Monitoring of power system dynamics under incomplete PMU observability condition

Correspondence Gabriel Ortiz, Faculty of Electrical Engineering and Information Technology, Institute of Energy Systems, Energy Efficiency and Energy Economics (ie3), TU Dortmund University, Emil-Figge-Straße 70, Dortmund, 44227, Germany. Email: gabriel.ortiz@tu-dortmund.de Abstract This work presents a hybrid state estimation procedure that allows power system dynamics associated to slow and fast transient events to be accurately monitored considering a limited amount of phasor measurement units. The proposal represents an enhanced version of a previously developed static state estimation approach. This time, two main changes are introduced. First, a weighted least absolute value estimator instead the conventional weighted least square technique is used to estimate bus voltages in transient regime. This makes the estimation process more robust at fast scan rates. Second, an extended Kalman filter based dynamic state estimator is integrated. Thus, the proposed scheme is able to estimate, along with static variables, dynamic variables of all generators and motors in the power system regardless of the availability of phasor measurement unit measurements at terminal buses. This is possible thanks to a novel data-mining based methodology for full phasor measurement unit observability restoration. Performance parameters such as accuracy, computing time and convergence properties are assessed by applying the estimation procedure to the New England benchmark system under different operating conditions.


INTRODUCTION
The continuously growing demand for electricity, driven by deregulated electricity markets, has forced modern power systems to operate closer to their secure operating limits. Furthermore, with the increasing penetration of renewable energy resource and other decentralized generation, more uncertainties will be brought to the operation of transmission systems [1]. Their intermittency and uncontrollability nature brings significant challenges to the secure operation of the power grid [2]. A contingency occurring in the system under such stressed conditions can lead to more complex system dynamics which increases the dynamic insecurity risk and even blackout risk.
In order to increase power system stability and security during and after disturbances, new strategies for enhancing operator situational awareness and power grid global and local controller must be developed. Enhancing system stability through a more sophisticated control requires accurate information about the power system dynamics to carry out online high-sampling rate dynamic security assessment (DSA) [3]. However, the system monitoring tool of the Energy Management Systems (EMS) This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. in charge of delivering such information, i.e. the state estimator, is based on a steady state system model, which cannot capture the dynamics of power system very well. This limitation is primarily due to the dependency of the EMS on slow update of Remote Terminal Units (RTUs) from the Supervisory Control and Data Acquisition (SCADA) system [4]. The sampling rate is too low (2 to 5 seconds) to reveal the electromechanical dynamics in power systems. In addition, RTU-SCADA data are not synchronized. As a result, most controllers for controlling the electromechanical dynamics only use local states to achieve their control objectives. Compatibility among controllers is only studied offline in planning models. Without global objectives and systematic coordination over wide areas, the influence of local controllers at the grid level may have adverse effects [5,6] Today, phasor measurement units (PMUs) are presented as a technological alternative capable of improving the existing estimators or even changing the paradigm of the estimation process [7]. Their high accuracy, reporting speed and synchronization capacity by means of the global positioning system (GPS) make them suitable for global monitoring of the power system dynamics [8].
In addition, if an estimate could be formed with only PMU data, then the issues of data scan (low rates) and time skew could be eliminated. Since the voltage and current measurements are linear functions of the system state, the estimation problem is a linear weighted least squares problem which requires no iterations. Given PMU data can be reported at rates as high as 60 times a second, a truly dynamic estimate would be available. However, the deployment of such an approach for a large transmission system is still very expensive ($ 40,000 to $180,000 per PMU installation) [9]. A more viable and economically sound method to exploit the benefits offered by PMUs is a hybrid state estimator that combines synchronized phasor measurements with conventional measurements [9][10][11].
