# The Effect of Power Flow Entropy on Available Load Supply Capacity under Stochastic Scenarios with Different Control Coefficients of UPFC

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## Abstract

**:**

## 1. Introduction

- An index of improved power flow entropy is defined. It can not only quantify the equilibrium of the power flow distribution throughout the whole power system, but also reflects the degree of the branch loading rate.
- The adjustment of the control parameters of UPFC is taken into account during the calculating procedure of RPF herein, which makes the calculation result of ALSC under the influence of UPFC more reasonable.
- Taking LHS as the basis of the PPF method, and combining it with the RPF method, the probabilistic repeated power flow (PRPF) calculation method is proposed in this paper. Thus, the intrinsic relationship between the improved power flow entropy and the ALSC in stochastic scenarios is deeply analyzed.

## 2. The Physical Model of UPFC

#### 2.1. The Structure and Control Strategy of UPFC

#### 2.2. The Steady-State Model of UPFC

#### 2.3. The Power Flow Calculation Method of AC System with UPFC

#### 2.3.1. The Mismatch Equations of UPFC

#### 2.3.2. The Mismatch Equations of the AC System

**P**and Δ

**Q**about the power injection are often written. As for the buses which are not connected to the UPFC (the buses except bus h and o in Figure 2), the regular mismatch equations can be written as follows:

#### 2.3.3. The Power Flow Calculation Method of the AC System with UPFC

## 3. The Basic Theories of LHS-MCS

#### 3.1. The Principles of the LHS Method

#### 3.1.1. The Sampling

**F**that the variable satisfies is $\mathit{Y}=\mathit{F}\left(\mathit{R}\right)$. If the sampling size is set to N for the random variable of the i-th dimension, the sampling value is ${{[R}_{i1}{,R}_{i2}{,\dots ,R}_{ij}{,\dots ,R}_{iN}]}^{T}$. At the same time, N non-overlapping equally spaced intervals can be provided as $[0,1/N]$, $[1/N,2/N]$, …, $\left[\right(N-1)/N,1]$. The length of each interval is $1/N$. When picking a number from each interval, the data set about ${Y}_{i}$ can be obtained as ${{[Y}_{i1}{,Y}_{i2}{,\dots ,Y}_{ij}{,\dots ,Y}_{iN}]}^{T}$.

#### 3.1.2. The Sorting

**ρ**, which can be expressed as follows:

**R**, expressed as follows:

**L**represents the permutation position of the corresponding data of the sampling matrix

**R**. The steps for constructing the permutation matrix by Cholesky decomposition are as follows:

- Set the initial value of
**L**; each row consists of a random permutation of the set of integers [1, 2, …, N]. - By using the Cholesky decomposition method to decompose the correlation coefficient matrix
**ρ**, a lower trigonometric matrix**D**can be obtained, which satisfies ${\mathit{\rho}=\mathit{D}\mathit{D}}^{T}$. - Obtain a sort matrix with a lower column correlation as ${\mathit{L}\u2019=\mathit{D}}^{-1}\mathit{L}$. It is worth noting that the elements in $\mathit{L}\u2019$ may not be positive integers, so each row of data in $\mathit{L}\u2019$ can be arranged in order from largest to smallest and reassigned to positive integers ranging from 1 to N.
- Repeat steps 1 to 3 until the column correlation of
**L**is less than a predetermined value. Then, according to the arrangement order, which can be represented in**L**,**R**is arranged to obtain the final sampling matrix.

