With the application of phasor measurement units (PMU) in the distribution system, it is expected that the performance of the distribution system state estimation can be improved obviously with the PMU measurements into consideration. How to appropriately place the PMUs in the distribution is therefore become an important issue due to the economical consideration. According to the concept of efficient frontier, a value-at-risk based approach is proposed to make optimal placement of PMU taking account of the uncertainty of measure errors, statistical characteristics of the pseudo measurements, and reliability of the measurement instrument. The reasonability and feasibility of the proposed model is illustrated with 12-node system and IEEE-33 node system. Simulation results indicated that uncertainties of measurement error and instrument fault result in more PMU to be installed, and measurement uncertainty is the main affect factor unless the fault rate of PMU is quite high.

State estimation (SE) is a function that takes advantage of the inherent redundancy among the available measurements in order to determine the best estimate of the system states defined as the set of all the bus voltage magnitudes and phase angles in the system [

For the SE performance, traditionally, there are three evaluation metrics: estimation accuracy, observability of the system, and numerical stability of the estimator. In the literature of SE in the transmission system, the objective of the meter placement is mainly to improve the network observability (e.g., [

With regard to the tradeoff between the estimation accuracy and cost, few papers address this problem. Pegoraro et al. [

As for the uncertainty of measure errors, most of the papers adopt Monte Carlo simulation based on a given probability distribution of the measure errors. For the reliability of the measure instruments, Liu et al. [

Addressing the above problems, i.e., i) How to quantify the estimation accuracy with an intuitive index taking account of the uncertainties in the process of SE? ii) How to make a tradeoff between the accuracy and cost? iii) How to address the reliability problem of the measure instruments), this paper proposed a framework for the optimal allocation of PMU in the SE of the distribution system based on the concept of efficient frontier and value at risk (VaR), and developed a Genetic Algorithm (GA) based approach to solve the proposed optimization model since GA is a widely used optimization algorithm and has been used in the optimization of PMU configuration [

The main innovations of this paper are as follows: i) VaR index is used to quantify the impact of uncertain factors such as PMU fault and measurement error on the accuracy of state estimation (compared with the Variance index used in most literatures, VaR can give an intuitive description of the uncertainty of estimation accuracy, which is helpful for decision makers to make a choice between Cost and Accuracy), ii) Based on the concept of Pareto optimality, a VaR-Cost effective frontier decision model is established to facilitate decision makers to select the Optimal PMU configuration scheme according to their own investment scale or estimation accuracy requirements (this scheme has fully considered PMU failure, measurement error and other uncertain factors).

In the following,

As discussed above, an optimal PMU allocation (i.e., the optimal PMU placement) is the one which has higher estimation accuracy with lower investment cost taking account of the associated uncertainties.

In order to make a tradeoff between the accuracy and cost, we apply the concept of efficient frontier to establish the optimization model of PMU placement. Efficient frontier has been widely used in the investment literature to solve portfolio issues [

In the process of forming the efficient frontier, the associated uncertainties, such as the uncertainty of measure errors and instrument failures, are modeled based on the Monte Carlo simulation approach. In the following, we introduce the uncertainties considered, the accuracy index taking into account of uncertainties, and the mathematical model of the PMU placement.

There are three types of uncertainties considered in the paper, i.e., i) random fluctuation of loads, ii) measure errors of the measurement instruments, and iii) possible fault occurring in measurement devices.

For those load nodes without real measurements, pseudo measurements are adopted in the SE of distribution system in order to meet the observability requirement. Since pseudo measurements are obtained through modeling or forecasting loads based on historical data and in the reality loads fluctuate randomly, pseudo measurements are therefore random variables. In the paper, we suppose pseudo measurements are normally distributed or uniformly distributed and analyze their affect on the optimal PMU allocation in

For those nodes with real measurements, their measure values are also random variables since there exists measure errors arising from i) inaccurate transducer calibration, ii) the effect of analogue-to-digital conversion, iii) noise in communication channels, and iv) unbalanced phases, etc. [

For the real measure devices, except for the uncertainty of measure errors, they have the uncertainty of device failure in the operation. In the paper, we suppose the fault probability of the measure device is known based on the operation data, and adopt Monte Carlo simulation to model the randomness of the device fault.

In order to judge the estimation error intuitively, in the paper, estimation error is expressed as the relative error in percentage. Therefore, estimation accuracy rate (

In the paper, the accuracy index,

There are numerous methods to calculate VaR, Jorion et al. [^{th} simulation.