Most of the hybrid estimators that have been proposed in the literature [9,[12][13][14][15][16] do not take into consideration the fact that PMUs update rate is remarkably higher than RTU-SCADA update rate, therefore they only provide states in stationary or quasi-stationary regime (normally defined as a set of bus voltages magnitudes and phase angles) at low sampling rates. They usually focus on different ways for including phasor measurements and how it impacts on accuracy, convergence properties and bad data detection capabilities; but leaving dynamics aside. In hybrid distributed approaches, such as those in [17][18][19], PMUs are most used to coordinate angles between subsystems thanks to their synchronization capacity. Once more, speed is not considered. Few are the works that put emphasis on dynamics monitoring by using hybrid approaches with incomplete PMU observability. In [20] a hybrid estimator to monitor bus voltages (also known as static state) under different operating conditions is proposed. The state is estimated with a modified weighted least square (WLS) formulation. Voltages at nonobserved buses are calculated from phasor measurements and an interpolation matrix, which is obtained from network admittances as a function of voltages and power injections at nonobserved buses and is updated whenever conventional measurements arrive or in contingencies that provoke a line outage or a significant load change. The author achieves good results regarding accuracy and computing time. Nevertheless, in the construction of the interpolation matrix loads are represented by static models when actually in real systems approximately 60-70% of loads are inductions motors (dynamic loads) [21]. In [22] a hybrid estimator that has been though to work in large systems using parallel computing by means of a graphic processing unit (GPU) is presented. Buses without PMUs are predicted using previous data. RTU-SCADA measurements are extrapolated at PMU update rate by means of a data collation method. The estimation procedure is based on the extended Kalman filter (EKF). Its performance is evaluated under both normal and contingency conditions. A linear state-transition model is used, which could represent a major source of error in the presence of sudden changes. Paper [23] proposes a multi-area hybrid estimator that operates at PMU speed. It estimates the state by using a cubature Kalman filter (CKF). The approach is run in different areas in parallel. The method is evaluated regarding its accuracy and computing time. Results prove that the method can run fast in large systems using all the available measurements.
Nevertheless, it is only capable of estimating with accuracy and quite fast the static state in stationary or quasi-stationary regime. In [11] a two-stage hybrid estimator capable of tracking static states is presented. Both stages are based on the WLS method. The second stage is solved in one iteration using a linear estimator. PMU observability is ensured by adding current pseudo-measurements from previous estimates. The method is prone to errors in the presence of sudden changes and presents limitations when electrical distance between PMUs and nonobserved buses is greater than two buses. Paper [24] introduces another hybrid scheme to track the static state at high sampling rate. Between two consecutive RTU-SCADA scans a limited number of phasor measurements are received. To maintain full observability a set of old conventional measurements are used. The WLS formulation is employed at instances when both kind of measurements are refreshed simultaneously. When only PMU measurements are refreshed, a weighted least absolute value (WLAV) estimator is utilized. The proposed approach can monitor voltage collapses, sags and swells but is not designed to detect faults, instantaneous events, or electrical transients.
From the works presented above, it can been seen that the integration of a few PMU measurements into an existing nonlinear estimator results mostly in a hybrid estimator that has most of the limitations of the conventional estimator when trying to estimate the power system dynamics, that is, only steady state models are used and the high PMU speed is not exploited. Few papers propose schemes at high speed with the aim of tracking the time evolution of the static state. Nevertheless, they may not perform well in the presence of sudden disturbances (transient regime) like faults and what is even more important, power system dynamic is completely ignored. They do not deliver information regarding those variables which have a great meaning in stability analysis, that is, dynamic states such as speed and rotor angle of generators.
In this paper a novel hybrid state estimation procedure that addresses and solves the shortcomings mentioned above is proposed. The main contribution of this paper is the ability of the developed scheme to accurately estimate at high speed power system dynamics associated to slow and fast transient events with a reduced number of PMUs. It is an enhanced version of a previously developed hybrid static state estimator [25]. This time, two changes are introduced: the first one aims at improving the robustness of the estimation process by using a WLAV estimator instead the WLS method to estimate grid bus voltages in transient regime at high speed; the second one involves the incorporation of an EKF based dynamic state estimator in charge of estimating dynamic states of all generators and motors in the power system, even if the unit is not observed by a PMU. Full PMU observability is restored through a novel data-mining based methodology, which defines, first, a PMU topology that allows monitoring the post-contingency bus voltage dynamics of the entire power system and, second, generates a number of bus voltage pseudo-measurements to extend the observability to the whole system. In this way, static and dynamic states are estimated globally, with a reasonable accuracy and as frequent as PMU sampling rate. Such an The rest of the paper is organized as follows. Section 2 presents the hybrid state estimation procedure. First an overview is given. Next, algorithms employed and mathematical models used to represent the static and dynamic behaviour of the power system are described. Section 3 gives a brief description of the data-mining based methodology for restoring full PMU observability. Section 4 includes simulation results and finally, Section 5 presents the conclusions of this paper.

