1. Introduction
In recent years, the development of renewable generations and power electronic devices has been accelerated [
1,
2]. The outputs of renewable generations are uncertain and fluctuant [
3]. In addition, the response velocity of power electronic devices is faster than conventional synchronous generators [
4]. With the development of renewable generations and power electronic devices, states of the power system are becoming more complicated and variable. The estimation of power system states is necessary under both the normal operation and contingencies [
5,
6]. Therefore, it is necessary to attain power system states for the security and economic operation.
State estimation (SE) serves as an essential part in the advanced application of the energy management system, whose main role is to provide a reliable and sufficient database for other applications, e.g., power flow analysis, economic dispatch, and stability control [
7,
8]. The main factors influencing the accuracy of SE include measuring devices and transmission line parameters [
9]. The rapid development of the phasor measurement unit (PMU) and advanced communication infrastructure has improved the precision of measured data [
10]. However, parameters of transmission lines are often regarded as constant values and the effect of line temperature is ignored, which will result in parameter inaccuracy and has become an obstacle restricting the precision of state estimation. Failure to modify line parameters according to temperature may lead to significant errors in the calculation of power flow and network loss, whose errors may even exceed 30% [
11,
12].
Electro-thermal coordination (ETC) that combines line temperature with electrical quantities is capable of decreasing temperature-dependent errors in power flow, SE, and related analysis [
13]. The basic ETC principle is to establish an analytical relationship of the line temperature, current, and weather conditions through the heat balance equation (HBE) [
14]. Line temperature is calculated by the HBE, and then line parameters can be corrected correspondingly.
Studies on ETC in the power system mainly focus on the thermal rating [
15,
16,
17], power flow analysis [
18,
19,
20], and estimation of power system states and temperature [
21,
22,
23,
24]. Based on dynamic thermal models, a real-time thermal rating method for the lines of the distribution network was proposed [
15]. Considering the scenarios of normal operation and contingency, the performance of different real-time line monitoring devices on dynamic thermal rating has been assessed [
16]. The dynamic thermal-line rating is conducted using online measurements so as to adjust the operational tripping scheme [
17]. In addition, a temperature-dependent power flow algorithm was proposed, where the augmented equation set is solved by the Newton–Raphson method [
18]. ETC was combined with optimal power flow, on which economic losses can be reduced [
19]. With the consideration of temperature-related resistance and thermal rating, a weather-based optimal power algorithm was designed [
20].
Moreover, the influence of temperature on SE performance was analyzed, and the results verify that the errors caused by temperature are not negligible [
21]. Using Monte Carlo simulation and a variance reduction method, the critical line temperature in the presence of fluctuating power flows was estimated probabilistically [
22]. Based on the analytical solution and numerical weather prediction, a simulation method for the evolution of line temperature was proposed [
23]. In addition, a constrained nonlinear optimization model for estimating both the voltage phasors and the temperature of transmission line conductors was established and solved by a predictor–corrector interior point method [
24].
With regard to SE, there are extensive efforts devoted to its performance modification techniques [
25,
26,
27,
28,
29,
30,
31,
32,
33]. A hybrid state estimator using the measurements of remote terminal units (RTUs) and PMUs was designed with bad data detection [
25]. Considering the parameter errors and bad measurements, a robust SE method was proposed, which is formulated as a tractable mixed-integer linear programming optimization problem [
26]. The distributed SE operates locally with minimal data exchange with neighbors and is applicable for multi-area power systems [
27]. Based on the iterative reweight least square algorithm, a distributed SE method was proposed, where the improved alternating direction method of multipliers is utilized to improve result accuracy and convergence speed [
28]. In addition, the placement of PMUs was optimized to improve the reliability of SE results [
29]. Considering bad data and missing measurements, a multi-objective PMU allocation method was presented for achieving the desired accuracy of SE [
30]. The unscented Kalman filter was used to estimate the states of a permanent magnet synchronous motor (PMSM) [
31]. In addition, a multi-area distributed SE method was proposed with the use of the data-driven Kalman filter [
32]. In order to estimate the states of electric vehicle batteries (EVBs), a robust adaptive filter, an adaptive smooth variable structure filter has been designed [
33]. The aforementioned SE methods are summarized in
Table 1.