Based on the above definition of the accuracy index, the efficient frontier can be formed by maximizing accuracy index,

In the SE, traditional measurements include voltage magnitude

p=a, b, c for three phases,

where

^{th} element of the complex bus admittance matrix,

For the solution of problem

where

This paper applies genetic algorithms (GA) to solving the above optimization problem

In this paper, an individual is a vector,

Step 1: let the cost equal to the minimal cost, i.e., the cost of installing one PMU;

Step 2: find the optimal PMU portfolio through the GA (i.e., the dashed area in

Step 3: increase the PMU cost to another feasible value (i.e., increase the installation quantity of PMU);

Step 4: repeat Steps 2 and 3 until the cost exceeds the maximum cost, i.e., the cost of installing PMU at every node;

Step 5: form the efficient frontier based on the above optimization results for each given cost. All figures and tables should be cited in the main text as

Case | Load error | Reliability | Amount | VaR(%) | PMU portfolio (combination of nodes) |
---|---|---|---|---|---|

of PMU | |||||

1–1 | no | no | 9 | 99.895 | 1, 2, 5, 6, 7, 8, 9, 10, 11 |

1–2 | normal | no | 10 | 99.703 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 |

1–3 | uniform | no | 10 | 99.698 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 |

1–4 | no | N-1 | 10 | 99.895 | 1, 2, 3, 5, 6, 7, 8, 9, 10, 11 |

1–5 | no | pf = 0.1 | 10 | 99.762 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 11 |

1–6 | no | pf = 0.01 | 10 | 99.857 | 1, 2, 4, 5, 6, 7, 8, 9, 10, 11 |

1–7 | no | pf = 0.001 | 9 | 99.895 | 1, 2, 5, 6, 7, 8, 9, 10, 11 |

1–8 | normal | pf = 0.01 | 10 | 99.706 | 1, 2, 3, 4, 5, 7, 8, 9, 10, 11 |

1–9 | normal | pf = 0.1 | 11 | 99.533 | 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11 |

The proposed GA mainly consists of seven procedures: parameters initialization, population initialization, fitness calculation, selection, crossover, mutation, and population updating. In order to meet the constraint of

In order to make an intuitive comparison, a simple 12-node distribution system with balance load is firstly simulated and analyzed. Then the IEEE33-node distribution system with unbalanced load is simulated to illustrate the effectiveness and scalability of the proposed method.

Based on the provided line data and load data, and let

Suppose Node 1 has the measurements from the SCADA system, i.e., voltage magnitude, power flow, and branch current magnitude. The corresponding measure error is set to 1% for voltage magnitude and 3% for power flow and branch current magnitude as in [

To model the uncertainties of the measurement errors and instrument fault, Monte Carlo simulation approach are adopted and the simulation time is 1,000 in the paper. The confidence interval of VaR is set as 95%.

For the parameter setting in GA, the population size is 50, chromosome length is 11, crossover probability is 0.5, mutation probability is 0.001, and the number of iterations is 50. Besides, in order to simplify the calculation, the cost of a PMU is supposed to one unit.

Nine cases are simulated in the paper in order to compare the effect of uncertainties of the measurement errors and instrument fault on the PMU placement, i.e.,

Case 1-1: This is a benchmark. In this case, we do not consider any uncertainties, i.e., we ignore the uncertainty of measurement errors and the possibility of instruments fault.

Case 1-2: Based on Case 1-1, this case considers the uncertainty of measure errors and supposes all the measurement errors are normally distributed.

Case 1-3: Based on Case 1-1, this case considers the uncertainty of measure errors and supposes that real measurement errors are normally distributed and pseudo measurement errors are uniformly distributed.

Case 1-4: Based on Case 1-1, this case considers the possibility of PMU fault with traditional reliability criterion, N-1 criterion, i.e., in each PMU portfolio, 1 PMU is supposed to fault.

Cases 1-5, 6, and 7: Based on Case 1-1, this case considers the possibility of PMU fault and supposes the fault rate of PMU is 0.1, 0.01 and 0.001, respectively.

Cases 1-8 and 9: Based on Case 1-1, this case considers both the uncertainty of measurement error and the possibility of PMU fault. Suppose that all the measurement errors are normally distributed and the fault rate of PMU is 0.01 and 0.1, respectively.

For each case, the efficient frontier is simulated and shown in

Suppose there is no cost limitation, the optimal PMU portfolio for each case is shown in

General conclusion

If there is no cost limitation, for each case, there is a optimal installation portfolio that maximize the SE accuracy, but the marginal improvement in SE accuracy is obviously reduced when the installation quantity is greater than about 50% of the total potential installation quantity, i.e., 5–6 PMUs in the simulation.