PROPOSED HYBRID STATE ESTIMATION PROCEDURE
The hybrid state estimation procedure described below consists of two phases depending on the system operating conditions as shown in the flowchart of the Figure 1. When the power system is in stationary regime only phase one takes place and the static state together with all the variables that come from it such as line currents, power flows and power injections, are estimated at SCADA speed by a WLS-based static state estimator. Conventional as well as PMU measurements are used together in one single estimation run. Once the estimation is completed, a post-estimation method is carried out in order to detect and identify bad measurements. Weights corresponding to bad measurements are decreased in order to minimize their effect on the result and the estimation is rerun. Input in phase one includes the grid topology, parameters of lines and transformers, mathematical models that represent the network behaviour in stationary regime and both kind of measurements.
Phase two is activated when a physical disturbance happens and the system is in a transient regime. In this case, two estimators work in sequence making use of all the information provided by the PMUs. This allows estimating the power sys-tem dynamics associated with any kind of transient event. First, a static state estimator is used to estimate grid bus voltages as soon as the PMU measurement set arrives. Here, a major drawback is the lack of available PMUs that makes full grid observability unfeasible. This issue is solved through a novel data-mining based methodology for restoring full PMU observability. This approach uses as input the estimated static state at the previous time, phasors of bus voltage provided by PMUs and information regarding the contingency such as failed element and kind of fault and generates voltage pseudomeasurements. Since a post-estimation bad data processing technique is unfeasible at fast scan rates such as those of PMU measurements, the static estimator in phase two uses the information provided by the bad data processor from phase one to adjust the measurement weights if needed. Additionally, in order to improve the algorithm performance in phase two when PMUs are affected by large errors, a robust WLAV based linear estimator is used instead of the WLS method used in the previous version of this work. Second, dynamic variables such as rotor angle and speed of generators and speed of motors are estimated by means of an EKF based dynamic state estimator. The estimator uses as input the estimated static state, mathematical models that represent the dynamic behaviour of the systems under consideration and their parameters. Phase two will run until the power system reaches the stationary regime again.
The hybrid state estimation approach proposed in this work was though and designed under the following assumptions: • Measurement and process errors follow a normal distribution, with standard deviation and mean = 0. • Uncertainty of networks parameters is considered and modelled by using a uniform distribution function with specified lower and upper bounds. • In stationary regime conventional measurements as well as phasor synchronized measurements are used together in the same estimation run. The time-skew between measurements is usually tolerated due to the slowly varying operating conditions of power systems under normal operating conditions [26]. • The contingency together with its time of occurrence is known. Despite the fact this assumption represents an appreciable simplification of the problem, advances in computer science have lately led to several promising works [27][28][29] in the field. • The power system operates under balanced condition and the load composition at each bus (static-dynamic load ratio) is known.

Phase one
Considering the following nonlinear model: where z is the measurement vector; x is the true state vector; h(.) is a nonlinear vector function relating measurements to states; and e is the measurement error vector and assuming m measurements and n states variables, where n < m, the WLS method will minimize the following objective function [26]: The state estimation problem is solved by the first order optimality condition and the estimated static statex is obtained by an iterative procedure at SCADA speed.
The estimator employed in this case has been modified in order to include PMU measurements together with conventional measurements. This allows, first, improving the accuracy of the estimated state [16] and, second, detecting and identifying bad PMU data so that this information can be used later in phase two.
The measurement vector z1 consists of conventional measurements of power injection S inj. , power flow S fl. and voltage magnitude V , PMU voltage measurements V PMU (M< ) (magnitude and angle), virtual measurements S virt. (zero injection), and power flow pseudo-measurements S ps. : The incorporation of PMU current measurements in traditional estimators introduces ill-conditioning problems [13]. That is why, current measurements are replaced by power flow pseudo-measurements, calculated from PMU voltage and current phasor measurements. For further information regarding the calculation of power flow pseudo-measurements readers can refer to [30]. The estimation process is followed by a bad data processor. Error detection is performed by assessing the target function J(x) by means of the Chi-square test and bad measurements are identified by means of the largest normalized

