# Credibility Theory-Based Available Transfer Capability Assessment

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## Abstract

**:**

## 1. Introduction

## 2. Credibility Theory

#### 2.1. Basic Concept

_{os}{A}:

_{i}} in P(Θ).

_{I}is a non-empty set, Posi{·}, i=1,2,…,n meets the first three axioms, and Θ=Θ

_{1}×Θ

_{2}×…×Θ

_{n}, for any set A∈P(Θ), ${P}_{os}\{A\}=\underset{({\mathrm{\theta}}_{1},{\mathrm{\theta}}_{2},\cdots ,{\mathrm{\theta}}_{n})\in A}{\mathrm{sup}}{P}_{os1}\{{\mathrm{\theta}}_{1}\}\wedge {P}_{os2}\{{\mathrm{\theta}}_{2}\}\wedge \cdots \wedge {P}_{osn}\{{\mathrm{\theta}}_{n}\}$.

_{OS}meets the first three axioms, P

_{OS}is defined as the possibility measure. (Θ, P(Θ), P

_{OS}) is a possibility space. If A

^{c}is the complement of A, the necessity measure N

_{ec}is defined as N

_{ec}{A}=1 − P

_{os}{A

^{c}}. Obviously, P

_{OS}and N

_{ec}are one pair of dual measures, so the credibility measure is defined as follows:

_{OS}) to the real line R. The triangle fuzzy variable and the trapezoidal fuzzy variable are commonly used ones.

_{OS}), the membership function of ξ is:

_{G}to represent its available output. Suppose the available output ξ

_{G}= (35, 50, 55) MW, we have the credibility measure of ξ

_{G}:

#### 2.2. Random Fuzzy Variable

_{OS}) to the set of random variables.

_{pro}

_{,G}(ε = ξ

_{G}) = 0.99, P

_{pro}

_{,G}(ε = 0) = 0.01. ξ

_{G}is a triangle fuzzy variable as defined in Example 1.

^{2}]

## 3. Credibility Theory-Based ATC Assessment Approach

#### 3.1. Modeling Uncertainties in ATC Calculation

_{G}with two-point distribution:

_{pro}

_{,G}is the state occurrence probability of the generator; ε

_{G}= ξ

_{G}refers to the normal on-state, and ε

_{G}= 0 is the off-state; the triangle fuzzy variable ξ

_{G}is used to represents the fuzzy available output of a generator, and F

_{fuz,G}represents its membership function; a

_{G}

_{,L}, a

_{G}

_{,M}, a

_{G}

_{,H}are the minimum possible value, the most likely possible value and the maximum possible value of ξ

_{G}, respectively. In this paper triangle fuzzy variables are used to represent the fuzzy states of the generator, transmission line and load, but other types of fuzzy variables such as trapezoidal fuzzy variables, can also be used according to specific conditions.

_{B}as follows:

_{pro}

_{,B}is the state occurrence probability of the transmission line; ε

_{B}= 1 expresses the normal on-state, and ε

_{B}= 0 is the off-state; the triangle fuzzy variable ξ

_{B}is used to represents the fuzzy failure rate of the transmission line, and F

_{fuz,B}represents its membership function; a

_{B}

_{,L}, a

_{B}

_{,M}, a

_{B}

_{,H}are the minimum possible value, the most likely possible value and the maximum possible value of ξ

_{B}, respectively.

_{L}, σ

_{L}) in traditional methods. Here the parameter β

_{L}is the expected value of the distribution, which usually takes the predicted value of the nodal load. The parameter σ

_{L}is the variance of the distribution, which shows the degree of deviation between the real value of the load and the forecasted one, and is usually determined according to the operator’s experience. Actually, σ

_{L}is not a fixed value, but a fuzzy one, so the fluctuation of nodal load has both randomness and fuzziness features, so the random fuzzy load ε

_{L}is represented as follows:

_{L}is used to represents the fuzzy variance of a nodal load, and F

_{fuz,L}represents its membership function; a

_{L}

_{,L}, a

_{L}

_{,M}, a

_{L}

_{,H}are the minimum possible value, the most likely possible value and the maximum possible value of ξ

_{L}, respectively.

