Further Improvement in Stability and Stabilization Margin of Micro-grid Load Frequency Control System with Constant Communication Delays

Abstract

In this paper, improved results are presented for networked micro-grid load frequency control system integrated with electrical vehicle using the classical transcendental characteristic equation approach. In the system topology dealt in this paper, the centralized controller of the micro-grid system is connected to the locally synchronized distributed generation units through a communication channel. This architecture introduces a time delay in the feedback path of the closed-loop control system. The network-induced delay exerts a serious negative influence on performance and stability of the system. If the delay margin soars beyond a critical limiting value, called stable delay margin, the closed-loop system loses stability, and subsequently, the micro-grid system trips from the conventional grid. This paper presents improved results on delay-dependent stability and stabilization of networked micro-grid load frequency control system for a specified gain-phase margin and relative stability index. By employing a standard benchmark system, the analytical results are validated through simulation results.

Appendices

Appendix 1: Delay-Dependent Stability Analysis

The transcendental characteristic equation of the closed-loop system is expressed as follows:

$$\begin{aligned} P(s)+Q(s)e^{-s\bar{\tau }}=0. \end{aligned}$$

(7)

Then, at the marginal stable condition ($s=\pm j\omega _{C}$), following pair of equations hold good:

$$\begin{aligned} P(j\omega _{C})+Q(j\omega _{C})e^{-j\omega _{C}\bar{\tau }}= & {} 0, \end{aligned}$$

(8)

$$\begin{aligned} P(-j\omega _{C})+Q(-j\omega _{C})e^{j\omega _{C}\bar{\tau }}= & {} 0. \end{aligned}$$

(9)

From the equation pair (8) and (9), the exponential term is eliminated, and the following augmented polynomial in $\omega _{C}^2$ is determined:

$$\begin{aligned} W(\omega _{C}^2)=P(j\omega _{C})P(-j\omega _{C})-Q(j\omega _{C})Q(-j\omega _{C})=0. \end{aligned}$$

(10)

The augmented characteristic equation $W(\omega _{C}^2)=0$ is solved for its positive roots $j=1,2,\ldots , m$, and using either (8) or (9), possible stable delay margin candidates $\bar{\tau }_{j}^{\star }$ are computed as follows.

$$\begin{aligned} \bar{\tau }_{j}^{\star }=\frac{1}{\omega _{C,j}}\bigg [tan^{-1}\bigg (\frac{Im\big [\frac{P(j\omega _{C,j})}{Q(j\omega _{C,j})}\big ]}{{-Re\big [\frac{P(j\omega _{C,j})}{Q(j\omega _{C,j})}\big ]}}\bigg )+2\pi n\bigg ], n=0,1,2,\ldots \end{aligned}$$

(11)

The stable delay margin of the system is then given by

$$\begin{aligned} \bar{\tau }^{\star }=\min _{j}\bar{\tau }_{j}^{\star }, \end{aligned}$$

(12)

provided it satisfies the following root sensitivity test:

$$\begin{aligned} sign\bigg (\frac{dW(\omega _{C}^2)}{d\omega _{C}^2}\bigg )=+1,~\forall ~j. \end{aligned}$$

(13)

Appendix 2: Delay-Dependent Stabilization

For a given time delay $\tau$, at the verge of instability, the following characteristic equation holds:

$$\begin{aligned} P(j\omega )+Q(j\omega )e^{-j\omega \tau }=0. \end{aligned}$$

(14)

Now, by replacing the exponential term with the Euler formula, we obtain the following equation:

$$\begin{aligned} P(j\omega )+Q(j\omega )(\cos (\omega \tau )-j\sin (\omega \tau ))=0. \end{aligned}$$

(15)

Now, by segregating the real and imaginary parts in (15) and equating them to zero, the stabilizing region in the PI controller parametric space can be readily obtained; refer [1].

For relative stability specification, the complex variable s is replaced with $(s+\sigma )$ and analyzed in the similar manner.