Phase two
Phase two operates in transient regime. First, the static state is estimated as soon as the PMU measurement set arrives. Only phasor measurements together with voltage pseudo-measurements are employed. Pseudo-measurements are obtained by means of a data-mining based methodology, which will be given later in detail. The problem is transformed from polar (M < ) to rectangular (x, y) coordinates in order to get a constant and numerical Jacobian H of the functions h. The measurement vector in the second stage can be expressed as follows: where V PMU (x,y) and I PMU i− j (x,y) are the PMU voltage and current measurement vector and V ps. (x,y) is the voltage pseudo-measurement vector. Standard deviations of the measurements which initially are expressed in polar coordinates must be also adapted to rectangular coordinates. This is accomplished by using the classical theory of propagation of uncertainty [13].
With the aim of improving the algorithm robustness in transient regime when PMUs are affected by large errors, a robust WLAV based linear estimator is used instead of the conventional WLS to estimate grid bus voltages since it can automatically reject bad measurements during the estimation process [31,32]. When only phasor measurements are available the measurement where r is the vector of measurement residuals and the Jacobian H is a constant matrix. The state estimatedx will be given by the solution of the following optimization problem [33]: where c is a weight vector. The optimization problem shown above is expressed as an equivalent linear programming (LP) problem [33] and solved using the interior point solver [34]. The estimated static state will be used as measurement data by an EKF based dynamic state estimator to estimate dynamic variables such as rotor angle and speed of generators and speed of induction motors. Estimators and measurements involved in phase two are depicted in the flowchart of Figure 3.
The dynamic estimator is implemented using a model decoupling technique called "event play back", which has the potential to decouple the EKF problem to better use of the measurement data at terminal buses. Measurements are separated in two groups. One group is treated as input signals to the dynamic model and the other group is treated as output or measurements in the estimation problem. Using such a technique, there is no longer the need to deal with the dynamic model of an entire system [35]. Rather, small-scale dynamic models of generators and motors can be used in the EKF in parallel.
The EKF minimizes the covariance of squared error between real states and estimated ones [36]. Assuming the following general and nonlinear discrete-time system model: where f is the vector of nonlinear function of the states and inputs, x represents the state vector, u is the control input vector, z is the output or measurement vector, w and v are the process and measurement noise. Q and R are the process and measurement noise covariance, and k is the time step for each iteration.
The filter is initialized defining the initial estimated statê x 0 . It is namedx + 0 , where the "+" superscript denotes that the estimated state is a posteriori (on the contrary the "-" superscript denotes a priori), despite not having any available measurement yet, since the first one is taken at t = k. The covariance of the initial estimated state P + 0 it is also defined here: If the initial state is perfectly known, then P + 0 = 0, otherwise P + 0 = ∞I. Then, for k = 1, 2,…, the following operations are performed: 1. Matrices from partial derivatives 2. Time update of the estimated state and its covariancê

Matrices from partial derivatives
. Measurement update of the estimated state and its covariance