For the sake of reducing the temperature-dependent errors, this paper proposes two ETC-SE approaches with the consideration of transmission line temperature. In the augmented Jacobian ETC-SE (AJ-ETC-SE) approach, line temperature is integrated into state variables, and the HBE serves as a pseudo-measurement. Compared to the weighted least square SE (WLS-SE), the dimension of the measurement and state vectors increases, but the equation set is still overdetermined. Additionally, the elements in the augmented Jacobian matrix are provided, and the equation set is solved using the Newton method. On the other hand, the improved two-step ETC-SE (ITS-ETC-SE) approach decouples the calculation process of voltage phasors and line parameters. Compared with the AJ-ETC-SE approach, the SE and temperature estimation are solved via alternate iteration so as to reduce the iteration number and computation time.
Therefore, the main contributions of this paper are three-fold. Firstly, an ETC-SE model integrating the HBE and WLS-SE is established to consider the impact of line temperature on power system states. Secondly, the AJ-ETC-SE approach is proposed to simultaneously solve the problems of SE and temperature estimation through an augmented Jacobian matrix. Finally, the ITS-ETC-SE approach is proposed to accelerate the solving process, in which the SE and temperature estimation are decoupled and solved via alternate iteration.
The rest of this paper is organized as follows.
Section 2 develops an ETC-SE model with the consideration of the HBE and meteorological data. The procedure of the AJ-ETC-SE approach is presented in
Section 3. Moreover,
Section 4 gives the ITS-ETC-SE approach. The case studies of the IEEE 14-, 39-, and 118-bus systems are conducted in
Section 5. Finally, conclusions are drawn in
Section 6.
5. Case Studies
The IEEE 14-, 39-, and 118-bus systems are utilized to verify the effectiveness of the proposed AJ-ETC-SE and ITS-ETC-SE approaches. The WLS-SM method in [
19] is used for comparison. Moreover, the algorithms are developed on the MATLAB R2019b. Further, the calculation is conducted on a 2.33 GHz Intel (R) Core (TM) 2 Quad CPU Q8200 processor with 16 GB of RAM. Additionally, the convergence threshold ξ for the Newton method is set as 0.0001.
Voltage amplitude, power injection, branch power flow, and currents are incorporated in the measurement vector. Then, 2% and 4% Gaussian-distributed white noise are added to the voltage and power measurements, respectively. Meteorological parameters remain constant, which is reasonable for the short simulation time.
Furthermore, the root mean square (RMS) error of estimation results
eRMS is calculated, and can be expressed as,
where
N is the group of measurement data,
M is the number of buses,
xj is the value of the
jth state variable, and
xi,j,r is the estimated value of the
jth state variable with the
ith group of measurement data.
5.1. Estimation Results of Proposed ETC-SE Approach
The simulation and estimation of the three systems are conducted 10,000 times. The probability density functions (PDFs) of the estimated voltage
U1 with the proposed AJ- and ITS-ETC-SE approaches are depicted in
Figure 5. For comparison, the PDF of the WLS-SE method is also depicted.
Table 2 shows the RMS and maximum estimated errors of the 10,000 samples with different methods. The errors of the proposed ETC-SE approaches are almost the same and smaller than the errors of the WLS-SE method, whose voltage amplitude and phase angle errors are merely 0.13% and 6.18%, respectively. In addition, the averaged voltage errors of the IEEE 39-bus system with the proposed ITS-ETC-SE approach and WLS-SE method can be depicted in
Figure 6. The temperatures are considered in the ITS-ETC-SE method but not in the WLS-SE method. Due to the influence of variable temperatures, the fluctuations of the WLS-SE method are more serious than the ITS-ETC-SE method.
Furthermore, the estimated temperature and power loss of transmission lines in the IEEE 39-bus system are demonstrated in
Table 3. It is obvious that the temperature varies greatly on different lines, ranging from 34 °C to 68 °C, which is affected by branch power flow. Similar results have been obtained in power loss, with an increase over 10% in the proposed ITS-ETC-SE approach compared with the WLS-SE method.
Estimated errors of line resistance and temperature in the IEEE 39-bus system are demonstrated in
Figure 7. Line resistance errors refer to the deviation between the actual resistance and its rated value. Compared with its rated values, the actual values of most line resistance increase over 10% when taking ETC into account. The resistance errors of the WLS-SE method are over 10% and the maximum error even reaches 19.1%. Estimated temperature error of most lines with the proposed ETC-SE approaches are less than 1%. Furthermore, the voltage and temperature errors of the IEEE 118-bus system are depicted in
Figure 8.