#### 3.2. The Procedure for LHS-MCS

## 4. Probabilistic Determination on the Power Flow Entropy and Available Load Supply Capability

#### 4.1. Definition of Some Indices

#### 4.1.1. Improved Power Flow Entropy

#### 4.1.2. Available Load Supply Capability

#### 4.2. The Repeated Power Flow

- Import basic system data. Set the initial values of h. Set $\lambda =1$ At this time, the load of each bus is ${P+jQ=P}_{\left(0\right)}{+jQ}_{\left(0\right)}$. The UPFC control parameters are ${P}_{UPFC}^{ref}{=P}_{UPFC}^{max}/k$ and ${Q}_{UPFC}^{ref}{=Q}_{UPFC}^{max}/k$.
- Calculate the power flow of the system and determine whether the active power of each branch exceeds the capacity. If no, turn to step 3; if yes, turn to step 4.
- Perform $\lambda =\lambda +h$, $P+jQ=\lambda (P+jQ)$, ${P}_{UPFC}^{ref}{=P}_{UPFC}^{max}\lambda /k$, and ${Q}_{UPFC}^{ref}{=Q}_{UPFC}^{max}\lambda /k$. Then, go back to step 2.
- Perform $\lambda =\lambda +h$, and order $h=h/2$. Judge whether h is smaller than the convergence accuracy. If yes, go to Step 5; if no, go back to step 2.
- Perform other necessary calculations and give out the results.

#### 4.3. The Step of the PRPF

**Z**

_{N}of samplings through the LHS method and Nataf transformation, in which there are N groups of samplings. That is, ${\mathbf{Z}}_{N}{=[z}_{1}{,z}_{2}{,\dots ,z}_{{i}_{PPF}}{,\dots ,z}_{N}]$. Set the iteration of probabilistic power flow calculation as i

_{PPF}, then for the i

_{PPF}-th probabilistic power flow calculation, the system input of each probabilistic power flow calculation is ${z}_{{i}_{\mathrm{P}\mathrm{P}\mathrm{F}}}$. Then, based on ${z}_{{i}_{\mathrm{P}\mathrm{P}\mathrm{F}}}$, the power flow entropy η and λ can be calculated.

## 5. Test Results

#### 5.1. The DPF Calculation of AC System with UPFC

#### 5.1.1. Validity of the Proposed Model of UPFC

#### 5.1.2. Robustness of the Proposed Algorithm

#### 5.2. Tests of Combining LHS-MCS and Repeated Power Flow

#### 5.2.1. Test System under Study

^{4}and RS-MCS-10

^{3}), and that of the LHS-MCS is set to 1000 (i.e., LHS-MCS-10

^{3}). The results of RS-MCS-10

^{4}are regarded as the reference. The convergence criterion of the repeated power flow calculation is assigned as ${10}^{-3}$.

#### 5.2.2. Analysis of the Results

^{4}is 2497.6 s, while that of LHS-MCS-10

^{3}and RS-MCS-10

^{3}is about 248 s.

^{3}, RS-MCS-10

^{3}, and RS-MCS-10

^{4}are obtained, as shown in Figure 12 and Figure 13, respectively.

^{3}, the curve calculated by LHS-MCS-10

^{3}is closer to the curve calculated by RS-MCS-10

^{4}. Therefore, it is proven that LHS-MCS has higher computational accuracy.

^{3}and RS-MCS-10

^{3}under different cases, which are shown in Table 7. The relative error is calculated by the following Equation (27):

^{3}or RS-MCS-10

^{3}. ${\sigma}_{{\mathrm{LHS}/\mathrm{RS}-\mathrm{MCS}-10}^{3}}^{2}$ represent the variances in the results calculated by LHS-MCS-10

^{3}or RS-MCS-10

^{3}.

#### 5.3. The Positive Impact of UPFC on the System under Probabilistic Scenarios

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 5.**Flow chart of the specific PPF calculation steps combining Nataf transformation and LHS-MCS.