Component modelling
This subsection introduces the models used by the proposed hybrid estimation procedure.
To mimic the reality that the available model is a simplified representation of the real world system, simulation data are generated from a complex model where generators are represented with a full sixth-order electromechanical model including transient and sub-transient dynamics in addition to governor and excitation systems and motors are represented with a seventh-order double cage rotor model.
The grid model employed by static estimators to represent the network behaviour in stationary regime entails a conventional two-port pi model for lines and transformers and equivalent power injections for loads and generators [26]. In transient regime synchronous machines are represented by using a fourth-order nonlinear model in d-q reference frame as follows [37]: are the open-circuit transient time constants. E ′ d , E ′ q and E fd are the transient voltages along the d and q axes and the field voltage. X d , X q , X ′ d , and X ′ q are synchronous and transient reactance. I d and I q are the d and q axis current. is the angle giving the position of rotor, 0 is the nominal synchronous speed and is the speed deviation. H is the inertia constant and D is the damping coefficient. P m , P e and Q e are the mechanical and electrical power (active and reactive) of the generator. V and are the terminal bus voltage magnitude and phase angle and V d and V q are the terminal bus voltage in the d and q axes. The state vector together with the input and output vectors are defined as follows: In Equation (14) the field voltage (E fd ) and the mechanical power (P m ) are assumed measurable and therefore known. V , , P e and Q e are obtained from the static estimator. The overall structure of the dynamic estimation process for generators is shown in Figure 4. Variables with a hat operator represent estimated quantities.
Loads comprise a static and a dynamic component (motors). During a transient event, the power consumption of the static part (P st and Q st ) is approximated using the following expression [21]: In Equations (15) and (16) P st 0 , Q st 0 , V 0 and f 0 represent power consumption of the static load, bus voltage magnitude and frequency at the initial operating condition. Coefficients p 1 to p 3 and q 1 to q 3 define the proportion of each component in the model. Typically, K pf ranges from 0 to 0.3, and K qf ranges from -0.2 to 0. The bus frequency f is obtained either directly from PMUs when the bus is measured by such devices or computed as follows: where Δ = − 0 is the bus voltage angle deviation and Δt PMU is the PMU refresh rate. f PMU_next is the frequency reported by the closest PMU to the bus under analysis.
The power injection at terminal bus (P e and Q e ) is estimated by the WLAV-based static state estimator. Then, the power consumed by motors during a transient event can be determined as the difference between power injection and power consumption of the static load: The acceleration of the induction motor is expressed as [38]: where In Equations (19) and (20) is the rotation speed; H is the inertia constant; T e and T m are the electromagnetic and mechanical torque; P mot is the active power consumption of the motor; R s and R r are the stator and rotor resistance; s is the slip; P mot0 , T m0 and 0 are active power, mechanical torque and rotation speed under nominal condition. T e is assumed to be equal to T m in steady state. State, input and output are defined as: The electromagnetic torque is approximated using the estimated bus voltage and power consumption of the motor and its parameters. Figure 5 gives an overview of the dynamic state estimation procedure employed in motors.

METHODOLOGY FOR FULL PMU OBSERVABILITY RESTORATION
The methodology for PMU observability restoration that is proposed in this work is based on the concept of bus voltage coherency. Buses that are coherent with each other show similar voltage behaviour during a transient event and, therefore, the behaviour in one of these buses represents approximately the behaviour of the remaining buses. The proposed approach relies on this idea to define bus voltage pseudo-measurements by means of voltage magnitudes and angles at PMU observed buses, the pre-contingency estimated state and information regarding the contingency. Since coherency depends mainly on

FIGURE 6
New England system. Measurement topology the last two characteristics mentioned before [39], a PMU topology, that is, number of units and their location, must be defined first so that all the coherent areas are observed with high probability. This is accomplished by means of an off-line knowledge discovery process based on data-mining techniques. The scheme works on a large set of scenarios obtained from numerical simulations that are representative of normal, during fault and post-fault behaviour in a broad range of conditions. Uncertainties in the type of contingency (following the N-1 criterion commonly used by the EMS in control centres for real-time security analysis [40]), the failed component and fault location and the load/generation scenario are added by using the "Monte Carlo" method [39]. Once the PMU topology has been defined, a classifier that is designed off-line will forecast coherency at high speed whenever a contingency happens and with that an assignment vector that shows the connection between observed and non-observed buses.
Finally, bus voltage pseudo-measurements will be obtained using information about the estimated state at the previous time, the assignment vector predicted by the classifier and PMU measurements at observed buses using the following equation: (22) where V ps. U i is the bus voltage pseudo-measurement (magnitude and angle) at the non-observed bus U at time i,V U i−1 is the estimated state (bus voltage magnitude and angle) at non-observed bus U at time i − 1 and ΔV PMU O∈U is the voltage deviation measured by the PMU at the observed bus O that is assigned to the non-observed bus U according to the assignment vector.
Since bus voltage pseudo-measurements are calculated from PMU measurements at coherent buses and the estimated state at previous time, they are considered less accurate than real measurements. Therefore, a standard deviation equal to ten times the standard deviation of phasor measurements is considered.
For further information regarding the scheme to define the PMU topology and the classifier in charge of forecasting bus voltage coherency readers can refer to [25].