#### 3.2. ATC Calculation Model

#### 3.3. ATC Assessment Indices

- (a)
- The expected value of random fuzzy ATC—E
_{ATC}—it comprehensively reflects the ATC of a power system.$$\begin{array}{l}{E}_{pro-fuzz,\mathrm{ATC}}={\displaystyle {\int}_{0}^{\infty}{C}_{r}\{\mathrm{\theta}\in \mathrm{\Theta}|E[{\mathrm{\epsilon}}_{\text{ATC}}(\mathrm{\theta})]\ge r\}dr}\\ \text{\hspace{1em}\hspace{1em}\hspace{1em}}-{\displaystyle {\int}_{-\infty}^{0}{C}_{r}\{\mathrm{\theta}\in \mathrm{\Theta}|E[{\mathrm{\epsilon}}_{\text{ATC}}(\mathrm{\theta})]\le r\}dr}\end{array}$$ - (b)
- The variance of random fuzzy ATC—V
_{ATC}—it expresses the fluctuation of ATC and reflects the impacts of uncertainties on ATC:$${V}_{pro-fuz\text{,ATC}}=E[{({\mathrm{\epsilon}}_{\text{ATC}}-{E}_{\text{ATC}})}^{2}]$$ - (c)
- Calculation time t: it reflects the efficiency of different ATC calculation approaches under the same initial conditions.

#### 3.4. Parallel Algorithm with Bootstrap Method

#### 3.5. Random Fuzzy Simulation Based ATC Assessment

- (1)
- Read the initial parameters of generators, transmission lines and loads, build basic system information and set e = 0, i = 1.
- (2)
- From the set Θ extract a θ
_{k}which meets P_{OS}{θ_{k}} ≥ ε (ε is a permissible small value making the sample space be bounded), get the variables of generators, transmission lines and loads, and produce a set of fuzzy sampling vectors: ${\mathrm{\xi}}_{i,G},{\mathrm{\xi}}_{i,B},{\mathrm{\xi}}_{i,L}$. - (3)
- According to ${\mathrm{\xi}}_{i,G},{\mathrm{\xi}}_{i,B},{\mathrm{\xi}}_{i,L}$ and the corresponding equipment random parameters, get the system state vectors: ${\epsilon}_{G}({\mathrm{\xi}}_{i,G}),{\epsilon}_{B}({\mathrm{\xi}}_{i,B}),{\epsilon}_{L}({\mathrm{\xi}}_{i,L})$, change the random fuzzy models of generators, transmission lines and loads to the random ones, then the fuzziness is eliminated. Then the Monte Carlo random simulation is applied M times, and the value of ATC can be calculated by the improved repeated power flow method for each simulation state.
- (4)
- By the bootstrap method re-sample in the above obtained ATC values, and calculate their expected value of ATC. Figure 2 illustrates the bootstrap method procedure.
- (5)
- Set sample counter i = i + 1, and repeat (2) to (4) for N times.
- (6)
- Set a = min
_{1≤i≤N}E_{pro}[ε_{i,ATC}], b = max_{1≤i≤N}E_{pro}[ε_{i,ATC}], and loop control variable w = 1. - (7)
- From the interval [a, b] randomly generate r
_{w}and calculate $e=e+{C}_{r}\{\mathrm{\theta}\in \mathrm{\Theta}|{E}_{pro}[{\epsilon}_{i,\text{ATC}}]\ge {r}_{w}\}$. - (8)
- Set w = w + 1, and repeat (7) for N times.
- (9)
- Lastly calculate the expected value and variance of ATC as follows:$${E}_{pro-fuz\text{,ATC}}={E}_{pro-fuz}[{\mathrm{\epsilon}}_{\text{ATC}}]=a\vee 0+b\wedge 0+e\times (b-a)/N,{V}_{pro-fuz\text{,ATC}}=E[{({\mathrm{\epsilon}}_{\text{i,ATC}}-{E}_{\text{ATC}})}^{2}].$$

