# Contribution Determination for Multiple Unbalanced Sources at the Point of Common Coupling

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## Abstract

**:**

## 1. Introduction

## 2. The Proposed Unbalance Contribution Determination Method

#### 2.1. Contribution Determination for Single-Point Unbalanced Source Problem

_{Si}(i = 0, 1, 2) is the self sequence impedance, and Z

_{Sij}(i = 0, 1, 2, and j = 0, 1, 2) is the mutual impedance. The POE phase voltage and current can be measured and decomposed into their symmetrical components: ${\dot{V}}_{1},{\dot{V}}_{2},{\dot{V}}_{0}$ and ${\dot{I}}_{1},{\dot{I}}_{2},{\dot{I}}_{0}$. In Figure 1a, ${Z}_{L}$ represents the equivalent impedance of the downstream system, which has a similar form as Z

_{S}.

_{Li}(i = 0, 1, 2) and Z

_{Lij}(i = 0, 1, 2, and j = 0, 1, 2) are the equivalent sequence impedance and mutual impedances for the downstream network. Similarly, the voltage unbalance caused by the downstream sources can also be represented by an equivalent voltage source ${\dot{V}}_{L2}$:

_{S}is the unbalance index for the upstream unbalanced source and UI

_{L}is the index for the downstream source. VUF is the simplification for the Voltage Unbalance Factor.

#### 2.2. Contribution Determination for Concentrated-Multiple Unbalanced Source System

^{th}unbalanced source on the POE negative sequence voltage can be calculated by the projection of ${\dot{V}}_{2}^{k}$ on ${\dot{V}}_{2}$; therefore, the unbalance index $U{I}_{k}$ can be calculated as below:

^{th}unbalanced source is calculated by:

_{k}can be calculated based on the “individual current”.

#### 2.3. Relationship Analysis between “Individual Current” and “Actual Current”

_{i}(i = 1, 2, 3). The impedance value of feeder 1 is used as the basis, and the ratios of Z

_{2}to Z

_{1}, Z

_{3}to Z

_{1}are defined as Z

_{21}, Z

_{31}, respectively. Different impedance combinations represent different load levels. The sub-cases of the impedance variation combinations are shown in Table 2. On the other hand, the parameter variations for the voltage source are shown in Table 3. Unbalance levels for both the feeders and the upstream source are varied randomly hundreds of times for each sub-case. Numerous simulation results have been obtained. However, due to the space limit, only the results for one sub-case are given here.

## 3. Negative Sequence Shunt Impedance Estimation

_{i}is the i

^{th}sampling time instant, ${\dot{V}}_{2}({t}_{i})$ and ${\dot{I}}_{2}^{k}({t}_{i})$ are the POE voltage and current in feeder k at the time instant t

_{i}(i = 1, 2, …, n), respectively. The n equations can be manipulated into a matrix form.

## 4. Main Steps and Flowchart of the Proposed Method

_{S}and downstream unbalance contribution index UI

_{L}can be calculated according to Equations (6)–(10). By comparing the unbalance contribution results, the main unbalanced source can be identified. If the main unbalanced source is determined to be located upstream of POE, the mitigation measures should be targeted to the upstream unbalanced sources. If the main unbalanced source is at the downstream of POE, the individual unbalance contribution for the multiple sources connected at the POE should be further determined based on the third step.

## 5. Case Study Result Analysis

_{self}= 0.4936 + 3.1026j (Ω) and Z

_{mutual}= −0.1976 + 0.1463j (Ω). The transmission line is un-transposed and the impedance matrix per kilometer is:

_{i}represents the unbalanced loads on feeder i. The rated load capacities for the four feeders are 0.8 MVA, 0.5 MVA, 0.6 MVA, and 0.1 MVA, respectively. The power factors for all loads are set to be 0.90. The unbalanced three-phase power flow was calculated based on the multiple-phase harmonic load flow (MHLF) program, and the results are used to validate the proposed method.

#### 5.1. Negative Sequence Impedance Estimation

_{S2}) and the parallel impedance for all other branches seen from feeder k (${Z}_{L2\_\mathrm{shunt}}^{k}$) are estimated. The estimation results are compared with their actual values as shown in Table 4 (feeder 3 is used as an example). From the results, it can be observed that the estimated impedances have a good consistency with their exact values.

#### 5.2. Unbalance Contribution Determination

**Case 1:**- Load of phase B on each feeder is set to 10% of its rated capacity, while loads of phase A and phase C take their rated capacities. The upstream three-phase voltage source is assumed to be ideally balanced.
**Case 2:**- The load unbalance condition is the same as that in Case 1, while the voltage source of phase B at the upstream side of POE is set to ${\dot{V}}_{SB}=0.98\angle {0}^{\circ}\mathrm{p}.\mathrm{u}.$
**Case 3:**- The upstream unbalance condition is the same as that in Case 2. For the downstream side, the load of phase B changes to 70% S
_{N}, and loads of phase A and C take their rated value.

