Property Analysis of Riccati Difference Equation for Load Frequency Controller of Time Delayed Power System Using IMMKF

In this paper, initially a mathematical model is formulated for transient frequency of power system considering time delays which occur while transmitting the control signals in open communication infrastructure. Time delay negligence in a power system leads to improper measurement of frequency variation in power system. The study of impact of time delays on the stability of power system is performed by estimating the decay rate of frequency wave form using Kalman Filter (KF). In power system, there is a possibility of multiple time delays. This paper also focusses on developing Interacting Multiple Model (IMM) Algorithm with multiple model space using Kalman Filter (KF) as state estimator tool. The multiple time delays in power system is considered as multiple model space. The result shows that KF provides better estimate of correct model for a particular input-set. The qualitative properties of Riccati difference equation (RDE) in terms of state error covariance of IMMKF are also analyzed and presented.


Introduction
In many industrial applications, direct measurement of the state variables is impossible which required state estimation based on the output in the course of the process of state controller model [1,3].Among the various filters used, Kalman filter is incorporated in several applications to evaluate the state for power system, signal processing and navigation [3,5] systems with measurement noises.This filter maintains an optimal recursive solution for the least square state estimation problem and its properties including stability.The Riccati equation and its variation, the Matrix Riccati equation appear in numerous scientific and engineering applications such as optimal control and filtering problems [10,20] which plays an important role in the stability analysis of different variations of the Kalman filter.The results are later extended to the non monotonic case [9].
The properties of the solutions of the Riccati equation convergence like monotonicity and stabilizability were discussed [6,7,8,12] by selecting appropriate initial covariance matrix through the monotonic properties [11,20,24].These assumptions were then composed of detectability and stabilizability by for continuous time systems [27] and for discrete-time systems [8].Interrelated results were acknowledged [3] for time-varying detectable and stabilizable systems.The detailed relations between various Riccati equations and the closed-loop stability of linear quadratic optimal control and estimation are illustrated in [6,7,12].
Power system is one of the most intricate system in the eminent field of engineering theory.
Enormous literature is available on the frequency measurement techniques for power system.
Frequency is significantly determined by active power whereas the voltage is determined by the reactive power.The obstacles in power system control are the frequency(active power)control and the voltage(reactive power) regulation.The active power control with frequency control is also known as the Automatic Load Frequency Control (ALFC).Time delays in voltage and frequency play a vital role in evaluating amplitude, phase of voltage and current signals.Time-delays in communication between a controller and generators are one of the eminent impediments in stability.
Minimal research has been performed in the field of power system stability due to time delay.In order to maintain the stability of power system, multifarious control loops are required.Proper control and maintenance of the frequency stability of a power system requires primary, secondary and tertiary frequency control loops.Generally, Primary Frequency Control (PFC)loop is important for intercepting the frequency decline before triggering the under/over frequency protection relays.
The PFC is performed by the governor droop resulting in the steady state errors.
The secondary frequency control (Load Frequency control (LFC) ) [4,22] is used to regulate the frequency in power systems into a desirable range and to control the interchange power between the different control areas through major tie-lines.Finally, tertiary control level is resending the generated units and secondary reserve after a distinct disturbance.An LFC model to consider the effects of frequency control loops for two-area power systems is already developed [13].
The main objective of LFC during load sharing is to restore the balance between load and generation in every control area under definite limits.The delay dependent stability of power system is analyzed using frequency domain approach in [15,16,17].The delay dependent stability of the LFC scheme by using the Lyapunov-theory based delay-dependent criterion and Linear Matrix Inequalities (LMIs) techniques was earlier investigated [19].The wide range of communication networks in power system control causes unavoidable time delays.The impact of these delays on the stability of one-area and two-area LFC systems and an analytical method to determine delay margins, for stability is proposed in [23].An LFC approach based on LMI theory to design a robust controller in delayed control signals is proposed in [2] and of the equivalent seven stability criteria is established in [25].The present estimation problem is based on the measured data to approximate the values of the unknown parameters used from the measured data.

