# Optimal Micro-PMU Placement Using Mutual Information Theory in Distribution Networks

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## Abstract

**:**

## 1. Introduction

_{2}. The integrations of DGs would also cause bidirectional power flow and great uncertainties, which makes the supervision and operation of distribution more complicated. It is necessary to use different strategies to improve the reliability, efficiency, and safety in planning and operation of distribution, such as fault analysis [1,2], dynamic operation and control strategies [3], and the improvement of transient stability [4]. Therefore, the distribution system needs powerful and accurate monitoring meting devices. Phasor measurement unit (PMU) is the current most advanced metering device of synchronized measurement technology which plays an important role in wide-area measurement system [5]. Phasor measurement unit can provide real-time and high-accurate magnitude and phase angle measurements of both voltage and current. Based on PMU measurements, many applications, such as state estimation, fault location, outage management, and event detection can be exploited [6,7]. For example, a hierarchical architecture for monitoring the distribution grid based on PMU data is proposed [8]. A linear model which considers PMU location for the observability assessment in different contingencies is presented [9]. Currently, PMU has been widely applied in the transmission network, but not in the distribution network. With the development of PMU technology and the integration of DGs, it is promising to deploy PMUs in distribution level. The number of nodes in the distribution network is far larger than that of the transmission network. So, it is important to study the optimal PMU placement (OPP) problem considering the characteristics of distribution network.

- (1)
- The 2PEM is proposed to solve the stochastic state estimation considering the measurement errors of distribution network caused by DGs and pseudo injection measurements.
- (2)
- The differential entropy of mutual information is proposed to evaluate the uncertainty of network which can be used in the AC power flow mode in distribution level.
- (3)
- The improved IENS is proposed to obtain the optimal μPMU placement for both complete and incomplete observability under the improvement of initial IENS.

## 2. Mathematical Formulation of Optimal μPMU Placement

#### 2.1. Differential Entropy for Assessing Uncertainty of Network

#### 2.2. Stochastic State Estimation Using Two-Point Estimation Method

- (1)
- Determine the number of uncertain variables of pseudo measurements as n, and the number of certain measurements obtained from PMU and SCADA as ${n}_{1}$.
- (2)
- Set $E\left(X\right)=0$ and $E\left({X}^{2}\right)=0$.
- (3)
- Set $t=1$, and carry out the following steps until $t=n$.
- (4)
- Calculate concentrations ${y}_{t,1},\text{}{y}_{t,2}$, locations of concentrations ${\xi}_{t,1},{\xi}_{t,2}$ and its probabilities ${P}_{t,1},\text{}{P}_{t,2}$$${\xi}_{t,1}=\sqrt{n},\text{}{\xi}_{t,2}=-\sqrt{n}\text{}$$$${P}_{t,1}={P}_{t,2}=\frac{1}{2n}\text{}$$$${y}_{t,1}={\mu}_{Y,t}+{\xi}_{t,1}{\sigma}_{Y,t}\text{}$$$${y}_{t,2}={\mu}_{Y,t}+{\xi}_{t,2}{\sigma}_{Y,t}\text{}$$

- (1)
- Run the deterministic state estimation for ${y}_{t,i}$ by using $Y=\left[{\mu}_{Y,1},{\mu}_{Y,2},\dots ,{y}_{t,i},\dots ,{\mu}_{Y,n},{y}_{n+1},\dots ,{y}_{n+{n}_{1\text{}}}\right]$.
- (2)
- Update $E\left(X\right)$ and $\text{}E\left({X}^{2}\right)$$$E(X)\cong {\displaystyle \sum _{t=1}^{n}{\displaystyle \sum _{i=1}^{2}\left({P}_{t,i}{h}^{\prime}\left(\left[{\mu}_{Y,1},{\mu}_{Y,2},\dots ,{\mu}_{t,i},\dots ,{\mu}_{Y,n},{y}_{n+1},\dots ,{y}_{n+{n}_{1}}\right]\right)\right)}}\text{}$$$$E({X}^{2})\cong {\displaystyle \sum _{t=1}^{n}{\displaystyle \sum _{i=1}^{2}\left({P}_{t,i}{h}^{\prime}{\left(\left[{\mu}_{Y,1},{\mu}_{Y,2},\dots ,{\mu}_{t,i},\dots ,{\mu}_{Y,n},{y}_{n+1},\dots ,{y}_{n+{n}_{1}}\right]\right)}^{2}\right)}}\text{}$$