SIMULATION RESULTS
This section presents the results obtained from the evaluation of the proposed hybrid state estimation procedure together with the methodology for restoring full PMU observability in the New England benchmark system. The software "PowerFactory" is used to generate simulation data that imitate responses of a real system. The simulation step is chosen to be 1 ms. An update rate of 2.4 s for conventional measurements and 40 ms for synchronized measurements are assumed.
Measurements are obtained by adding a random error with normal distribution and zero mean to results from simulation. Standard deviations are computed from the maximum uncertainty for each kind of measurement (see Table 1).
Transmission line parameters are usually assumed to be exact and known. However, in practice this assumption do not hold true. In this work, the uncertainty of line parameters is considered and modelled using a uniform distribution function with typical upper and lower bounds of ± 2%.
In order to obtain more realistic scenarios and to test the estimator under more real conditions, model inadequacy and process noises are considered. This is accomplished by using a more complex model to generate the simulation data in relation to the model used by the hybrid estimation procedure to estimate the states.
Because in phase two estimators operate in sequence, computing times introduce a delay of the order of a few tens of milliseconds (as will be shown later). In order to take this aspect into account, the dynamic estimator in phase two uses as input the estimated static state at the previous PMU time step, that is, the dynamic estimator uses estimated data from 40 ms ago.

Measurement topology
The location of conventional measurements is defined by means of a topological observability analysis [41], so that the system is observable with a level of redundancy [42] of 1.13. In order to determine the PMU topology, a database made up of 10,800 operating scenarios is generated from numerical simulations using the "Monte Carlo" method. Three kind of contingencies are simulated: three-phase fault, load curtailment and generator outage. The database comprises stable and unstable scenarios. A fault clearing time of 0.2 s and a simulation time of 10 s are considered. Then, coherency is evaluated in each scenario using the "k-medoids" method [43], which is applied to time series of voltage magnitude and angle. Similarity is calculated employing the "Euclidean" distance between wavelet coefficients. Next, data pre-processing techniques are applied. They entail the following tasks: (1) reduction of dimension by means of "Principal Component Analysis" [43] and clustering of scenarios through "k-means" [43] considering the precontingency operating state; (2) normalization and reduction of the number of classes (by approximating similar scenarios regarding their coherency) and; (3) removal of scenarios whose frequency of class (in relation to the total number of simulated scenarios) is less than three. Once the data have been adapted, the number of classes for voltage magnitude and angle is considerably reduced (by a factor of 11 and 12 respectively) and, in addition, 17% of the total amount of scenarios are dismissed. Finally, the PMU topology is obtained from a medoid-based iterative exploring process. For the New England system, the best option involves seven PMUs at buses 2, 6, 9, 14, 20, 23 and 29 (see Figure 6). This topology allows seeing all the coherent areas with a 96% of probability.

Coherency forecasting and accuracy analysis of voltage pseudo-measurements
First, a classification matrix is built from a sample that includes approximately 60% of simulated scenarios from the PMU topology. Scenarios are selected so that the sample includes all the classes for voltage magnitude and angle. For each scenario in the matrix, an assignment vector is determined. After that, a "Random Forest" [43] classifier is trained using data from the classification matrix with the aim of predicting the cluster associated to the pre-contingency operating state of a new and unknown scenario. Test scenarios used to assess the classifier performance are obtained from data that have not been used before in the classification matrix. Approximately 90% of test data are well predicted considering voltage magnitude and angle simultaneously. The average computing time for each scenario is 12 ms. Hence, results confirm the classifier accuracy and its ability to work at high speed.
The mean absolute deviation is used as a measure to evaluate the accuracy of voltage pseudo-measurements: where Err Vps is the voltage pseudo-measurement error (magnitude or angle). N is the number of time periods under analysis. Vps i and V i are the voltage pseudo-measurement and the real voltage (magnitude or angle) at time i. With the help of box plots the voltage pseudo-measurement accuracy at non-observed bus 35 is depicted in Figure 7. As noticed pseudo-measurements are less accurate in scenarios that involve three-phase fault. The error remains within 3% and 7 • in most scenarios for voltage magnitude and angle respectively. As an example, Figure 8 shows a bus voltage pseudo-measurement at bus 35 under a three-phase fault on line 25-26.

