In this section, the IEEE 14, 30, 118 bus test systems are used to verify the presented approaches. To easily calculate and intuitively display the results, usually it is better to transform the active and reactive power injection measurements, Pi, Qi, and active and reactive power flows Pij, Qij to per-unit values. In this paper, the reference value as 100 MW or 100 MVAR is selected to transform the above data to per-unit system, and the units of phase angle are degree. All other measurement units are p.u.
Measurements from PMUs have a high accuracy, and their amplitude errors are from 0.1% to 1%, and the range of phase angle is from to . Then the covariance matrix of traditional measurement errors Rt and PMU voltage phasor measurement errors can be calculated by .
There is the relation, i.e.,
. According to the error propagation theory [
37], the standard deviations of
and
can be calculated by Equation (24):
so, the covariance matrix of PMU branch current phasor measurement errors
RI can be calculated as well. The mixed error covariance matrix
Rm defined by Equation (24) can be obtained [
36].
In order to check the estimation accuracy and computational efficiency of the proposed state estimation method, this paper compares it with results from three common methods. Method 1: the traditional weighted least squares (WLS) state estimation method with only SCADA measurements. Method 2: PMU installed at partial nodes of power systems, but the PMU measurements with voltage phasors without participating in state estimation. In this method, the PMU measurement results are used as the state solution of test system. Method 3: PMU equipped at partial nodes of power systems, and the PMU measurements with voltage phasors participates in state estimation.
In this paper, the tolerance to converge is set to 10
−4, meaning that if there is:
the simulation program calculation will finish its iteration and we get the state estimation results.
3.1. IEEE 14 Bus System
In the IEEE14 bus test system, the parameter configuration includes traditional measurements and PMU measurements.
Figure 8 is the measurement configuration diagram of the IEEE 14 bus system [
38]. The PMU is installed on buses 2, 7 and 9. In addition, traditional measurements include, the power flows in 1–2, 2–3, 4–7, 4–9, 5–6, 6–13 and 13–14, power injections in buses 2–11 and the voltage amplitude in bus 1. Four methods based on
Figure 8 are designed to complete the state estimation, and simulation is carried out by MATLAB. The results are shown in
Table 1 and
Table 2. It should be noted that the proposed method is a fixed reference angle method, but in the calculation results, in order to intuitively reflect all the state of the entire system, the phase angles of all the buses are listed in this paper.
Table 1 presents the bus voltage amplitude estimation results of the four methods in the 14 bus system. The second column indicates the true value of the bus voltage amplitude, and the third, fourth, fifth and the sixth columns show voltage state estimated values using different methods.
Table 2 is the bus voltage angle estimation results of the four methods in the 14 bus system.
The state estimation average relative error absolute value can be defined as [
39]:
where
yi is the real value,
yei is the estimation value.
Then in this report, the state estimation average accuracy can be obtained as follows:
Figure 9 and
Figure 10 are the result of percentage bus voltage amplitude and angle estimation errors using the data from
Table 1 and
Table 2. As can be seen from
Table 3, the maximum percentage voltage error of the proposed method is 4.8849%, which represents a great improvement when compared with the three other methods. On the whole, we can see that Method 1 has the worst state estimation effect and the proposed method has the best estimation effect. It also should be pointed out the elapsed time of the proposed method shows no lessening, but we can accept it. This shows that when adding PMU data to conventional SCADA measurement inputs, the state estimator will raise its redundancy and improve its accuracy. Especially, the proposed mixed state estimation has more marked improvement.
In the field of numerical analysis, on the one hand, the condition number of a function with respect to an argument measures the asymptotically worst case of how much the function can change in proportion to small changes in the argument. The “function” is the solution of a problem and the “arguments” are the data in the problem. Usually, the condition number is described as:
If complex square matrix
A is normal, i.e.,
A*A =
AA*,
A* is the conjugate transpose of
A, then the condition number can be described as:
where
are maximal and minimal eigenvalues of
A, respectively.
A problem with a low condition number is said to be well-conditioned, while a problem with a high condition number is said to be ill-conditioned.
On the other hand, quality of an estimate is inversely related to it is variance. Estimation quality is very important to state estimator. There are many parameters which can be looked as a reference. In this report, we select the following equation to describe the quality of the estimator:
Here numerically computed error covariance matrix
P can be defined as:
where
X is the true state vector,
is estimation state vector and
is the gain matrix.
Sometimes, it becomes difficult to compute the determinant of error covariance matrix numerically. Hence, trace of error covariance matrix can be utilized to quantify the quality of estimation as follows:
The gain matrix condition number
ka, quality
and
of the four methods are shown in
Table 4 and these results are plotted in
Figure 11.
From the above analysis, we can see the proposed method has better condition number and estimation quality than the other three methods. In order to further validate the effectiveness, the following five cases with different PMU measurement configurations will be discussed to illustrate the performance of the proposed method. The simulation results of different cases are shown in
Table 5. It can be observed from
Table 5 that as the number of PMU configurations increases, we can get better results, which include voltage accuracy (
Va), angle accuracy (
Aa), the condition number (
Ka) and quality (
Qt).
In statistics, bias of an estimator is the difference between estimator expected value and the true value of the parameter being estimated [
40]. Based on mathematical theory, one has:
where
Pk is the distribution probability. In this paper, Monte Carlo simulations were utilized to research the statistical characteristic, and the estimation step number
N is 500 steps. The measurement inputs are randomly produced by the following equation:
where
produces a normal distribution random number. The symbol
is set to “1”, and the symbol
is set to “
”.
Figure 12 is the bias analysis of the proposed method using the Monte Carlo method. It’s clear from the figure that the bias of proposed mixed state estimator is low.
3.3. IEEE 118 Bus System
In the IEEE 118 bus test system, the PMU is installed at buses 14, 20, 21, 22, 42, 43, 44, 45, as shown in
Figure 17. All the simulation conditions are the same as those of the IEEE 14 bus system. The results of state estimation of the different methods are observed from
Figure 18 and
Figure 19. It can be seen from
Figure 18 and
Figure 19 that the overall accuracy of the proposed estimator is better when compared with the others. As can be seen from the
Table 8, although the percentage of angle maximum error of proposed method shows no lessening relative to Method 3, it drops a lot when compared with Methods 1 and 2. Furthermore, the accuracy and the number of iterations using the proposed method are greatly improved in a large scale system. Similarly, it can be observed from the
Table 9 and
Figure 20 that the proposed method the effectiveness and accuracy of the calculation model is also verified by the 118 bus system.
Note that the convergence time of the proposed method is slightly bigger than that of the other three methods in
Table 10. This is because of the proposed method utilizes more PMU measurements in the calculation, but the convergence time can meet the basic assessment requirements of the State Grid, i.e., the calculation time of a single state estimate is less than 5 seconds. The computer configuration of the simulations is as follows: processor i5-4460, CPU 3.20 GHz, internal memory 4 GB. The computer system has Windows 10 64-bit operating system. With improved online computer memory capacity and processing speed, the convergence time of the proposed method could be reduced greatly.