#### 2.3. Information Entropy Evaluation and Node Selection Strategy for μPMU Sets

#### 2.3.1. Information Entropy Evaluation and Node Selection Strategy

- (1)
- Define the set of candidate buses from which to choose for the installation of new μPMU: $\text{}{\mathit{B}}_{c}=\left\{{b}_{{1}_{c}},{b}_{{2}_{c}},\dots ,{b}_{{n}_{c}}\right\}$. The location of new μPMU is selected from the buses in$\text{}{\mathit{B}}_{c}$. It is assumed to contain all the buses in the network if there is no mandatory μPMU allocated beforehand. The bus to be installed with new μPMU will be discarded from $\text{}{\mathit{B}}_{c}$ after the selection of new μPMU.
- (2)
- Define the set of buses for the installation of μPMU as ${\mathit{B}}_{s}=\left\{{b}_{{1}_{a}},{b}_{{2}_{a}},\dots ,{b}_{{n}_{a}}\right\}$. The buses in ${\mathit{B}}_{s}$ would be installed with μPMUs. ${\mathit{B}}_{s}$ is null if there is no μPMU allocated beforehand. The bus to be installed with new μPMU will be added into ${\mathit{B}}_{s}$ after the selection of new μPMU.
- (3)
- Set the number of μPMUs to be installed in the network as$\text{}{n}_{s}$.

- (1)
- Run the following part:
For $l=1,2,\dots ,{n}_{c}$: (a) Build a new set: ${\mathit{B}}_{s}^{l}=\left[{b}_{1},{b}_{2},\dots ,{b}_{{n}_{a}}|{b}_{l}\right]$ where first ${n}_{a}$ columns are ${n}_{a}$ buses already installed with μPMUs and last column means the lth bus candidate for the location of μPMU. (b) Add μPMU measurements of ${\mathit{B}}_{s}^{l}$ into initial measurement configuration as new measurement configuration. Then run stochastic state estimation by using 2PEM under lth measurement configuration and calculate its differential entropy ${E}_{l}$ using Equation (17). End - (2)
- Find bus k which maximizes the improvement in information gain of differential entropy.$${b}_{k}=\mathrm{arg}\left(\underset{l}{\mathrm{max}}\left(\left|{E}_{0}-{E}_{l}\right|\right)\right)\text{}\text{}$$Then $\text{}{E}_{0}={E}_{k}$, excludes bus k from ${\mathit{B}}_{c}$, adds bus k into ${\mathit{B}}_{s}$,$${\mathit{B}}_{c\text{}}\leftarrow {\mathit{B}}_{c\text{}}\backslash \left\{{b}_{k}\right\}\text{}and\text{}{\mathit{B}}_{s\text{}}={\mathit{B}}_{s}\cup \left\{{b}_{k}\right\}$$

#### 2.3.2. Selection Rules to Be Noticed

#### 2.3.3. Improved Information Entropy Evaluation and Node Selection Strategy

## 3. Case Studies

#### 3.1. Optimal Placement for Full Observability by Improved IENS

_{54}, the μPMU would also be located at b

_{54}instead of b

_{53}.

#### 3.2. Incomplete Observability Analysis

#### 3.3. Effects of Two Rules

#### 3.4. Limitations of the Improved IENS

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

Sets and Indices | |

${\mathit{B}}_{c}$ | The set of candidate buses where the installation of new micro-phasor measurement unit (μPMU) is selected from. |

${\mathit{B}}_{s}$ | The set of buses for the installation of μPMU, the location of new μPMU will be added in this set. |

${\mathit{B}}_{s}^{l}$ | The set of buses ${\mathit{B}}_{s}$ at $l$th iteration. |

${\mathit{B}}_{ak}$ | The set of buses which contains the buses adjacent to bus k on two-bus branches. |

${l}_{i-j}$ | The line connected between bus i and bus j. |

${b}_{i}$ | The ith bus. |

${b}_{k}$ | The bus k which maximizes the improvement in information gain of differential entropy. |

${b}_{add}$ | The selected bus to be installed with new μPMU in current round. |

Parameters | |

$\sigma $ | The standard deviation of variable x. |

$\mu $ | The mean of variable x. |

$z$ | Vector of measurements. |

$\epsilon $ | Error vector of measurements. |

${\sigma}_{i}^{2}$ | Variance of ith measurements. |

$H\left(x\right)$ | The Jacobian matrix. |

${m}_{i}$ | The number of measurements. |

${W}_{i}$ | The covariance matrix of measurements. |

$W$ | The block diagonal matrix. |

$q$ | The number of iteration in weighted least square (WLS) state estimation. |

$R$ | The general rotation matrix. |

$Y$ | The measurements vector in state estimation. |

${y}_{i}$ | The ith measurement in state estimation. |

$n$ | The number of uncertain variables of pseudo measurements. |

${n}_{1}$ | The number of certain measurements obtained from phasor measurement unit (PMU) and supervisory control and data acquisition (SCADA) system. |