4.3
Performance of the proposed hybrid state estimation procedure Scenarios from the methodology for full PMU observability restoration are used to test the performance of the hybrid estimation scheme.
Static estimators initializes assuming flat start condition. The initial dynamic state vector can be set from the estimated static state as in [36,44]. In addition, speed of machines will likely be close to its nominal value when the power system is in stationary regime and under normal operating conditions; nevertheless, EKF can converge to the real values even with zero initial states. The initial covariance matrix for generators and motors is set to be P 0 = diag ([5 5 5 5]) and P 0 = 5 respectively.
As mentioned before, in this work measurement as well as structure (line parameters) uncertainties are taken into consideration. Furthermore, process noises from model inadequacy and linearization (integration error) are also included. The process and measurement covariance matrices for generators are defined as Q = 0.06 2 ×(I 4×4 ) and R obs = 0.06 2 ×(I 4×4 ) or R unobs = 0.6 2 ×(I 4×4 ) respectively depending on whether the bus associated with the machine is observable by a PMU or not. Process and measurement variances for motors are set to be w k = 0.9 2 and v k = 0.06 2 respectively. Uncertainties of measureable inputs in generators (field voltage and mechanical power) are modelled in the same way as conventional measurements. The mean absolute deviation is employed as a measure of error to evaluate the estimation accuracy.
The performance of the proposed hybrid state estimation procedure is investigated by means of three case studies.
A. Case 1 (Base case): The first case is also defined as "Base case" since the algorithm operates following the assumptions made above regarding noises, initialization, mathematical models, etc. In this case the hybrid estimator is statically characterized. Test scenarios correctly classified by the methodology for PMU observability restoration are used to this end. Aspects such as convergence, accuracy and computing time are evaluated.
Convergence is analysed by measuring the time evolution of the estimation error. The EKF based dynamic estimator in phase 2 fails to find a solution in approximately 10% of test scenarios. It was found that most cases with convergence problems imply unstable scenarios. Strong nonlinearities could be the cause for the malfunction of the algorithm since the first order Taylor series expansion used by the EKF may induce large estimation errors if the system behaviour is highly non-linear. Nevertheless, an in-depth analysis must be carried out in order to find the real cause behind this issue.
Box plots of Figures 9 and 10 show the distribution of the estimation error in respect of the generator at bus 35 and the motor at bus 4. Only stable scenarios are analyzed. Results are grouped together according to the kind of contingency. It can be noticed that the hybrid estimator provides accurate results on stable condition. Generator speed is estimated more accurately than motor speed despite the fact that the estimated static state is more accurate at bus 4 than at bus 35. This is mainly due to the additional estimation error of the static load linked to the The average computing time is determined in order to check whether the proposed approach can keep up with PMUs update rate of 25 samples per second. Tests are performed in a computer configured with an Intel Core i7 2.2 GHz processor and 8 GB RAM. Results in table 2 prove that the execution time is lower than the PMU sampling period, enabling the proposed scheme to track fast dynamics. Although a centralized approach like the one proposed in this paper might not work in large scale power grids due to the higher computing time involved, it can be extended to a decentralized scheme typically employed for monitoring and control of large scale systems, where the system is divided into smaller areas and local estimators run in each of them. In addition, the problem could be adapted to be solved by means of parallel computing methods. Successful applications in this field can be found in [31,45].

A. Case 2:
This case aims to test the algorithm robustness against higher noise levels. For this purpose measurement variances and noise boundaries in grid parameters are varied one at a time. In the first instance, they are increased by a factor of two with respect to their initial value and then by a factor of five. Results are compared with the ones from the base case. Figures 11 and 12 show how different noise levels in measurements and grid parameters impact on the estimated dynamic states of the generator at bus 35 and the motor at bus 4 respectively considering a tree-phase fault on line 8-9. Due to space-constraints, estimated transient voltages of the generator are not depicted. Two aspects can be highlighted from this example: first, the proposed estimator remains robust against higher noise levels and performs with acceptable accuracy and second, it is able to estimate well enough the states even if the starting point is not exactly known. Under such kind of conditions, the algorithm operates properly and within acceptable margin of error.