- (1)
- Estimation Error$$EE=\left|\frac{U{I}^{\mathrm{exact}}-U{I}^{\mathrm{estimate}}}{U{I}^{\mathrm{exact}}}\right|\times 100\%$$
- (2)
- Average Accuracy (AA)$$AA=(1-\frac{1}{n}{\displaystyle \sum _{k=1}^{n}E{E}_{k}})\times 100\%$$
- (3)
- Highest Accuracy (HA)$$HA=(1-\mathrm{min}\{E{E}_{k}\})\times 100\%,\text{}k=1,2,\dots ,n$$

#### 5.3. Sensitivity Analysis

#### 5.4. Field Data Analysis

## 6. Conclusions

- First, a new method is proposed to identify the unbalance contribution of each unbalanced source connected at the POE, including the upstream unbalanced sources and the downstream multiple unbalanced sources.
- Second, the paper proposed a method to estimate the negative sequence equivalent impedance for the system seen from each feeder at the POE.
- Third, the current flowing in each load feeder (“actual current”) is proposed to be used for the unbalance contribution estimation instead of the current actually emitted by the unbalanced source (“individual current”).

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 1.**Analysis Circuit for the single-point unbalanced source problem: (

**a**) simplified schematic circuit; (

**b**) Thevenin equivalent analysis circuit; and (

**c**) vector projection diagram for the contribution determination. PCC: point of common coupling.

**Figure 3.**Relationship analysis between “individual current” and “actual current”: (

**a**) amplitude variation results for all simulation cases; (

**b**) enlarged view for the current results of cases 61–90; and (

**c**) unbalance contribution estimation results for all cases.

**Figure 5.**Flowchart for the unbalance contribution determination method. POE: point of evaluation; and UI: unbalance index.

**Figure 7.**Sensitivity analysis results for the “individual current” and “actual current”: (

**a**) comparison of the “individual current” and “actual current” for the four-feeder system; and (

**b**) relationship between the actual current accuracy and unbalance responsibility.

**Figure 8.**Root mean square (RMS) variation for the negative sequence voltage and feeder current. (

**a**): The negative sequence votlage for the measured bus; (

**b**): the negative sequence current for feeder 1; (

**c**): the negative sequence current for feeder 2 and (

**d**): the negative sequence current for feeder 3.

Parameter | Source | Feeder 1 | Feeder 2 | Feeder 3 |
---|---|---|---|---|

R (Ω) | 1.48 | 6.2 | 6.6 | 5.8 |

X_{L} (Ω) | 5.29 | 27.8 | 37.7 | 29.3 |

V_{2} (V) | 100∠50° | 350∠45° | 400∠61° | 200∠49° |

Case No. | Z_{21} | Z_{31} |
---|---|---|

1 | 0 ≤ Z_{21} ≤ 1 | 0 ≤ Z_{31} ≤ 1 |

2 | 0 ≤ Z_{21} ≤ 1 | Z_{31} ≥ 1 |

3 | Z_{21} ≥ 1 | 0 ≤ Z_{31} ≤ 1 |

4 | Z_{21} ≥ 1 | Z_{31} ≥ 1 |

Case No. | ${\dot{\mathit{V}}}_{\mathit{S}2}\left(\mathbf{V}\right)$ | ${\dot{\mathit{V}}}_{\mathit{L}2}^{1}\left(\mathbf{V}\right)$ | ${\dot{\mathit{V}}}_{\mathit{L}2}^{2}\left(\mathbf{V}\right)$ | ${\dot{\mathit{V}}}_{\mathit{L}2}^{3}\left(\mathbf{V}\right)$ |
---|---|---|---|---|

1 | 100∠50° | 350∠45° | 400∠61° | 500∠49° |

2 | 100∠50° | 350∠45° | 400∠61° | 800∠49° |

3 | 100∠50° | 350∠45° | 400∠61° | 1100∠49° |

4 | 100∠50° | 350∠45° | 400∠61° | 1500∠49° |

5 | 100∠50° | 350∠45° | 400∠61° | 3300∠49° |

6 | 400∠50° | 350∠45° | 400∠61° | 200∠49° |

7 | 800∠50° | 350∠45° | 400∠61° | 200∠49° |

8 | 1200∠50° | 350∠45° | 400∠61° | 200∠49° |

9 | 1800∠50° | 350∠45° | 400∠61° | 200∠49° |

10 | 2300∠50° | 350∠45° | 400∠61° | 200∠49° |

Parameters (Ω) | Values | t = 0–10 (s) | t = 10–20 (s) | t = 20–30 (s) |
---|---|---|---|---|

