate the control variable increment $\Lambda {u}_{k}^{h}$.

**Step 5**: Update the control variable by ${u}_{k+1}^{h}={u}_{k}^{h}+\Delta {u}_{k}^{h}$ and calculate the new active power loss ${P}_{loss,k+1}$.

**Step 6**: If ${P}_{\mathrm{loss},k+1}<{P}_{\mathrm{loss},k}$, let k = k + 1 and continue the iteration process to

**Step 3**. Otherwise, output the final control variable ${u}_{k}^{h}$.

## 5. Experiments and Results

#### 5.1. Experiments Setting and Descriptions

#### 5.2. Experiment on Standard Test Case

#### 5.2.1. Calculation of Minimum Load Similarity Margin

#### 5.2.2. Three test cases

#### Case 1: Test of a Random Day

_{F1}and C

_{F2}at several points shown in Figure 8a,b. The optimization results of the selected day are shown in Table 5.

_{F1}and C

_{F2}, as shown in Figure 8a,b. The BDO method shows less action times at C

_{F1}and presents less compensation capacity at C

_{F2}compared with the MGA method.

#### Case 2: Test of Some Random Days

#### Case 3: Test of Typical Days

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Three types of typical daily load curves in different seasons: (

**a**) residential load; (

**b**) commercial load; and (

**c**) industrial load.

**Figure 8.**Capacities of capacitor units of different methods: (

**a**) capacitor unit C

_{F1}at Node 4; and (

**b**) capacitor unit C

_{F2}at Node 9.

$c(\mathsf{\lambda},\mathsf{\mu})$ | Workday, $\mathsf{\mu}=1$ | Weekend, $\mathsf{\mu}=1$ |
---|---|---|

Spring, $\mathsf{\lambda}=1$ | $t\in c(1,1)$ | $t\in c(1,2)$ |

Summer, $\mathsf{\lambda}=2$ | $t\in c(2,1)$ | $t\in c(2,2)$ |

Autumn, $\mathsf{\lambda}=3$ | $t\in c(3,1)$ | $t\in c(3,2)$ |

Winter, $\mathsf{\lambda}=4$ | $t\in c(4,1)$ | $t\in c(4,2)$ |

Device | Setting |
---|---|

ULTC | 70/10 kV, −10% to 10% regulation with 32 steps |

C_{S} | Substation capacitors: 2.5 Mvar each |

C_{F} | Feeder capacitors: 1.5 Mvar each |

Load Type | Node |
---|---|

Residential load | 3, 6, 8, 14 |

Commercial load | 4, 5, 7, 9, 13 |

Industrial load | 2, 10, 11, 12 |

**Table 4.**Optimization result of MGA (multi-population genetic algorithm) and BDO (big data reactive power optimization) without fine adjustment.

No. | Loss of MGA (MWh) | Loss of BDO (MWh) | Similarity | Error (%) |
---|---|---|---|---|

1 | 7.9886 | 7.9971 | 0.9741 | 0.1074 |

2 | 7.8271 | 7.8406 | 0.8888 | 0.1734 |

3 | 15.3140 | 15.3225 | 0.9779 | 0.0561 |

4 | 8.5332 | 8.5356 | 0.9618 | 0.0282 |

5 | 7.1424 | 7.1453 | 0.9731 | 0.0408 |

6 | 11.5872 | 11.6022 | 0.9595 | 0.1288 |

7 | 7.4006 | 7.4100 | 0.9332 | 0.1268 |

8 | 15.1935 | 15.2026 | 0.9780 | 0.0599 |

9 | 8.6196 | 8.6207 | 0.9856 | 0.0131 |

10 | 14.0853 | 14.0928 | 0.9857 | 0.0529 |

11 | 8.9968 | 8.9973 | 0.9557 | 0.0054 |

12 | 8.7716 | 8.7723 | 0.9597 | 0.0080 |

13 | 10.5138 | 10.5196 | 0.9754 | 0.0551 |

14 | 7.0980 | 7.0980 | 0.9688 | 0.0000 |

15 | 7.3734 | 7.3806 | 0.9382 | 0.0976 |

16 | 6.6822 | 6.6869 | 0.9509 | 0.0700 |

17 | 9.0290 | 9.0409 | 0.9552 | 0.1311 |

18 | 9.4201 | 9.4521 | 0.9515 | 0.3393 |

19 | 8.9582 | 8.9799 | 0.9665 | 0.2428 |

20 | 8.0961 | 8.1206 | 0.9350 | 0.3029 |

Optimization Method | Action Times | Loss (MWh) | Computing Time (s) | ||||
---|---|---|---|---|---|---|---|

ULTC | C_{S} | C_{F1} | C_{F2} | C_{F3} | |||

MGA | 4 | 2 | 2 | 2 | 2 | 14.4631 | 141.6 |

BDO | 4 | 2 | 1 | 2 | 2 | 14.4805 | 0.283 |

No. | Load Similarity | $s({t}_{\text{max}})\ge {s}_{\text{min}}$ | Loss (MWh) | Loss Error (%) | Computing Time (s) | |||
---|---|---|---|---|---|---|---|---|

