# Voltage Stability Index Calculation by Hybrid State Estimation Based on Multi Objective Optimal Phasor Measurement Unit Placement

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Power System State Estimation

#### 2.1. Conventional State Estimation by Weighted Least Square

**z**is a measurement vector,

**x**is a state vector,

**h**(

**x**) is a nonlinear function with regard to state vector, and

**ε**is a measurement error vector. Since (1) is the nonlinear equation, Weighted Least Square (WLS) is employed to obtain the optimal value of state vector

**x**. WLS minimizes the sum of weighted squares of residuals by:

**R**= diag{σ

_{1}

^{2}, …, σ

_{nm}

^{2}} is a diagonal covariance matrix of measurement error formed by the measurement error variance σ

_{i}

^{2}, and i is the measurement number ranging from 1 to nm. Since

**h**(

**x**) is a nonlinear function, it is linearized around an equilibrium point as

**H**= ∂

**h**(

**x**)/ ∂

**x**. Thus, the solution of the optimization problem in (1) can be obtained by iterating the following equation:

**G**=

**H**

^{T}

**R**

^{−1}

**H**is the gain matrix, and k is the present iteration number. WLS iteratively obtains the estimation value Δ$\widehat{x}$ by minimizing (3), until a terminal condition is satisfied by the following condition:

#### 2.2. Hybrid State Estimation with Measurment Uncertainty Propagation of Phasor Measurement Unit (PMU) Measurements

**z**’ is the new measurement vector composed of the bus voltage phasor state vector previously obtained by SCADA SE and the bus voltage phasor measurement vector.

**x**is the new voltage phasor state vector,

**H’**is the new linear Jacobian matrix, and

**ε’**is the new measurement error vector. The detail of the equation can be represented by the following equation:

_{R}, V

_{I}, R, and I mean the real and imaginary value of the complex voltage, and the measurement error for real and imaginary voltage, respectively. Direct measurement is the measurement value obtained by PMU directly. Pseudo measurement is the measurement value obtained by calculation using several direct/pseudo measurements already obtained. In the pseudo measurements, the measurement error can be bigger than the direct measurement by measurement uncertainty propagation. The authors have proposed the MOOPP formulation to consider the influence of the uncertainty propagation which occurs in the use of zero injection, the detail of which is explained in our previous paper [19]. Hereby, the measurement equation is linear, and is easily solved by the following equation without any iterations:

**G**′ =

**H**′

^{T}

**R**′

^{−1}

**H**′ is the new gain matrix, and

**R**′ = diag{

**R**,

**R**

^{PMUd},

**R**

^{PMUp}} is the extended diagonal covariance matrix of measurement error.

**R**

^{PMUd}and

**R**

^{PMUp}are formed by measurement error variance of direct measurement and pseudo measurement with uncertainty propagation, respectively.

**p**(k) is the maximum uncertainty specified by the meter manufacturer in the measurement

**p**(k). This paper determines the maximum measurement uncertainty of each meter with reference to an article by Valverde et al. [21]. By the classical uncertainty propagation theory [24], uncertainty of pseudo measurement is affected by the set of measurement uncertainty that calculates it. Thus, measurement uncertainty of pseudo measurements is generally given by:

**p**is a measurement vector used to compute the pseudo measurements in this case, and u(n) is the standard uncertainty of measurement n. m is the length of vector

**p**.

## 3. Voltage Stability Index Calculation

- Sensitivity analysis by Jacobian
- Bus VSIs
- Line VSIs

#### 3.1. Critical Boundary Index

_{l}and Q

_{l}are active and reactive power flow at the receiving end, respectively. V

_{l}and V

_{k}are voltage magnitude at the receiving and sending ends, respectively. θ

_{l}and θ

_{k}are voltage angle at the receiving and sending ends, respectively. r

_{kl}and x

_{kl}are resistance and reactance on line k-l. By separating the real and imaginary parts of (11), the following equation can be derived:

^{2}θ + cos

^{2}θ = 1:

_{0},Q

_{0}) and the nearest voltage stability critical boundary point C(X,Y) is a function of f(X,Y). The minimum distance between them is given by:

_{kl}is the Critical Boundary Index on the line k-l. CBI approaches from a certain value to 0, which means the voltage stability limit.

