The problem of calculating the steady-state mode of the electrical network according to the measurements was called the state estimation [30]. Because of its cyclic solution, almost all software applications for controlling the operating mode of an electric power system are performed: mode forecasting, stability analysis, the determination of control actions for emergency automation, etc. The use of synchronous vector measurements for the task of state estimation (SE) opens up the possibility to significantly improve the accuracy and speed of calculations, and therefore, in general, to improve the quality of power system management. Provided that the power system is sufficiently supplied with devices, the transition from a nonlinear formulation of the SE problem with an iterative solution algorithm to a linear formulation associated with a one-time solution of a system of linear equations (SLE) is possible. Due to the high cost, the amount of PMUs in the energy system is still limited; therefore, the integration of PMUs into the classical formulation of the SE problem based on telecommunications and their joint use is a promising area of research. In addition, there are other problems that need to be reviewed, taking into account the data received from the PMU; for example, the choice of weighting coefficients for conventional and vector measurements, the placement of measurements and the analysis of the observability of the network, and the identification of bad data. The assessment of the reliability of the PMU is of independent importance, since with the development of these measurement systems, the scope of their application in the control algorithms of normal and emergency modes is expanding. Measurement errors for these algorithms should be detected reliably and quickly. To date, many studies have been conducted on the possibility of including PMUs in SE algorithms. We consider both options for evaluating information from the PMU together with classical tele-measurements and solving the SE problem only on the basis of the PMU. The works analyze the features, efficiency, advantages, and problems that arise when introducing new types of measurements into static and dynamic SE algorithms. Some studies are devoted to the issues of the optimal arrangement of the PMU [31], including those aimed at improving the observability and minimizing the number of critical measurements [32]. In [33,34], the use of PMUs is considered for a robust SE based on the method of smallest modules. In [35], the SE of PMUs is solved on the basis of the method of control equations. In [36,37], approaches to the SE based on the use of the Kalman filter are presented, which, in addition to the current electrical mode, make it possible to determine the synchronous rotation angle of the generator rotor and the rotor rotation frequency. However, most of these approaches are very demanding in terms of equipping the PMU power system.

The active implementation of PMUs in the task of assessing the state of the EPS is possible due to the constant reduction in the cost of electronic components and the unification of devices and approaches to production. Thus, it is feasible to introduce a significant number of PMUs into the EPS.

The most common practice is the implementation of the SE based on the weighted least squares method. Many researchers considered its adaptations for the purpose of inclusion in the composition of the used measurements and PMUs. There exist several methods in the literature that were developed to account for ultrasound data in the SE based on the weighted least squares method (WLSM). These methods can also act as a basis for the development of SE algorithms based on other methods of minimizing measurement errors. Next, the main ways of including the PMU in the task of the SE based on the WLSM are considered.

In [38,39], an approach is presented in which the PMU is directly included in the class setting and, together with the tele-measurements, participates in the WLSM SE. This expands the list of measurement types that can be used in the task.

Advantages:

• A single set of measurements is used in one procedure, which increases the accuracy of the initial data and can improve the accuracy of the assessment.

Disadvantages:

• The advantages of the accuracy of the PMU data, their synchronization, and linking to the moment of time will be partially lost due to their use in conjunction with classical telemetry;
• There is no certainty in the choice of weight coefficients;
• As the dimension of the problem increases, modern approaches to the SE aim to solve this problem, but it may persist, for example, when searching for bad data, analyzing observability, and searching for topological errors;
• Depending on the formulation of the problem, there may be some difficulties associated with taking into account currents, the angle of the base node, and measurements of voltage angles;
• The implementation of this approach will require the modification of existing SE programs, both in terms of introducing new types of measurements, and ensuring optimal design characteristics.

Based on standard telemetry, the WLSM-based SE problem is solved iteratively since the equations for most measurements are nonlinear. When only complex values of currents and voltages act as measurements, and when writing the state vector in the form of voltages in a rectangular shape, a linear dependence of the estimated functions can be obtained. Thus, if the observability of the network is provided only by ultrasound, then the SE can be performed linearly by a single solution of the SE problem.

Advantages:

• Increases the speed of operation and reliability of obtaining a solution due to the exclusion of the iterative process;
• Only high precision measurements are involved;
• The matrices used in the calculations do not change as long as the repair scheme of the network and the composition of measurements are preserved;
• All measurements have a timestamp, which can be taken into account at the preprocessing stage.

Disadvantages:

• For the correct operation of the SE algorithm and the associated search for bad data, an ultrasound redundancy PMU is required, which is extremely rare in practice.

With a two-level SE based on the framework of the PMU [40], the measurements are divided into two groups. The first group of measurements forms a certain framework for the model under study, a single tree-like connected network. The second group makes it possible to obtain the most probable electric mode in nodes that are not observed by the PMU but have other measuring complexes. In this approach, at the first stage, a linear SE is performed on more accurate data supplied by the PMU. Next, a nonlinear SE is carried out while fixing the results of the first stage as equality-type constraints.

Advantages:

• With a certain framework, the remaining network can be divided into islands, in each of which an independent SE is performed, which significantly speeds up the calculations.

Disadvantages:

• Measurements are evaluated in groups, which reduces the efficiency of the algorithm to reduce the overall error of the measurement set;
• The PMU should be arranged in a certain way, which is difficult to achieve given the variety of the repair schemes of the network.

In [41,42] it is proposed to perform a two-level SE with a post-processing stage. It is postulated that most power systems are not observable only by using the PMU. To ensure the observability of the linear SE, it is possible to use the values of the state vector obtained by solving the SE on the basis of classical telemetry as pseudo-measurements represented by stress complexes. The authors propose to keep the existing SE software unchanged and use its results for new linear SE software complexes. As the results of calculations [41] showed, the proposed two-level SE with a post-processing stage leads to equivalent results to the classical SE of the extended PMU, where all information from different measuring complexes is evaluated within a single iterative procedure.