MGA | BDO (a) | BDO (b) | MGA | BDO | ||||

1 | 0.9657 | Yes | 7.2708 | 7.2710 | - | 0.0020 | 138.91 | 0.2473 |

2 | 0.9682 | Yes | 7.3735 | 7.3747 | - | 0.0153 | 133.34 | 0.2810 |

3 | 0.9536 | Yes | 7.8297 | 7.8472 | - | 0.2241 | 155.77 | 0.2966 |

4 | 0.9332 | No | 7.4009 | 7.4017 | 7.4012 | 0.0048 | 155.51 | 0.3130 |

5 | 0.9465 | No | 7.4885 | 7.4908 | 7.4897 | 0.0162 | 108.40 | 0.3263 |

6 | 0.9668 | Yes | 6.1113 | 6.1120 | - | 0.0127 | 141.80 | 0.2826 |

7 | 0.9797 | Yes | 7.3538 | 7.3563 | - | 0.0334 | 136.40 | 0.2353 |

8 | 0.9787 | Yes | 8.0612 | 8.0658 | - | 0.0581 | 168.11 | 0.2577 |

9 | 0.9670 | Yes | 9.4710 | 9.4823 | - | 0.1193 | 186.73 | 0.2482 |

10 | 0.9264 | No | 13.3107 | 13.3258 | 13.2658 | −0.3370 | 187.37 | 0.3505 |

11 | 0.9852 | Yes | 14.9465 | 14.9511 | - | 0.0304 | 108.45 | 0.2615 |

12 | 0.9796 | Yes | 13.0150 | 13.0156 | - | 0.0046 | 146.09 | 0.3024 |

13 | 0.9358 | No | 14.7954 | 14.8173 | 14.7873 | −0.0550 | 94.44 | 0.2742 |

14 | 0.9399 | No | 8.6167 | 8.6319 | 8.6119 | −0.0546 | 98.46 | 0.3067 |

15 | 0.9783 | Yes | 9.6760 | 9.6763 | - | 0.0030 | 142.73 | 0.2571 |

16 | 0.9802 | Yes | 9.0971 | 9.1012 | - | 0.0456 | 204.31 | 0.2379 |

17 | 0.9509 | Yes | 6.6868 | 6.6906 | - | 0.0571 | 112.21 | 0.2373 |

18 | 0.9765 | Yes | 8.3112 | 8.3162 | - | 0.0599 | 157.86 | 0.2946 |

19 | 0.9632 | Yes | 10.3017 | 10.3087 | - | 0.0674 | 133.98 | 0.2747 |

20 | 0.9765 | Yes | 6.9651 | 6.9654 | - | 0.0045 | 187.69 | 0.2741 |

Load Property | Forecasting Date | Historical Date | Load Similarity | Loss (MWh) | |
---|---|---|---|---|---|

BDO | MGA | ||||

Workday, spring | 17 March 2014 | 5 March 2012 | 0.9576 | 7.0678 | 6.8871 |

Weekend, spring | 16 March 2014 | 7 April 2013 | 0.9618 | 6.1504 | 6.0030 |

Workday, summer | 18 June 2014 | 10 June 2013 | 0.9644 | 16.1209 | 15.7828 |

Weekend, summer | 15 June 2014 | 7 July 2013 | 0.9531 | 14.7790 | 14.3192 |

Workday, autumn | 15 September 2014 | 13 September 2011 | 0.9701 | 12.9096 | 12.7573 |

Weekend, autumn | 14 September 2014 | 15 September 2012 | 0.9665 | 12.4243 | 12.1118 |

Workday, winter | 17 December 2014 | 9 December 2009 | 0.9636 | 7.8655 | 7.6621 |

Weekend, winter | 14 December 2014 | 16 December 2012 | 0.9547 | 6.1919 | 5.8347 |

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**MDPI and ACS Style**

Sheng, W.; Liu, K.; Pei, H.; Li, Y.; Jia, D.; Diao, Y.
A Fast Reactive Power Optimization in Distribution Network Based on Large Random Matrix Theory and Data Analysis. *Appl. Sci.* **2016**, *6*, 158.
https://doi.org/10.3390/app6060158

**AMA Style**

Sheng W, Liu K, Pei H, Li Y, Jia D, Diao Y.
A Fast Reactive Power Optimization in Distribution Network Based on Large Random Matrix Theory and Data Analysis. *Applied Sciences*. 2016; 6(6):158.
https://doi.org/10.3390/app6060158

**Chicago/Turabian Style**

Sheng, Wanxing, Keyan Liu, Hongyan Pei, Yunhua Li, Dongli Jia, and Yinglong Diao.
2016. "A Fast Reactive Power Optimization in Distribution Network Based on Large Random Matrix Theory and Data Analysis" *Applied Sciences* 6, no. 6: 158.
https://doi.org/10.3390/app6060158