#### 3.2. Active and Reactive Power Estimation by Obtained Bus Voltage Phasor

_{kl}is the voltage phase difference between nodes k and l. The conceptual procedure of CBI estimation using voltage phasor obtained by HSE is depicted in Figure 3.

## 4. Multi Objective Optimal Phasor Measurement Unit (PMU) Placement

#### 4.1. Formulation

_{VC}indicates the total PMU placement cost and TVE

_{max}is the maximum mean value of Total Vector Error (TVE) in all system buses for all SE scenario sets. Both details will be given later.

**y**is the PMU placement decision variable vector given by zeros and ones. If y

_{i}as an element of

**y**is 1, a PMU is placed at bus i, otherwise

**y**= 0. Thus, (29) is set to avoid placing none of the PMUs in the system. V

_{i}and θ

_{i}are voltage magnitude and angle in polar form, respectively. The hat (^) upon them indicates the estimated values whereas those without a hat mean true values ${E}_{max}^{mag}$ and ${E}_{max}^{ang}$ are limits of voltage magnitude and angle error, respectively. nb is the number of buses in the power system, np is the considered number of power flow scenarios. As a prior condition, RTUs are already placed at the system redundantly, according to the two-step HSE procedure in Section 2. Hence, the binary decision variable of this MOOPP only deals with PMU voltage/current channel placement.

_{VC}is represented by the following equation:

**b**= [1, …, 1] with the length of nb.

**D**is the decision variable for the current channel selection, in which the element d

_{ij}is 1 if the current channel is placed at line i-j otherwise 0. Since

**D**is based on the bus connectivity, it is a diagonal matrix. w

_{v}and w

_{c}are cost weight coefficients for a PMU itself with a voltage measurement channel and a current measurement channel, respectively.

_{VC}and TVE

_{max}are in a trade-off relationship with each other. Thus, in order to improve the objective function value, it is necessary to obtain a solution which has at least no choice but to deteriorate the other one. This solution is called the Pareto optimal solution. The set of Pareto optimal solution is called a Pareto front.

#### 4.2. Optimization Method

#### 4.3. The Best Compromised Solution Selection

_{i}is the objective function value of objective i, and ${f}_{i}^{\mathrm{min}}$ and ${f}_{i}^{\mathrm{max}}$ are the minimum and the maximum objective function value of objective i obtained from the Pareto optimal solutions, respectively. The satisfaction degree can be calculated as follows [27]:

_{obj}is the number of objectives, and equal to 2 in the case of this paper, which the first objective is set to K

_{VC}, and the second one is set to TVE

_{max}. The BCS is selected with the largest value of S. The meaning of the BCS selected by Equations (34) and (35) is the most balanced solution in those objectives which it can be the criteria solution for the decision of the system operator.

## 5. Numerical Simulation Results and Discussions

#### 5.1. Configuration

#### 5.2. Pareto Optimal Solutions Obtained by Non-Dominated Sorting Genetic Algorithm II (NSGA-II)

- Method I: no consideration of the current channel selectivity. The current channel is placed at all lines incident to the PMU placed bus in calculation of K
_{VC}. The decision variable is only**y**. - Method II: no consideration of measurement uncertainty propagation in PMU pseudo measurement. Measurement uncertainty in HSE is always constant given by Table 3 regardless of the use of pseudo measurement.
- Method III: with consideration of both the current channel selectivity and measurement uncertainty propagation in PMU pseudo measurements.

_{VC}> 8 (p.u.). Figure 4b shows the comparison of Method II and Method III. In the optimization, some Pareto solutions in Method II are better than Method III, whereas the actual Pareto front in Method II stays apart from Method III. This is caused by ignoring the influence of measurement uncertainty of PMU pseudo measurement especially in case that many PMUs are placed in the power system, meaning many possibilities to use PMU pseudo measurement. Additionally, Method III is numerically proven to be better than the others using the Ratio of Non-dominated Individuals (RNI) [29]. RNI in two methods is calculated by taking the number of non-dominated solutions divided by the sum set of all solutions in the Pareto front of both methods. If RNI > 0.5, the method is better than the other. Table 4 shows the RNI, and therefore, the Pareto front obtained by Method III is quantitatively the best method of the three. Hence, now we have obtained good multiple PMU solutions in terms of SE accuracy and the PMU placement cost.