Effect of measurement uncertainty

If there is no cost limitation, the optimal installation quantity of PMU in Case 1-1, Case 1-2 and Case 1-3 is 9, 10, 10, respectively, and the optimal PMU portfolio in Case 1-2 and Case 1-3 is the same, i.e., the optimal installation site is Nodes 1, 2, 3, 5, 6, 7, 8, 9, 10, and 11. It indicates that more PMUs are needed when we consider the measure uncertainty, but the specific distribution characteristic of the pseudo measurements does not affect the optimal PMU portfolio for a given number of PMU.

If the expected SE accuracy rate is set to 99% with a confidence interval of 95%, the optimal installation quantity of PMU is 2, 5, and 7 for Cases 1-1, 1-2, and 1-3, respectively, which indicates that the SE accuracy is affected by the distribution characteristics of the pseudo measurements and more PMUs are needed to meet a SE accuracy target when the pseudo measurements are not normal distributed (e.g., uniform distribution in the simulation).

Effect of PMU fault uncertainty

If there is no cost limitation, with the possibility of PMU fault into consideration, more PMUs are needed when we consider the possibility of PMU fault (e.g., Cases 1-4, 1-5, and 1-6) compare to the case without considering PMU fault (i.e., Case 1-1). Of cause, if the fault probability of the PMU is very small (e.g., Case 1-7), the optimal PMU portfolio does not change compared to Case 1-1.

If the expected SE accuracy rate is set to 99%, compared to Case 1-1, more PMUs are needed (e.g., Cases 1-4 and 1-5). If the fault probability of the PMU is smaller (e.g., Cases 1-6 and 1-7), the optimal PMU portfolio does not change compared to Case 1-1.

Simulation results indicate that more PMUs are installed if we consider the possibility of the PMU fault, which is consistent with our intuitive judgment. Furthermore, compared to the N-1 criterion, probability method can more accurately model the effect of PMU fault on the SE. Higher fault probability results in more PMU installed and vice versa.

Effect of uncertainties of measurement and PMU fault

The optimal PMU portfolio of Case 1-8 is the same as that of Case 1-2 since the fault rate of PMU is quite lower (i.e., 0.01 in Case 1-8). If the fault rate of PMU is higher (i.e., 0.1 in Case 1-9), more PMUs are needed compared to Case 1-2.

Case | Load error | Reliability | Amount | VaR(%) | PMU portfolio (combination of nodes) |
---|---|---|---|---|---|

of PMU | |||||

1-1 | no | no | 2 | 99.063 | 2, 6 |

1-2 | normal | no | 5 | 99.047 | 1, 2, 4, 7, 10 |

1-3 | uniform | no | 7 | 99.139 | 1, 2, 4, 5, 7, 8, 11 |

1-4 | no | N-1 | 3 | 99.063 | 2, 6, 11 |

1-5 | no | pf = 0.1 | 4 | 99.298 | 1, 4, 6, 10 |

1-6 | no | pf = 0.01 | 2 | 99.044 | 2, 6 |

1-7 | no | pf = 0.001 | 2 | 99.063 | 2, 6 |

1-8 | normal | pf = 0.01 | 5 | 99.139 | 1, 2, 4, 7, 10 |

1-9 | normal | pf = 0.1 | 7 | 99.184 | 1, 2, 3, 6, 7, 8, 11 |

The single-line diagram of the IEEE 33-node system is shown in

Case 2-1: This case considers the uncertainty of measure errors and supposes all the measurement errors are normally distributed.

Case 2-2: This case considers the uncertainty of measure errors and supposes that real measurement errors are normally distributed and pseudo measurement errors are uniformly distributed.

Cases 2-3 and 4: This case considers the possibility of PMU fault and supposes the fault rate of PMU is 0.1 and 0.01, respectively.

Cases 2-5 and 6: This case considers both the uncertainty of measurement error and the possibility of PMU fault. Suppose that all the measurement errors are normally distributed and the fault rate of PMU is 0. 1 and 0.01, respectively.

The simulation results of Cases 2-1 to 2-6 are shown in

Compared with

Base on the concept of efficient frontier and VaR, this paper proposed a GA based optimization model for optimal placement of PMU (i.e., PMU portfolio). This model considers the measurement uncertainty, statistical characteristics of the pseudo measurements, reliability of the measurement instrument. The reasonability and feasibility of the proposed model is illustrated with a 12-node system and IEEE33-node system. Simulation results indicated that:

Uncertainties of measurement error and instrument fault result in more PMU to be installed;

Measurement uncertainty is the main affect factors unless the fault rate of PMU is quite high, and different measurement error distributions have different effects on the optimal configuration of PMU;

The proposed model can make a more reasonable PMU allocation scheme according to the risk preference of decision-maker (given confidence level), compared with the traditional multi-objective weighted optimization and N-1 criterion.

In the future, how to determine the measurement error distribution and the failure rate of PMU will be studied.