A. Case 3
As mentioned above, the estimation scheme proposed in this work incorporates a modification with respect to its previous version. In order to improve the algorithm robustness in transient regime when PMUs are affected by gross errors, the WLS based linear estimator in phase two is replaced by a robust WLAV formulation. This change aims at improving the algorithm performance taking advantage of the bad data rejection capability of robust approaches.
To demonstrate the improvement of the estimator performance that comes from the change mentioned before, a large error is simulated by adding a constant signal to the PMU voltage magnitude measurement at bus 20 on t = 0 s with a value equal to 40% of the reported value. A three-phase fault on line 5-8 is simulated. The algorithm response is compared with the one from its previous version. Figures 13 and 14 depict the estimated static and dynamic state at buses 34 and 20 respectively. As can be noticed, the inclusion of the WLAV estimator in phase two reduces the effect of PMU large errors on the estimated results and improves the algorithm performance.

CONCLUSIONS
This paper presents a novel hybrid state estimation procedure capable of accurately monitoring the power system dynamics associated to fast and slow phenomena using a reduced number of PMUs. The proposed approach allows estimating with accuracy even the dynamic states of units which are not equipped with a PMU at their terminals. The approach is an extension of a hybrid static state estimator previously developed. On this occasion, two changes are introduced: the first aims to increase the algorithm robustness by using a WLAV based linear static estimator instead of the conventional WLS method to estimate bus voltages at high speed, and the second one involves the inclusion of an EKF based dynamic state estimator with the aim of estimating the dynamic states of generators and motors.
The power system is assumed to be partially observed by PMUs. To circumvent the problem of inadequate observability, a data-mining based methodology to restore full PMU observability is proposed. First, it defines a PMU location that allows observing all the coherent areas of the system with high probability and, second, a classifier that is trained offline and forecasts coherency with the aim of computing bus voltage pseudomeasurements. Both algorithms have been tested and evaluated in the New England system. The PMU location implies seven units from which it is possible to observe with 96% of probability all the coherent areas of the system. The classifier accurately predicts at high speed 90% of test scenarios. Pseudo-measurement errors in transient regime remains acceptably low, being less accurate in scenarios concerning three-phase fault.
The performance of the hybrid state estimator is investigated by means of three case studies. In the first one the estimator is statically assessed regarding accuracy, computing time and convergence. Static and dynamic states are accurately estimated and within timescales that allow keeping up with the high PMU update rate and at the same time observing the dynamics involved in the phenomena under analysis. The EKF based dynamic estimator fails to converge in events that lead the system to an unstable and therefore highly no linear behaviour. The second case aims to test the estimator robustness against additional noise in measurements and grid parameters. Results prove that the proposed scheme can perform well under such conditions. The estimator is also capable of performing well when the initial state is not exactly known. Finally, last case prove that the incorporation of a WLAV based linear estimator instead the WLS method in phase two improves the accuracy of the estimated state in transient regime.
In conclusion, a new monitoring tool like the one presented in this paper provides the right conditions for dynamic wide-area monitoring considering the current limitation regarding the amount of available PMUs that can be installed in the network. Through a better knowledge of the actual network condition, emergency conditions can be more easily recognized and possibly avoided or during their occurrence better analysed and remedial action taken in a quicker and more controlled fashion.
Next, some aspects that should be further investigated and enhanced regarding the hybrid state estimation procedure presented in this work: • The proposed scheme has been evaluated in a small benchmark system. Its implementation in large scale interconnected power grids represents a challenge due to the increased size and complexity of the system model and measurement volume. Nevertheless it can be adapted to work in a decentralized scheme. In addition, highperformance computing and parallel computing techniques could be employed to make possible its application in large systems. • The performance of the EKF is degraded when the system behaviour becomes unstable. The strongly nonlinear behaviour of the system could be the reason behind the convergence issues. However, a detailed analysis should be perform in this regard with the aim of discarding other causes. More robust options like the Unscented Kalman Filter (UKF) and the Particle Filter (PF) to cite a few should be tested and evaluated. Data driven methods represent another direction for further research in this sense.