#### 5.3. CBI Estimation Using Bus Voltage Phasor Obtained by the Hybrid State Estimation (HSE) Based on the PMU Placement

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 7.**Statistical box plot of voltage magnitude estimation error in the BCS. (

**a**) SCADA SE, (

**b**) HSE.

**Figure 9.**CBI in case of bus 38 loading. (

**a**) CBI on line 38–39, (

**b**) Voltage magnitude at bus 38, (

**c**) Voltage angle difference between buses 38 and 39, (

**d**) Active power at receiving end on line 38–39, (

**e**) Reactive power at receiving end on line 38–39.

**Figure 10.**Decrease ratio of the Mean Absolute Percentage Error (MAPE) from SCADA SE for all Pareto solutions in Methods II and III. (

**a**) Discarding estimated state by PMU, (

**b**) Without discarding estimated state by PMU.

Parameter | Value |
---|---|

The number of buses | 39 |

The number of lines | 52 |

The number of load buses | 19 |

The number of the current channel placement candidates | 104 |

The length of decision variable in MOOPP | 143 |

Parameter | Value/Method |
---|---|

The population size | 70 |

Crossover rate | 0.95 |

Mutation rate | 0.05 |

The number of generations | 1000 |

The crossover method | Uniform crossover |

**Table 3.**Multi Objective OPP (MOOPP), Supervisory Control and Data Acquisition State Estimation (SCADA SE), and Hybrid State Estimation (HSE) parameters.

Class | Parameter | Value |
---|---|---|

MOOPP problem | A PMU and a voltage channel cost w_{V} [28] | 1.0 (p.u.) |

A current channel cost w_{C} [28] | 0.15 (p.u.) | |

Estimation error limit for voltage magnitude | 15 (%) | |

Estimation error limit for voltage angle | 10 (deg) | |

The number of power flow scenarios | 1000 | |

Maximum measurement uncertainty [21] | SCADA injection | 2 (%) |

SCADA flow | 2 (%) | |

PMU voltage magnitude | 0.02 (%) | |

PMU current magnitude | 0.03 (%) | |

PMU phase angle | 0.01 (deg) |

Method I | Method II | |
---|---|---|

Method III | 0.8125:0.1875 | 1:0 |

Class | Value |
---|---|

PMU placement buses | 2, 5, 16, 23, 26, 39 |

Current channel placement lines | 2-11, 2-19, 5-30, 16-1, 16-15, 16-21, 23-22, 23-24, 26-25, 26-27, 26-29, 26-31, 26-34, 39-9, 39-36, 39-38 |

K_{VC} | 8.40 |

TVE_{max} | 0.0286 |

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**MDPI and ACS Style**

Matsukawa, Y.; Watanabe, M.; Abdul Wahab, N.I.; Othman, M.L.
Voltage Stability Index Calculation by Hybrid State Estimation Based on Multi Objective Optimal Phasor Measurement Unit Placement. *Energies* **2019**, *12*, 2688.
https://doi.org/10.3390/en12142688

**AMA Style**

Matsukawa Y, Watanabe M, Abdul Wahab NI, Othman ML.
Voltage Stability Index Calculation by Hybrid State Estimation Based on Multi Objective Optimal Phasor Measurement Unit Placement. *Energies*. 2019; 12(14):2688.
https://doi.org/10.3390/en12142688

**Chicago/Turabian Style**

Matsukawa, Yoshiaki, Masayuki Watanabe, Noor Izzri Abdul Wahab, and Mohammad Lutfi Othman.
2019. "Voltage Stability Index Calculation by Hybrid State Estimation Based on Multi Objective Optimal Phasor Measurement Unit Placement" *Energies* 12, no. 14: 2688.
https://doi.org/10.3390/en12142688