The behaviour of a dynamically isolated power system for a dispatchable generator system connected with a load can be described by the state space averaging of all conceivable states [29][30]. A system’s state is described as a variable that is reliant on time. The time derivative is then represented in terms of the system’s inputs and state variables. On the other hand, a system’s output is its response to any changes in its state variable. The following is an expression for the time invariant state space equation:

ρ ∈ R n state vectors consisting of frequency deviations ω e , mechanical power P m and governor power P g v ; y ∈ R r r measured output; u d ∈ R m input vector (Load demand). The mathematical state space equation which explains the governor, turbine, and frequency output behaviour can be written as in equation (2). The prime mover model relates changes in steam position Δ P g v to variations in mechanical power output Δ P m . The most basic prime mover model, which may be roughly represented by a single time constant T c h , was employed in this study.

H = the inertia constant; T c h = the turbine time constant; T g v = governor time constant; R = governor speed regulation. Note that, for the simplification purpose, the initial condition of system is assumed zero and all the corresponding vectors are measured in per unit.

It should be emphasized that rather than the state variable, the power deficit in the isolated power system model given in Equation (2) is related to the change in electrical demand that can be achieved from the input state. For the linear estimator solution to be useful, the power deficit related to the input state needs to be converted into a state variable. Figure 1 displays a block diagram used to formulate the estimator design problem. The power deficit input condition was changed into a variable state using the low-pass filter, h, without affecting the observability of the frequency dynamic. The generator model was then added to this state. Additionally, the loop’s augmentation of h serves to guarantee an appropriate estimator transfer function.

Figure 1. Block configuration for an estimator design.

With h in the feedforward loop, the full state-space realizations can be written as follows

The matrix size changes and becomes a fourth-order system with x N = [ x x h ] T , while the matrices A N ∈ R n × n , B N ∈ R n × m , C N ∈ R k × n and C g ∈ R k × n are defined as

Note that the state vector is a separate system and is not dependent on the other state vectors. Hence, the fourth row in the matrix can be neglected during the verification of the observability system. Furthermore, an additional noise denoted as D N is neglected in this work. The primary goal is to create an estimator to estimate P ^ d of P d , which is provided by P ^ d = ℱ ⋅ y , where ℱ is a member of a linear estimator with state space realization in the form of

Hence, T M ( s ) : = C ˜ ( s I − A ˜ ) − 1 B ˜ = G E − h is the transfer function that relates the disturbance input ω d to the estimate error z ˜ . The matrices A ˜ , B ˜ and C ˜ of compatible dimensions are given by

The suggested setup to calculate the power deficit is depicted in Figure 2. It should

Figure 2. The proposed configuration to estimate the power deficit.

be noted that this configuration’s primary objective is to determine the power deficit with minimal estimation inaccuracy. The isolated power system dynamic produced the frequency dynamic variations Δ f after receiving the electrical power demand deviations Δ P e . After feeding this Δ f output to the estimator transfer function’s input, the feasible objective function over the bounded real γ that satisfies the Lyapunov and LMI constraints yields the estimated power deficit P ^ d .

The purpose of the H₂ estimation issue is to establish a guaranteed estimation performance index γ that provides a realistic upper bound over the estimator state-space realisation in Equation (5) is produced by ‖ G E − h ‖ 2 2 ≤ γ . The Lyapunov equation solution in Schur complement form, as shown in Equation (8), can be used to solve this H₂ norm.

The power deficit estimator was designed to adapt with two isolated power system conditions which are nominal parameter and uncertain parameter.

Design for Uncertain Parameter Condition Case

The isolated power system model is thought to be tainted by parameter uncertainty in a robust instance. The inertia constant H is the uncertain quantity that has been taken into account in this instance. This is the sole factor that has a substantial impact on both the power deficit estimation performance and the slope of dynamical frequency. The inertia constant H parameter dependence can be clearly seen at the first row of matrix A N and B N in Equation (4). In this research, the parameter uncertainty is confined to a given polytope satisfying m = ∑ i = 1 v λ i m i for some λ i > 0 such that ∑ i = 1 v λ i = 1 and m belongs to the state matrix in Equation (4). Since the uncertain inertia value denoted by H i is the only parameter that is considered as uncertainty, every point of uncertain inertia value is confined between two vertices.

For the H₂ estimation problem, Equation (9) produces a viable estimator that minimizes γ over the estimator state-space realization, resulting in guaranteed estimation performance that matches the objective function such that ‖ G E − h ‖ 2 2 ≤ γ . As in Equation (9), the Schur complement matrix for the H 2 norm in the sense of Lyapunov stability given in Equation (8) can be expressed as an uncertain state matrix.