5.2. Performance Analysis
A comparison of the computation time and iteration numbers on the proposed ETC-SE approaches and WLS-SE method was performed, as shown in
Table 4. The ETC-SE approaches need more iteration numbers than the WLS-SE method. Line temperature converges more slowly than state variables. As for the computation time, the proposed ITS-ETC-SE approach is shorter than the AJ-ETC-SE approach.
Table 5 shows the computational complexity of the AJ-ETC-SE approach and WLS-SE method. With the consideration of line temperature, the number of measurements and state variables, and the dimension of the Jacobian matrix increase sharply and even over 100% compared with the WLS-SE method.
Figure 9 shows the maximum unbalance of state variables ‖Δ
x‖
∞ during each iteration. It can be seen that the index ‖Δ
x‖
∞ first declines linearly, and then nearly remains constant.
Furthermore, different scenarios are conducted in order to explore the convergence performance of the proposed ETC-SE approaches, including varying environmental conditions and the existence of ill-conditioned branches. For each scenario, 1000 simulations are performed in the IEEE 118-bus system.
The environmental conditions vary with the ambient temperature
Ta and wind velocity
Vw. More specially, the ambient temperature
Ta increases from 25 °C to 50 °C and the wind velocity
Vw increases from 1 m/s to 20 m/s, which will lead to changes in convective heat loss
qc. The iteration numbers of the proposed ETC-SE approaches can be summarized in
Table 6 and
Table 7, respectively.
Furthermore, the ill-conditioned branches are added to the IEEE 118-bus system. Moreover, the ill-conditioned branches refer to the high-resistance branches whose
R = 0.5
X. The iteration numbers of the proposed ETC-SE approaches are shown in
Table 8.
6. Discussion
The proposed approach is to estimate the power system states considering electro-thermal coordination, and the calculation results include the power system states and transmission line temperature. The performance of the proposed approach in terms of efficiency, effectiveness, and accuracy can be verified by the comparison analysis in the above case study.
From
Figure 5, it is clear that the PDFs with the proposed ETC-SE approaches are nearly identical and more concentrated compared with the WLS-SE method. Additionally, compared to the WLS-SE method, the results in
Table 2 and
Figure 6 show that the proposed AJ- and ITS-ETC-SE approach is effective to decrease the errors. Moreover, the iteration numbers and computation time are compared in
Table 4. The results show that the proposed ITS-ETC-SE approach is effective in accelerating the solving process. Since the dimension of the measurement vector, state vector, and Jacobian matrix increases to a larger extent after integrating line temperature into the SE model, the computation time of the AJ-ETC-SE approach is much longer than the ITS-ETC-SE approach and WLS-SE method.
Figure 9 verifies the superior computational performance of the ITS-ETC-SE approach when compared to the AJ-ETC-SE approach. Furthermore, the sensitivity analysis is conducted by considering different scenarios in terms of ambient temperature, wind velocity, and ill-conditioned branches. The iteration numbers of different methods are summarized in
Table 6,
Table 7 and
Table 8, and the results show that the proposed ETC-SE approaches show good convergence performance with environmental condition variation and ill-conditioned branches. The iteration numbers increase as the ambient temperature increases and wind velocity declines. Moreover, reliable convergence can also be achieved when ill-conditioned branches exist, and there are merely tiny increases in iteration numbers, which will facilitate the utilization of the proposed ETC-SE approaches in practical application.
7. Conclusions
The SE considering the influence of line temperature is studied, and the ETC-SE approaches are proposed to reduce the temperature-dependent errors. An ETC-SE model integrating the HBE and WLS-SE is established. In addition, the AJ-ETC-SE approach is presented to simultaneously solve the problems of SE and temperature estimation through an augmented Jacobian matrix. For the sake of accelerating the solving process, the ITS-ETC-SE approach is proposed, in which the SE and temperature estimation are decoupled and solved via alternate iteration.
The effectiveness, efficiency, and convergence performance of the proposed ETC-SE approaches are verified through the IEEE 14-, 39-, and 118-bus systems. Results show that the proposed ETC-SE approaches can reduce estimated errors evidently and estimate line temperature precisely. The accuracy of the AJ-ETC-SE approach is slightly higher than the ITS-ETC-SE approach. Further, the ITS-ETC-SE approach is able to accelerate the calculation process. Furthermore, the ETC-SE approaches possess good convergence performance with varying environmental circumstances and ill-conditioned branches.
In future studies, the influence of the measurements of PMUs on the SE will be considered, and an ETC-SE approach will be provided based on hybrid measurement data.