Motivation and Contribution
Extended Kalman Filter has been applied for estimating various parameters in PFC [21], whereas there is no report in literature on the tracking and estimation of frequency decay in Secondary Frequency Controller (SFC) with this kind of deriving mathematical model and discretization of the system.This paper focuses on the application of an IMMKF for estimating the decay rate in the transient frequency of power system by considering various time delays as different models.This helps as a guide to analyze the impact of time delays on powers system.
The current research article is organized as follows.In section 2, the model of power system incorporated with communication delay is developed and discussed and also the mathematical formulation for transient frequency variation of power system with time delay is presented.Section 3 proposes an IMMKF algorithm for estimating the state of the system and the evaluation of the properties of the Riccati Difference equation through state error covariance is also presented.A detailed discussion on the IMMKF algorithm and the Properties of the Riccati Difference equation such as monotonicity and stability are stated in Section 4. Results of estimating the measurement vector using Kalman filter and identifying the correct input set in multiple model space using IMMKF is presented in section 5. Conclusions are drawn in Section 6. (2.1) is the reference set power which acts as a control input.
Here, ∆() is fed to PI controller which controls the speed changer position.From the Figure 1, ∆()can be written as Here, , where  is the magnitude of step change in load.
Then , ∆() = Inverse Laplace transformation of ∆() yields the answer in the form

Mathematical Formulation of Transient Frequency Deviation
The mathematical model for the transient frequency variation of power system is formulated as The above can be expressed as where  1 () =  1  − 1  , (exponentially decaying component) 1 () and  2 () are the solutions of the following differential equations Using state-space representation, we get With the definitions of the equations (3.5) -(3.7), equations (3.3) and (3.4) can be rewritten as Kalman filter is used to estimate the states x1, x2 and x3 and the parameters a1, a2, α1, α2, ω.These parameters are evaluated as additional states invariant with time.Using Forward Euler discretization method ,the augmented state equations is given as, where Here, k is the sampling instance and  is the sampling interval.The general form of the above equations is The measurement vector () is given by The coefficient matrix  ˇ,  ˇ are the Jacobian matrices are as follows; In the next section, the estimation of state variables of the system with the help of Kalman filter is discussed and the development of interacting multiple model is presented.

The IMMKF algorithm
The IMM algorithm incorporates r interacting filters functioning simultaneously with every filter corresponding to a model of the following stochastic hybrid system [5] where ℜ  is the state of every system and εℜ  is the measurement vector where •

• State Estimate and Covariance combiner
The state estimator and the covariance combiner are respectively calculated as where ℜ  is the state of every system and εℜ  is the measurement vector.
This section is concerned with the stability and monotonicity properties of solutions �� � �� of the Riccati Difference Equation (RDE) of optimal filtering [6,7].The associated Riccati Difference Equation is To obtain the results of monotonicity and stability in this section , the following fundamental theorems are needed.

Monotonicity
The subsequent theorems characterize the monotonic property of the solution of the RDE , which performs an important features in the stability analysis of optimal filters.
To conclude that, a real symmetric nonnegative definite solution  * of the RDE is called a strong solution if the corresponding state transition matrix  � ≜  � −  �  *  �  ( �  *  �  + ) −1  � has all its eigen values inside or on the unit circle and hence it is stabilizing and by comparing the RDE (5.10) with eqn.(5.9) we have  � � � � =   + � � � − � � + 1�, which leads to monotonicity.