Advantages:

• The approach can have all the advantages of a linear SE with a sufficient level of observability of the PMU network and the correct choice of weighting coefficients for the measurements;
• Instead of classical measurements with reduced accuracy, a state vector obtained at the stage of solving the iterative SE problem is added as the estimated parameters, which is better than adding traditional measurements directly;
• The existing software for the SE and processes based on the results of its calculations is preserved. In this case, the transfer of these processes to a linear SE can be done after a sufficient number of calculations have been performed by the power company, guaranteeing the best result for the new algorithm in comparison with existing SE modules.

Disadvantages:

• Post-processing is performed much more often than the first stage, which means that in most calculations, it contains outdated information; moreover, it is obtained on the basis of traditional measurements, which contain a large error in comparison with the vector measurements;
• There is no certainty in the choice of weight coefficients; this problem requires careful study to properly account for heterogeneous information.

In [43,44], an approach to a two-level SE is proposed, where at the first stage, for each voltage class, the PMU currents are individually refined at the object level, which is performed on the basis of the first Kirchhoff law. After this procedure, based on the calculated values of the currents, the states of the switching equipment are clarified. Next, the weighted average voltage is calculated. This sequence of actions allows the calculations to be carried out without considering the resistances of network elements. At the second stage, the obtained parameters are transmitted to the dispatch center, where a full-fledged SE is performed based on a proportion of vector measurements estimated at the first stage.

Advantages:

• Parallelization of the SE process at the level of objects where the PMU is installed, due to its execution for each voltage class of each object in isolation;
• Only the results of the SE performed at the facility can be transmitted to the dispatcher center level; there is no need to transfer a complete set of data from the PMU;
• Switch states can be clarified at the object level and incorrect measurements can be rejected;
• At the object level, the resistances of the elements do not participate; the SE is performed for each voltage class based on the first Kirchhoff law as well as the calculation of the weighted average voltage.

Disadvantages:

• With the SE at the object level, the redundancy is insignificant, which worsens the quality of evaluation;
• It is required that all objects of the dispatch center model are observed by the PMU;
• At the object level, it is required that the PMU provide measurements of the currents of absolutely all connections in the object of the voltage classes under consideration;
• There are problems with finding bad data in the circuit variety; for example, if one voltage level of an object is modeled by a node with two branches, then bad data in the measurement of such an object cannot be identified since both measurements can relate to it with the same probability.

The analysis of the literature allows us to conclude that most researchers consider the SE to be promising solely on the basis of PMUs, when other types of measurements are absent. This approach significantly reduces the calculation time and makes the task linear, and providing data at some synchronized moment can allow using the results of the SE to solve problems related not only to steady-state modes but also to transients [45]. However, it is also generally recognized that it will not be possible to cover the entire energy system with vector measurements in the near future. It is widely viewed that hybrid SE algorithms, when the analysis of the power system is performed on the basis of PMUs and conventional tele-measurements, will remain useful in the years of transition from SCADA systems to measurement systems based on PMUs. As can be seen from the analysis presented, due to the differences in the quality of information provided by PMUs and classical telemetry, as well as in the frequency of its receipt, many hybrid SE algorithms focus on the separate processing of PMUs and classical telemetry. The following sections explain in detail the features of using PMUs in the SE area.

The use of PMUs enables switching to a linear solution of the SE problem, eliminating the iterative procedure, which will help to increase the speed of SE execution and will allow avoiding the problem of the divergence of the iterative process. On the other hand, the linear formulation of the SE problem introduces additional restrictions on the initial data necessary for its solution, and the transition to it may require the processing of existing algorithms.

Currently, many operating SE algorithms perform the solution of the problem based on the state vector, represented in polar form. When using such a formulation, the stress complexes obtained by the PMU can be seamlessly included in the task. Then, the relationship between the source data and the desired information will be linear. The problem of the transition to a linear SE in this formulation lies in the measurements of current complexes since their calculated functions have a nonlinear dependence on the state vector. The solution to the presented problem may be to change the coordinates of the state vector by moving from the polar form of its record to a rectangular one. In this case, the dependences of both the current and voltage measurements on the state vector will be linear. The transition to this form of recording will preserve the possibility of using it on a par with the vector measurements of classical telemetry. The need for its addition may arise as a result of the low level of observability of the analyzed repair scheme of the power system or at the initial stages of the implementation of the PMU. However, the introduction of power measurements will require an iterative solution to the problem since their even functions will retain a nonlinear dependence on the state vector.

One of the most important tasks in SE is the search for bad data. The problem of detecting bad data in the PMU set, as well as identifying specific unreliable parameters, is considered in [46]. Currently, the point-by-point implementation of PMUs in power systems is practiced, which leads to a large number of critical measurements in which errors cannot be minimized by SE and bad data are identified. In such conditions, even some redundant measurements can become critical when the repair scheme of the network is changed. All this prevents the search for bad data and the minimization of measurement errors and can lead to obtaining electrical modes that differ greatly from the actual ones during the SE. For the proper operation of algorithms searching for bad data and the subsequent SE, when working exclusively on data from the PMU, it is required to provide the network with an excessive number of measurements.

3. The Use of PMUs for Relay Protection

3.1. Classification of Directions of Development of Protection Functions with PMUs

3.2. Solutions in the Field of Integration of PMUs into Traditional Protection Algorithms

3.3. Solutions in the Field of New Principles of Damage Detection