Equation (9) shows that the uncertain inertia constant H i only influences the state and input matrix. However, this inequality equation is not considered as a form of LMI due to the multiplication of multiple variables. conversion is required in order for the filter design to become a convex that can be solved extremely effectively by the numerical processes. The non-LMI equation can be decomposed

by partitioning the matrix P ˜ : = [ X U ■ ′ X ^ ] , satisfying the inequality Equation (9) and multiplying to the left by j ′ : = diag [ j ˜ ′ , I ] and to the right by j and j ˜ � = [ X − 1 Y 0 V ′ ] . Then, introduce the new variable as described in [28], the inequalities are equivalent to Equation (10).

As a result, the estimator design problem is equivalent to the following programming problem on the determination of variable positive definite matrices W = W ′ , Z = Z ′ and Y = Y ′ , as well as matrices Q , and F , given in terms of the LMI in Equation (10). The estimator matrices are defined by A f = − Y − 1 Q ( I − Y − 1 Z ) − 1 ; B f = − Y − 1 F ; C f = C g . Hence, the transfer function of H₂ Norm estimator can be written as T f ( s ) : = C f ( s I − A f ) − 1 B f .

The simulation is divided into two situations which are the variation of load demand and the response under different inertia constant parameter values. In addition to that, the isolated power system network model does not involve load frequency management via Automatic Generation Control (AGC) which allowing the influence of frequency droop response to be readily recognized. The dynamical frequency is the vital state to be observed as it will be the input for the designed estimator. Table 1 shows the list of parameters to set up the simulation.

Table 1. Isolated power system network model parameters setup.

The feasible solution for H₂ norm estimator is guaranteed by utilizing the LMI constraint in Equation (10) for robust case. The parameter settings for the isolated power system network model are shown in Table 1. Figure 3 illustrates a 15-second simulation with a sudden electrical load demand change of 0.2 per unit at 3 seconds. In addition, the H₂ norm estimation performance was analyzed through the Integral Absolute Error (IAE), Integral Square Error (ISE) and Root Means Square Error (RMSE) data collections. The IAE and ISE performances were identified to analyse the size of the estimation error over time, while the performance analysis through RMSE on the other hand, was identified to know the size of the data distribution over time.

Figure 3. Frequency dynamical response in per unit towards the variation of load demand.

The value of the inertia constant is supposed to be known, although it has uncertain minimum and maximum values ranging from 4.5 to 5.5, assuming the generator’s inertia varies as the load size changes. The resilience H₂ estimation was ensured in accordance with the target function sup ‖ G E − h ‖ 2 2 ≤ γ under the 0.2 per-unit rapid electrical load demand shift at time 3 seconds. The LMI constraint in Equation (10) was feasible, resulting in an upper bound γ = 159.17 . Table 2 displays the error performance at steady state response using the IAE, ISE, and RMSE. The data shows that the estimation distributions over time are under-conservative

Table 2. The estimation error performance at steady-state utilizing the H₂ norm approach under the uncertain inertia parameter condition.

for each inertia constant value. Compared to the reference of 0.2 per-unit, the steady-state estimated power deficit has a significant inaccuracy of roughly 29% at 0.1413 per-unit as shown in Figure 4. This value is the trade-off of the H₂ norm optimization.

Figure 4. Estimated power deficit using H₂ norm with 0.2 per-unit rapid electrical load demand variation and uncertain inertia constant values.

In a conventional way, the power deficit was estimated by observing the initial slope of the frequency drop upon the disturbance. The approach utilized the swing equation as depicted in Equation (1). Table 3 shows the analysis of steady-state estimation performance through the IAE, ISE and RMSE.

Table 3. The error performance of power deficit steady-state estimation using the initial slope approach with H = 5.0 and 0.2 per-unit load demand as reference.

The result shows that when the inertia constant value deviates from the reference, the estimation error is larger. The IAE, ISE, and RMSE results are the error performance index that proved the weaknesses of the initial slope method. The estimation error becomes worse when the inertia constant changes larger and away from reference. Figures 5(a)-(c) shows the illustration of the error performance data.

Figure 5. The comparison of error performance between H₂ Norm estimator and slope method under uncertain inertia parameter value.

In the nominal scenario, the isolated power system model parameters are assumed to be exactly known. The objective function in Equation (8) results in a slight norm below the optimal γ = 0 . Choosing H = 5.0 as a known inertia parameter, the LMI constraint in Equation (10) was feasible and conformed with the estimation performance index γ , resulting in sup ‖ G E − h ‖ 2 2 ≤ γ for an upper bound of γ = 12.64 . Table 4 displays tabulated statistics on the estimation error performance at steady-state response using the (IAE), (ISE), and (RMSE). The results

Table 4. The estimation error performance at steady-state using H₂ norm method under nominal case.

show that the estimation error is modest. Hence, the anticipated power shortfall is accurate.