Results and Observations
In this paper, Kalman filter is introduced as a tool to estimate the state variables of power system.
The Multiple Kalman filter is in parallel comparing how well each Kalman Filter matches true measurement from the actual measurements of system.Implementing the IMM allows for an arbitrary number of Kalman Filters to be operated each one modeling a different point with different time delays.Although number of Kalman Filters in IMM framework is conceptually unlimited, the computational requirement of IMM increase linearly as more models are evaluated.
In this section, the mathematical formulation of IMM is established.The parameters that define IMM and the results of IMM is presented with KF as the estimator of individual models.Although IMM was developed as a tool for target tracking environment, the basic formula can also be applied to the estimation of impact of time delay on the frequency waveform of power system.The set of time delayed power system models are defined as the model space in IMM.Each model is defined by the decay rate (α 1 ) of exponential component and decay rate (α 2 ) of oscillatory component which indirectly related to the study of impact of time delays on the frequency wave form of power system.
In this research, model space consists of 2 models.These models are chosen in order to mean the power system is subjected to different time delays while transmitting the control signals in LFC system.The operation of KF as the estimation for IMM is considered.
To demonstrate the application of IMMKF algorithm for the estimation of frequency decay in Power system, a single area LFC system is considered.The corresponding nomenclature is given in The initial covariance matrix , which is of the form of (6x6) matrix is assumed to have all elements as 15 and the noise vector  ˇ is given a value of 0.05 and  ˇ as 0.001.The coefficient matrix  ˇ,  ˇ are the Jacobian matrices are as follows For different h values, the initial values of the state vector are calculated and presented in table1 below.When h increases the decay rate decreases which means that for the higher value of time delays, the system may go unstable.This is verified through simulation.Initially, Kalman filter is applied for estimating the measurement vector of the system .It is presented in the following figures    The work is further extended by applying IMMKF by considering two different time delays with h = 5s and h = 35s as two different models.The operation of KF as the estimation for IMM is considered.The result shows that KF provides better estimate of correct model for a particular input set. Figure 4(c) shows the model estimate of IMMKF when the input data set is for h=5s.
The properties of the RDE such as stability and monotonicity are also analyzed and verified.
Using the eqn.(5.7), the solution of RDE (5.6) is Therefore ,  0 ≥ 0 and  0 ≥  *  * is a strong solution and it is also found that the all the eigen values are inside or on the unit circle.
Hence it is stabilizing.
In the same manner,  ˇ0 =   +  0 −  1 Using this it is observed that the state error covariance matrix of Ricatti equation is monotonic in nature.

Conclusion
In this paper, the transient frequency deviation in the power system following a load disturbance is mathematically formulated.And the Kalman filter was introduced as a means to estimate the decay rate of frequency waveform which indirectly related to the study of impact of time delay on the stability of power system.The application of the IMMKF technique for estimation of frequency decay, in one area LFC of power system with different time delay, is considered and the results of IMM when using KF as a state estimator for each model is also presented.KF provides a better estimate of the correct model.The impact of time delays in power system on the decay rate of frequency variation is also analyzed and presented.The variation of decay rate for various time delays is also shown in results.Moreover, the stability and monotonicity property of Riccati Difference Equation is analyzed.
the limitations of communication network is shown in Figure1.One of the main limitations in communication network is the time delay arising during the transmission of control signals.This consists of governor, turbine and load generator which are modeled by a transfer function of first order.Multiple factors contribute to the time delay in LFC system.All these delays are appended as a single delay denoted by  − , where τ is the time delay.The Transfer function of governor GH is 1 1+  where   is a governor time constant (s), the Transfer function of non reheat turbine GT is 1 1+  where   is a turbine time constant (s), the Transfer function of PI controller is   + ) is the frequency deviation which acts as a output signal.Let ∆  () =   +    .∆()

Figure 1 .
Figure 1.Block diagram of Load Frequency Control

Figure 2
Figure 2(c) shows the Frequency deviation at a particular bus following a step load distribution in the power system.This figure is the combination of the exponential component and oscillatory component which is the measurement vector.The figure shows that the Kalman filter is more effective in estimating the frequency decay of the power system.Similarly, for increasing value of the time delay h = 10s, estimation of the state vectors for exponential and oscillatory components is presented in Figure 3(a) and Figure 3(b).Also, tracking the frequency decay using Kalman Filter is shown in Figure 3(c).For different values of h, Figure 4(a) shows exponential component and oscillatory component is presented in Figure 4(b).