## 4. Numerical Example

#### 4.1. IEEE-30-bus System

#### Part 1: Compatibility analysis between the proposed approach and the conventional Monte Carlo random simulation.

Method | Case | Generators | Transmission Lines | Loads | |
---|---|---|---|---|---|

λ_{G} | ξ_{G} | ξ_{B} | ξ_{L} | ||

Monte Carlo random simulation (10,000 times) | A | 0.01 | 1 | None | None |

B | None | None | 0.02 | None | |

C | None | None | None | 0.02 | |

Random fuzzy simulation | D | 0.01 | (0.9999, 1, 1.0001) | None | None |

E | None | None | (0.0199, 0.0200, 0.0201) | None | |

F | None | None | None | (0.0199, 0.0200, 0.02001) |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

A | 8.5883 | 3.8085 |

D | 8.4657 | 3.6602 |

Error (%) | −1.4275 | −3.8939 |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

B | 9.7541 | 121.4598 |

E | 10.8670 | 123.5610 |

Error (%) | 11.4096 | 1.7300 |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

C | 11.3496 | 117.2293 |

F | 11.7530 | 117.3173 |

Error (%) | 3.5543 | 0.0751 |

#### Part 2: The comparison between the proposed assessment method and the traditional Monte Carlo simulation approach.

Case | Generators | Transmission Lines | Loads | |
---|---|---|---|---|

λ_{G} | ξ_{G} | ξ_{B} | ξ_{L} | |

J | 0.01 | (0.700, 1.0000, 1.100) | None | None |

H | None | None | (0.0100, 0.0200, 0.0600) | None |

I | None | None | None | (0.0100, 0.0200, 0.0600) |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

A | 8.5883 | 3.8085 |

J | 8.4212 | 3.8310 |

Error (%) | 1.9457 | 0.5908 |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

B | 9.7541 | 121.4598 |

H | 10.6504 | 168.7090 |

Error (%) | 9.1890 | 38.9011 |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

C | 11.3496 | 117.2293 |

I | 14.3061 | 261.7169 |

Error (%) | 26.0494 | 123.2521 |

#### Part 3: The sensitivity analysis to the fuzzy influencing factors of ATC.

_{B}= (0.01,0.02,0.06)) and the others are simulated only as random variables by the failure rate ξ

_{B}= 0.02.

_{L}= (0.01, 0.02, 0.06)) and the others the random variables with feature (ξ

_{L}= 0.02), while on the basis of Case I, Case L reduces the fuzzy range of the load variance (ξ

_{L}= (0.01, 0.02, 0.04)).

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

H | 10.6504 | 168.709 |

J | 10.5893 | 141.8906 |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

I | 14.3061 | 261.7169 |

K | 14.3047 | 261.6058 |

L | 13.0457 | 170.9183 |

#### Part 4: The comparison about the processing efficiency.

Case | Bootstrap Method | Dual-core Parallel Computing Technique |
---|---|---|

M | √ | √ |

N | × | √ |

O | √ | × |

P | × | × |

#### 4.2. An Actual Power System in Northwest China

Method | Case | Generators | Transmission Lines | Loads | |
---|---|---|---|---|---|

λ_{G} | ξ_{G} | ξ_{B} | ξ_{L} | ||

Monte Carlo random simulation (10,000 times) | Q | 0.01 | 1 | 0.02 | 0.02 |

Random fuzzy simulation | R | 0.01 | (0.9400, 1, 1.1400) | (0.0100, 0.0190, 0.0400) | (0.0100, 0.0190, 0.0400) |