R_{S2} | Exact | 0.96 | 1.23 | 1.54 |

Estimated | 0.99 | 1.19 | 1.56 | |

X_{S2} | Exact | 4.41 | 5.12 | 6.03 |

Estimated | 4.30 | 5.14 | 6.11 | |

${R}_{L2\_\mathrm{shunt}}^{3}$ | Exact | 0.78 | 0.94 | 1.29 |

Estimated | 0.81 | 0.96 | 1.40 | |

${X}_{L2\_\mathrm{shunt}}^{3}$ | Exact | 3.59 | 3.98 | 4.86 |

Estimated | 3.63 | 4.13 | 5.07 |

**Table 5.**Estimation results for the unbalance contribution. ICM: the results for the “individual current”-based method; and ACM: the results for the “actual current”-based method.

UI (%) | Feeder 1 | Feeder 2 | Feeder 3 | Upstream | Total | |
---|---|---|---|---|---|---|

Case 1 | Accurate | 41.59 | 25.83 | 31.81 | 0.77 | 100 |

ICM | 42.76 | 26.15 | 32.68 | −1.59 | 100 | |

ACM | 43.51 | 27.07 | 33.19 | −3.77 | 100 | |

Case 2 | Accurate | 39.11 | 21.81 | 25.29 | 13.79 | 100 |

ICM | 41.06 | 22.75 | 26.08 | 10.11 | 100 | |

ACM | 42.14 | 23.50 | 26.72 | 7.64 | 100 | |

Case 3 | Accurate | 43.59 | 13.58 | 25.64 | 17.19 | 100 |

ICM | 45.51 | 13.91 | 25.31 | 15.27 | 100 | |

ACM | 46.71 | 14.52 | 25.07 | 13.70 | 100 |

EE (%) | Feeder 1 | Feeder 2 | Feeder 3 | Upstream | |
---|---|---|---|---|---|

Case 1 | ICM | 2.81 | 1.24 | 2.73 | - |

ACM | 4.62 | 4.80 | 4.34 | - | |

Case 2 | ICM | 4.99 | 4.31 | 3.13 | 26.69 |

ACM | 7.75 | 7.75 | 5.65 | 44.60 | |

Case 3 | ICM | 4.40 | 2.43 | 1.29 | 11.17 |

ACM | 7.16 | 6.92 | 2.22 | 20.30 |

Index (%) | Case 1 | Case 2 | Case 3 | |||
---|---|---|---|---|---|---|

ICM | ACM | ICM | ACM | ICM | ACM | |

AA | 92.04 | 87.35 | 89.40 | 85.96 | 89.45 | 85.35 |

HA | 93.66 | 93.19 | 93.66 | 93.20 | 93.67 | 93.21 |

UI (%) | Feeder 1 | Feeder 2 | Feeder 3 | Feeder 4 | Upstream | |
---|---|---|---|---|---|---|

Accurate | 44.01 | 20.67 | 22.33 | 2.26 | 10.73 | |

ICM | 44.95 | 22.14 | 22.69. | 0.33 | 9.89 | |

ACM | Case1 | 50.03 | 21.33 | 31.25 | −5.98 | 3.37 |

Case2 | 45.13 | 20.72 | 22.07 | - | 12.08 |

Estimated Parameters | R_{2} (Ω) | Z_{2} (Ω) |
---|---|---|

Upstream | 1.19 | 2.47 |

Feeder 1 | 1.21 | 2.39 |

Feeder 2 | 1.08 | 2.31 |

Feeder 3 | 1.12 | 2.09 |

UI (%) | Upstream | Feeder 1 | Feeder 2 | Feeder 3 |
---|---|---|---|---|

Accurate | 12.81 | 40.62 | 26.90 | 19.67 |

ACM | 15.43 | 39.13 | 27.66 | 17.78 |

© 2017 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/).

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**MDPI and ACS Style**

Sun, Y.; Li, P.; Li, S.; Zhang, L. Contribution Determination for Multiple Unbalanced Sources at the Point of Common Coupling. *Energies* **2017**, *10*, 171.
https://doi.org/10.3390/en10020171

**AMA Style**

Sun Y, Li P, Li S, Zhang L. Contribution Determination for Multiple Unbalanced Sources at the Point of Common Coupling. *Energies*. 2017; 10(2):171.
https://doi.org/10.3390/en10020171

**Chicago/Turabian Style**

Sun, Yuanyuan, Peixin Li, Shurong Li, and Linghan Zhang. 2017. "Contribution Determination for Multiple Unbalanced Sources at the Point of Common Coupling" *Energies* 10, no. 2: 171.
https://doi.org/10.3390/en10020171