**Figure 9.**The comparison on voltage amplitude and voltage phase angle with or without UPFC: (

**a**) The comparison on voltage amplitude; (

**b**) The comparison on voltage phase angle.

**Figure 10.**The comparison between active and reactive power for each branch with or without UPFC: (

**a**) The comparison on active power; (

**b**) The comparison on reactive power.

**Figure 11.**Robustness verification of the system with UPFC when the loads and control parameters of UPFC change: (

**a**) The loads change; (

**b**) The control parameters of UPFC change.

**Figure 12.**The CDF of power flow entropy η: (

**a**) Case B1; (

**b**) Case B2; (

**c**) Case B3; (

**d**) Case B4; (

**e**) Case B5; (

**f**) Case B6.

**Figure 13.**The CDF of ${\lambda}_{\mathrm{max}}$: (

**a**) Case B1; (

**b**) Case B2; (

**c**) Case B3; (

**d**) Case B4; (

**e**) Case B5; (

**f**) Case B6.

**Figure 15.**The relationship between the η of the initial state and the ${\lambda}_{\mathrm{max}}$ under different control coefficients: (

**a**) k = 1.1; (

**b**) k = 1.2; (

**c**) k = 1.3; (

**d**) k = 1.4; (

**e**) k = 1.5; (

**f**) k = 1.6; (

**g**) k = 1.7; (

**h**) k = 1.8; (

**i**) k = 1.9; (

**j**) k = 2.0; (

**k**) k = 2.1; (

**l**) k = 2.2; (

**m**) k = 2.3; (

**n**) k = 2.4; (

**o**) k = 2.5; (

**p**) k = 2.6; (

**q**) k = 2.7; (

**r**) k = 2.8; (

**s**) k = 2.9; (

**t**) k = 3.0.

**Figure 16.**The mean and variance of the η and ${\lambda}_{\mathrm{max}}$ with different control coefficients: (

**a**) The mean and variance of the η; (

**b**) The mean and variance of the ${\lambda}_{\mathrm{max}}$.

The Active Control Variable | The Reactive Control Variable | |
---|---|---|

The parallel side | ― | The reactive power injection or the AC bus voltage amplitude |

