Frequency Regulation in Smart Microgrids Based on Load Estimation

Abstract

The desired frequency is maintained in Smart Microgrid (SMG) when the generated power matches the grid load. Variability of wind power and fluctuations of the load are the main obstacles for performance improvement of frequency regulation in SMG. Active Power Control (APC) services provided by wind power generators is one of the main sources for performance improvement in frequency regulation. New coordinated APC architecture, which involves simultaneous speed and pitch control actions delivers desired power to the grid despite significant variations of the wind power. A tool-kit with discrete-time input estimation algorithms, which estimate input quantity using output measurements is presented. Unmeasurable load fluctuations are estimated with input estimation method using measurements of grid frequency deviation. Desired power for APC is driven by estimated and a priori known loads. This observer-based control method reduces the risk of overshoots and oscillations in frequency regulation loop compared to PID controllers driven directly by the frequency deviation. The stability of the closed loop frequency control system is proved, and simulation results show that observer-based control architecture provides significant improvement of the frequency regulation in SMG.

Appendix: A Tool-Kit for Discrete-Time Input Estimation Algorithms

1.1 Problem Statement

Consider the following system :

$$\begin{aligned} x_k= & {} a x_{k-1} + \underbrace{z_{k-1}}_\text {known input} - \underbrace{d + \xi _{k-1}}_\text {unknown input} \end{aligned}$$

(26)

where $x_k$ is measurable output of the system, $z_k$ is known input, d is unknown constant input to be estimated, $\xi _k$ is unmeasurable zero mean white Gaussian noise,  $k=1,2,...$. The system parameter $0< a < 1$ is known.

The problem is to find estimator for unknown constant input d, using measurements of the system output $x_k$.

1.2 A Simple Data-Driven Estimator

Data-driven estimator $\hat{d}$ for unknown input d can be written as follows:

$$\begin{aligned} \hat{d}_k = \left\{ a^k x_{0} + \sum _{j=0}^{k-1} a^{k-j-1} z_{j} - x_k \right\} \frac{(1-a)}{(1 - a^k)} \end{aligned}$$

(27)

This estimator is suitable for tracking of unknown time varying input $d_k$. Accuracy of estimation is associated with the following estimation error

$$\begin{aligned} \hat{d}_k - d = - \frac{(1-a)}{(1 - a^k)} \sum _{j=0}^{k-1} a^{k-j-1} \xi _{j} \end{aligned}$$

(28)

which is sufficiently small if a is close to one.

Notice that the estimator (27) provides better performance compared to the performance of the simplest estimation technique, which follows directly from (26) $\hat{d}_k = z_{k-1} + a x_{k-1} - x_k$.

Notice also that estimator (27) is a discrete-time counterpart of continuous time estimator proposed in Stotsky and Kolmanovsky (2002), which is widely used in automotive applications. Discrete-time estimators similar to (27) can also be found in Ljung (1999).

1.3 Least Squares Estimator

1.3.1 Description of the Estimator

Equation (26) can be written in the following form:

$$\begin{aligned} y_k = \varphi _k~d + \zeta _k \end{aligned}$$

(29)

where $\displaystyle y_k = x_k - a^k x_{0} - \sum _{j=0}^{k-1} a^{k-j-1} z_{j}$ is the synthetic output, $\displaystyle \varphi _k = - \frac{(1-a^k)}{(1 - a)}$ is the regressor, d is unknown parameter, and $\zeta _k$ is input noise associated with the noise $\xi _k$.

Introduction of the following model

$$\begin{aligned} \hat{y}_k = \varphi _k~\theta _k \end{aligned}$$

(30)

for system (29) together with minimization of the following performance index $\displaystyle E_k = \sum _{j=1}^{k} w_j (y_j - \hat{y}_j)^2$ with respect to the parameter $\theta _k$ yields:

$$\begin{aligned} \theta _k = \left[ \sum _{j=1}^{k} w_j \varphi ^2_j \right] ^{-1} ~ \sum _{j=1}^{k} w_j \varphi _j y_j \end{aligned}$$

(31)

where $w_j$ is a weighting sequence. Assigning weighting factor to one in step k and to $\lambda _0$ in the previous steps the least squares estimate (31) is written in the following recursive form:

$$\begin{aligned} \gamma _k= & {} \frac{\gamma _{k-1}}{\lambda _0 + \gamma _{k-1} \varphi ^2_k}, ~~~~\gamma _0 > 0 \end{aligned}$$

(32)

$$\begin{aligned} \theta _k= & {} \theta _{k-1} + \gamma _k ~\varphi _k ~(y_k - \theta _{k-1} \varphi _k) \end{aligned}$$

(33)

where $0< \lambda _0 < 1$ is a forgetting factor. Notice that stability of the system (29), (30), (32) and (33) is proved in Stotsky (2013) for general case.

1.3.2 Limiting Form of the Estimator

Least squares estimator (32), (33) can be simplified for implementation via substitution the limiting form of the gain $\gamma _{\infty } = ( 1- \lambda _0 ) (1- a)^2$ in (33) as follows :

$$\begin{aligned} \theta _k = \theta _{k-1} - ( 1 - \lambda _0 ) (1- a) ~ (1 - a^k) ~ (y_k - \theta _{k-1} \varphi _k) \end{aligned}$$

(34)

The estimator is driven by the synthetic output $\displaystyle y_k = x_k - a^k x_{0} - \sum _{j=0}^{k-1} a^{k-j-1} z_{j}$ associated with the system (26).

Substituting regressor in (34) and neglecting for simplicity the transient component associated with $a^k$ the error model is presented in the following form:

$$\begin{aligned} \tilde{\theta }_k = \lambda _0^k \tilde{\theta }_0 - (1- \lambda _0)~(1-a)~ \sum _{j=1}^{k} \lambda _0^{j-1} ~\zeta _{k - j + 1} \end{aligned}$$

(35)

where $\displaystyle \tilde{\theta }_k = \theta _k - d$ is estimation error, $k = 1,2...$. Accuracy of estimation is determined (after some transient) by the second term in Eq. (35), which is associated with the noise. This term can be made sufficiently small, if forgetting factor $\lambda _0$ is close to one. The same factor $\lambda _0$ determines the convergence rate of estimated parameter $\theta _k$ to its true value d, and the convergence is slow, if $\lambda _0$ is close to one. In other words the choice of the forgetting factor represents a tradeoff between the tracking performance of fast varying $d_k$ and accuracy of estimation associated with amplification of input noise.

The estimator (34) may provide better performance compared to simple estimator (27) provided that the parameter $\lambda _0$ is chosen properly.

Notice that algorithm (34) is a discrete-time counterpart of the continuous time turbine inertia moment estimation algorithm described in Stotsky et al. (2013), see also Stotsky and Egardt (2013).