$E\left(X\right)$ | The expectation of state variables vector. |

$E\left({X}^{2}\right)$ | The expectation of square of state variables vector. |

${y}_{t,i}$ | The concentration of measurement at step t. |

${\xi}_{t,i}$ | The location of concentration of measurement at step t. |

${P}_{t,1}$ | The probability of concentration of measurement at step t. |

${y}_{t,i}$ | The concentration of measurement at step t. |

${Y}_{t}$ | The measurements vector at step t. |

${\mu}_{Y,t}$ | The mean value of ${Y}_{t}$, obtained from measurement information. |

${\sigma}_{Y,t}$ | The standard deviation of ${Y}_{t}$, obtained from measurement information. |

${\mu}_{X}$ | The mean value of state variables $X$. |

${\sigma}_{X}$ | The standard deviation of state variables $X.$ |

${E}_{0}$ | The initial differential entropy of the network. |

$E$ | The differential entropy of the network. |

$N$ | The number of all buses in the network. |

${\sigma}_{{V}_{i}}$ | The standard deviation of voltage amplitude at bus i. |

${\sigma}_{{\theta}_{i}}$ | The standard deviation of voltage phase angle at bus i. |

${n}_{c}$ | The number of candidate buses which can be the location for new μPMU. |

$l$ | The number of round in the information entropy evaluation and node selection strategy (IENS). |

${E}_{l}$ | The differential entropy of the network at $l$th iteration. |

${n}_{s}$ | The number of μPMUs decided to be installed in the network according to the budget. |

${n}_{TM}$ | The number of optimal placement calculated by topological method for network full observability. |

Variables | |

$x$ | State variables of network, including magnitude and phasor angle of voltage. |

$X$ | The state variables vector in state estimation. |

$h\left(x\right)$ | Nonlinear function of state variables. |

I(x) | Differential entropy for the continuous variable x. |

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**Table 1.**Minimal micro-phasor measurement unit (μPMU) numbers for full observability with and without feeder terminal unit (FTU) measurements.

With FTU Measurements | Without FTU Measurements | |
---|---|---|

Topological method | 45 | 47 |

Improved IENS | 45 | 47 |

Genetic method | 46 | 48 |

**Table 2.**Observability capability of improved IENS and topological method under different circumstances.

Number of μPMUs | 90% Pseudo Measurement Configurations | 80% Pseudo Measurement Configurations | ||
---|---|---|---|---|

Topological Method (Mean) | Improved IENS | Topological Method (Mean) | Improved IENS | |

40 | 90.08% | 97.00% | 65.66% | 88.40% |

35 | 78.93% | 82.50% | 38.26% | 42.40% |

30 | 66.55% | 73.20% | 18.79% | 26.70% |

Method | Optimal μPMU Placement | Tested by Numerical Method |
---|---|---|

Topological method | 2, 4, 6, 9, 15, 16, 20, 22, 24, 28, 29, 31, 32, 37, 39, 41, 43, 46, 48, 52, 54, 56, 59, 61, 64, 66, 68, 71, 75, 79, 83, 85, 88, 90, 92, 94, 96, 98, 101, 104, 107, 109, 111, 114, 122 | observable |

IENS | 2, 14, 68, 53, 61, 77, 106, 41, 27, 90, 9, 55, 15, 79, 24, 111, 48, 122, 82, 94, 65, 37, 16, 99, 70, 46, 75, 96, 20, 30, 101, 103, 51, 59, 114, 6, 124, 121, 19, 58, 123, 109, 88, 5, 28 | unobservable |

Improved IENS | 2, 9, 20, 61, 22, 68, 56, 79, 107, 109, 41, 88, 32, 24, 28, 59, 71, 92, 48, 75, 15, 101, 111, 43, 54, 106, 83, 85, 46, 94, 37, 64, 66, 4, 104, 31, 90, 52, 114, 16, 6, 96, 39, 99, 29 | observable |

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## Share and Cite

**MDPI and ACS Style**

Wu, Z.; Du, X.; Gu, W.; Ling, P.; Liu, J.; Fang, C. Optimal Micro-PMU Placement Using Mutual Information Theory in Distribution Networks. *Energies* **2018**, *11*, 1917.
https://doi.org/10.3390/en11071917

**AMA Style**

Wu Z, Du X, Gu W, Ling P, Liu J, Fang C. Optimal Micro-PMU Placement Using Mutual Information Theory in Distribution Networks. *Energies*. 2018; 11(7):1917.
https://doi.org/10.3390/en11071917

**Chicago/Turabian Style**

Wu, Zhi, Xiao Du, Wei Gu, Ping Ling, Jinsong Liu, and Chen Fang. 2018. "Optimal Micro-PMU Placement Using Mutual Information Theory in Distribution Networks" *Energies* 11, no. 7: 1917.
https://doi.org/10.3390/en11071917