The series side | The active power | The reactive power |

Branch | 1–2 | 1–5 | 2–3 | 2–4 | 2–5 | 3–4 | 4–5 |

P^{max}/p.u. | 3 | 1.5 | 1.5 | 1.5 | 2 | 1.5 | 2 |

Branch | 4–7 | 4–9 | 5–6 | 6–11 | 6–12 | 6–13 | 7–8 |

P^{max}/p.u. | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

Branch | 7–9 | 9–10 | 9–14 | 10–11 | 12–13 | 13–14 | |

P^{max}/p.u. | 1 | 1 | 1 | 1 | 1 | 1 |

Parallel Side/p.u. | Series Side/p.u. | |
---|---|---|

UPFC1 | ${V}_{5}^{ref}=1.02$ | ${\tilde{S}}_{5-6}^{ref}=-0.3+j0.2$ |

UPFC2 | ${V}_{2}^{ref}=1.03$ | ${\tilde{S}}_{2-3}^{ref}=0.8+j0.4$ |

UPFC3 | ${V}_{4}^{ref}=0.98$ | ${\tilde{S}}_{4-5}^{ref}=-0.7+j0.3$ |

UPFC1 | Vse_{5-6} | θse_{5-6} | Vsh_{5} | θsh_{5} |

0.1580 | 1.6738 | 0.9950 | −0.0577 | |

UPFC2 | Vse_{2-3} | θse_{2-3} | Vsh_{2} | θsh_{2} |

0.1529 | −2.4690 | 1.0231 | −0.0321 | |

UPFC3 | Vse_{4-5} | θse_{4-5} | Vsh_{4} | θsh_{4} |

0.1008 | 2.2749 | 0.9395 | −0.0923 |

Parallel Side/p.u. | Series Side/p.u. | |
---|---|---|

UPFC1 | ${V}_{5}^{ref}=1.045$ | ${\tilde{S}}_{5-6}^{max}=-0.95+j0.2$ |

UPFC2 | ${V}_{2}^{ref}=1$ | ${\tilde{S}}_{2-3}^{max}=1.45+j0.1$ |

UPFC3 | ${V}_{4}^{ref}=1.07$ | ${\tilde{S}}_{4-5}^{max}=-1.95+j0.2$ |

Case | Modification |
---|---|

Case B1 | Basic case |

Case B2 | Basic case but not including UPFC |

Case B3 | Change the standard deviation of each load from 5% to 15% |

Case B4 | Change the standard deviation of each load from 5% to 25% |

Case B5 | Change the correlation coefficient between loads from 0.2 to 0.5 |

Case B6 | Change the correlation coefficient between loads from 0.2 to 0.8 |

**Table 7.**The mean and variance relative error of power flow entropy η and ${\lambda}_{\mathrm{max}}.$

Case B1 | Case B2 | Case B3 | Case B4 | Case B5 | Case B6 | |||
---|---|---|---|---|---|---|---|---|

η | Mean | LHS-MCS-10^{3} | 1.87 × 10^{−4} | 3.29 × 10^{−4} | 1.63 × 10^{−3} | 2.33 × 10^{−3} | 4.49 × 10^{−4} | 6.44 × 10^{−4} |

RS-MCS-10^{3} | 7.98 × 10^{−3} | 5.03 × 10^{−3} | 2.59 × 10^{−3} | 3.51 × 10^{−3} | 3.04 × 10^{−3} | 1.61 × 10^{−3} | ||

Variance | LHS-MCS-10^{3} | 7.98 × 10^{−3} | 1.25 × 10^{−2} | 1.98 × 10^{−2} | 1.41 × 10^{−2} | 3.11 × 10^{−2} | 5.53 × 10^{−2} | |

RS-MCS-10^{3} | 5.30 × 10^{−2} | 4.17 × 10^{−2} | 7.85 × 10^{−3} | 6.12 × 10^{−2} | 1.64 × 10^{−2} | 1.00 × 10^{−1} | ||

${\lambda}_{\mathrm{max}}$ | Mean | LHS-MCS-10^{3} | 1.91 × 10^{−5} | 3.46 × 10^{−3} | 1.13 × 10^{−3} | 2.10 × 10^{−3} | 5.63 × 10^{−4} | 5.75 × 10^{−4} |

RS-MCS-10^{3} | 1.81 × 10^{−3} | 8.73 × 10^{−3} | 6.65 × 10^{−3} | 6.88 × 10^{−3} | 4.62 × 10^{−3} | 2.61 × 10^{−3} | ||

Variance | LHS-MCS-10^{3} | 4.25 × 10^{−3} | 1.54 × 10^{−2} | 4.05 × 10^{−3} | 4.60 × 10^{−4} | 2.24 × 10^{−3} | 8.07 × 10^{−3} | |

RS-MCS-10^{3} | 2.37 × 10^{−1} | 4.06 × 10^{−2} | 2.02 × 10^{−2} | 1.67 × 10^{−2} | 6.89 × 10^{−2} | 2.91 × 10^{−2} |

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## Share and Cite

**MDPI and ACS Style**

Ou, Z.; Lou, Y.; Wang, J.; Li, Y.; Yang, K.; Peng, S.; Tang, J.
The Effect of Power Flow Entropy on Available Load Supply Capacity under Stochastic Scenarios with Different Control Coefficients of UPFC. *Sustainability* **2023**, *15*, 6997.
https://doi.org/10.3390/su15086997

**AMA Style**

Ou Z, Lou Y, Wang J, Li Y, Yang K, Peng S, Tang J.
The Effect of Power Flow Entropy on Available Load Supply Capacity under Stochastic Scenarios with Different Control Coefficients of UPFC. *Sustainability*. 2023; 15(8):6997.
https://doi.org/10.3390/su15086997

**Chicago/Turabian Style**

Ou, Zhongxi, Yuanyuan Lou, Junzhou Wang, Yixin Li, Kun Yang, Sui Peng, and Junjie Tang.
2023. "The Effect of Power Flow Entropy on Available Load Supply Capacity under Stochastic Scenarios with Different Control Coefficients of UPFC" *Sustainability* 15, no. 8: 6997.
https://doi.org/10.3390/su15086997