Case | E_{pro-fuzz,}_{ATC} (MW) | V_{pro-fuzz,}_{ATC} (MW^{2}) |
---|---|---|

Q | 4417 | 4,702,842 |

R | 4145 | 5,241,506 |

Error (%) | −6.1591 | 11.4540 |

## 5. Conclusions

## Author Contributions

## Conflicts of Interest

## Nomenclature

Θ | Nonempty set. |

ϕ | Empty set. |

P(Θ) | Power set of Θ. |

∧ | Minimum operator. |

∨ | Maximum operator. |

P_{os} | Possibility measure of fuzzy event. |

N_{ec} | Necessity measure of fuzzy event. |

C_{r} | Credibility measure of fuzzy event. |

μ | Membership function of fuzzy variable. |

B | Borel set. |

sup | Supremum. |

E_{fuz} | Expected value of fuzzy variable. |

E_{pro} | Expected value of random variable. |

E_{pro}_{-fuz} | Expected value of random fuzzy variable. |

R | Set of real numbers. |

(Θ,P(Θ),P_{OS}) | Possiblity space. |

P_{pro}_{,G} | State occurrence probability of generator. |

P_{pro}_{,B} | State occurrence probability of transmission line. |

ε_{G} | Random fuzzy state of generator. |

ε_{B} | Random fuzzy state of transmission line. |

ε_{L} | Random fuzzy nodal load. |

λ_{G} | Forced outage rate of generator. |

ξ_{G} | Fuzzy available output of generator. |

ξ_{B} | Fuzzy failure rate of transmission line. |

ξ_{L} | Fuzzy variance of a nodal load. |

F_{fuz,G} | Membership function of ξ _{G}. |

F_{fuz,B} | Membership function of ξ _{B}. |

F_{fuz,L} | Membership function of ξ _{L}. |

a_{*}_{,L} | Minimum possible value. |

a_{*}_{,M} | Most likely possible value. |

a_{*}_{,H} | Maximum possible value. |

β_{L} | Load forecasting value. |

f | Electricity purchase cost. |

P_{g} | Active power output of the generator g. |

P_{g}^{max}, P_{g}^{min} | Upper and lower limits of P _{g}. |

Q_{g} | Reactive power output of the generator g. |

Q_{g}^{max}, Q_{g}^{min} | Upper and lower limits of Q _{g}. |

P_{d} | Active load of the node d. |

Q_{d} | Reactive load of the node d. |

V_{z} | Voltage of the node z. |

V_{z}^{max}, V_{z}^{min} | Upper and lower limits of V _{z}. |

S_{l} | Apparent power of the transmission line l. |

S_{l}^{max} | Maximum value of S _{l}. |

G_{xy} | Conductance of the branch from node x to y. |

B_{xy} | Susceptance of the branch from node x to y. |

δ_{xy} | Voltage phase angle difference of the branch from node x to y. |

ε_{ATC} | Random fuzzy value of ATC. |

E_{pro-fuz,}_{ATC} | Expected value of random fuzzy ATC. |

V_{pro-fuz,}_{ATC} | Variance of random fuzzy ATC. |

t | Calculation time. |

N, M, W | Sampling times. |

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**MDPI and ACS Style**

Zheng, Y.; Yang, J.; Hu, Z.; Zhou, M.; Li, G. Credibility Theory-Based Available Transfer Capability Assessment. *Energies* **2015**, *8*, 6059-6078.
https://doi.org/10.3390/en8066059

**AMA Style**

Zheng Y, Yang J, Hu Z, Zhou M, Li G. Credibility Theory-Based Available Transfer Capability Assessment. *Energies*. 2015; 8(6):6059-6078.
https://doi.org/10.3390/en8066059

**Chicago/Turabian Style**

Zheng, Yanan, Jin Yang, Zhaoguang Hu, Ming Zhou, and Gengyin Li. 2015. "Credibility Theory-Based Available Transfer Capability Assessment" *Energies* 8, no. 6: 6059-6078.
https://doi.org/10.3390/en8066059