Dynamic harmonics–interharmonics identification and compensation through optimal control of a power conditioning application

Abstract

This paper proposes an online identification and compensation scheme for a distorted waveform’s harmonic and interharmonic content in electrical circuits. The proposed novel identification scheme allows the simultaneous estimation of the harmonic components’ frequency, amplitude, and phase. One of the main characteristics of the proposed online identification scheme is how the harmonics and interharmonics are estimated using a fast-convergent state estimator based on the state-space representation of the harmonic content. The proposed estimator has a decentralized approach that relies on efficient implementation, a relevant result for real-time feedback control applications, such as power conditioners control. The decentralized structure allows, if needed, selective harmonic suppression. The paper exposes a case study of identifying and suppressing the harmonic and interharmonic content produced by a three-phase load connected to the electrical power grid to demonstrate the proposed estimator’s efficiency. Hardware-in-the-loop and simulation results validate the identifier effectiveness to determine in real time the harmonic content and how it can be used to reduce the total harmonic distortion through a nonlinear optimal tracking control scheme.

Appendices

Signal state-space representation

Table 1 Estimator gains for phase A

Consider the signal (4) rewritten as

$$\begin{aligned} s = A\cos {\theta } \bar{\omega }_{1}+A\sin {\theta }\bar{\omega }_{2} \end{aligned}$$

(A1)

where $\bar{\omega }_{1}=\sin {\alpha t}$ and $\bar{\omega }_{2}= \cos {\alpha t}$. Then, by taking the corresponding time derivative, it is possible to state that $\dot{\bar{\omega }}_{1}= \alpha \bar{\omega }_{2}$ and $\dot{\bar{\omega }}_{2}= -\alpha \bar{\omega }_{1}$. By proposing the change of variable $\omega _{1}=\frac{\bar{\omega }}{\alpha }$ and $\omega _{2}= \bar{\omega }_{2}$, the dynamics of (A1) becomes

$$\begin{aligned} \begin{aligned} \dot{\omega }_{1}&= \omega _{2}\\ \dot{\omega }_{2}&= -\alpha ^{2}\omega _{2}\\ s&= A\alpha \cos {\theta }\omega _{1}+A\sin {\theta }\omega _{2}. \end{aligned} \end{aligned}$$

(A2)

Introducing a parameter $\lambda$ to scale the magnitude of the signal, and defining $z_{1}= \lambda \omega _{1}$ and $z_{2}= \omega _{2}$, one obtains

$$\begin{aligned} \begin{aligned} \dot{z}_{1}&= \lambda z_{2}\\ \dot{z}_{2}&= -\frac{\alpha ^{2}}{\lambda }z_{1}\\ \dot{z}_{3}&=0 \end{aligned} \end{aligned}$$

(A3)

where the new variable $z_{3}$ is included to model the angular frequency, which is considered to be constant, with initial condition $z_{3}(0)= \alpha ^{2}$. Finally, signal (A1) can be obtained from (A3) through the output

$$\begin{aligned} s = \frac{k_{1}}{\lambda }z_{1}+k_{2}z_{2} \end{aligned}$$

(A4)

where $k_{1}=A \alpha \cos {\theta }$ and $k_{2}= A \sin {\theta }$.

Estimator gains

Table 2 Estimator gains for phase B

Table 3 Estimator gains for phase C

1.1 B.1: Estimator parameter selection guidance

• $\lambda$ Weights the rising time, improving the tendency to the estimated variable.

• $\gamma$ Weights the convergence to the estimated frequency.

• $\xi$ A larger value improves the tendency to the estimated variable.

• $k_{1}$ Improves the tendency to the estimated variable. However, its value must be smaller than those selected for $\lambda$ and $\gamma$.

• $k_{2}$ Weights the amplitude of the oscillations in the estimate of the variable in a steady state.

Proof of Theorem 1

The estimator stability is based on the analysis of interconnected systems, as exposed in [42]. To this end, let us consider that the nth sub-system associated with the generation of the n-signal is described by

$$\begin{aligned} \begin{aligned} \dot{z}_{n1}&= \lambda _{n} z_{n2}\\ \dot{z}_{n2}&= -\frac{z_{n3}}{\lambda _{n}} z_{n1}\\ \dot{z}_{n3}&=0\\ s_{n}&= \frac{k_{n1}}{\lambda _{n}}z_{n1}+k_{n2} z_{n2}. \end{aligned} \end{aligned}$$

(C5)

Then, the corresponding state estimator structure for (C5) is defined as

$$\begin{aligned} \dot{\hat{z}}_{n1}= & {} \lambda _{n} \hat{z}_{n2}+\frac{\lambda _n}{k_{n2}} (s_n-\hat{s}_n) \nonumber \\ \dot{\hat{z}}_{n2}= & {} -\frac{\hat{z}_{n3}}{\lambda _{n}}\,\hat{z}_{n1}+\xi (s_n-\hat{s}_n)\nonumber \\ \dot{\hat{z}}_{n3}= & {} -\gamma _{n} \hat{z}_{n1} (s_n-\hat{s}_n) \nonumber \\ \hat{s}_n= & {} \frac{k_{n1}}{\lambda _{n}}\hat{z}_{n1}+k_{n2}\hat{z}_{n2}. \end{aligned}$$

(C6)

By defining the estimator error variables as $e_{n1}=z_{n1}-\hat{z}_{n1}$, $e_{n2}=z_{n2}-\hat{z}_{n2}$ and $e_{n3}=z_{n3}-\hat{z}_{n3}$, thus $e_n = [e_{n1}~~e_{n2}~~e_{n3}]^T$, then their corresponding time derivatives become

$$\begin{aligned}&\begin{aligned} \dot{e}_{n1}&= \lambda _{n}e_{n2}-\frac{\lambda _{n}}{k_{n2}}\left( \frac{k_{n1}}{\lambda _{n}}e_{n1}+ k_{n2}e_{n2} \right) \end{aligned} \end{aligned}$$

(C7)

$$\begin{aligned}&\begin{aligned} \dot{e}_{n2}&= -\left( \frac{z_{n3}}{\lambda _{n}}+\frac{\xi _{n}k_{n1}}{\lambda _{n}} \right) e_{n1}-\xi _{n}k_{n2}e_{n2}+\frac{z_{n3}}{\lambda _{n}}\hat{z}_{n1}\\&\quad -\frac{\hat{z}_{n1}e_{n3}}{\lambda _{n}}-\frac{z_{n3}}{\lambda _{n}}\hat{z}_{n1} \end{aligned} \end{aligned}$$

(C8)

$$\begin{aligned} \dot{e}_{n3}&= \dot{z}_{n3}+\gamma _{n}\hat{z}_{n1} \left( \frac{k_{n1}}{\lambda _{n1}}e_{n1}+k_{n2}e_{n2} \right) . \end{aligned}$$

(C9)

Therefore, the nth error dynamics becomes

$$\begin{aligned} \dot{e}_{n1}&=-\frac{k_{n1}}{k_{n2}}e_{n1} \end{aligned}$$

(C10a)

$$\begin{aligned} \dot{e}_{n2}&= -\frac{1}{\lambda _n}(\alpha _n+\xi _n k_{n1})e_{n1}-\xi _{n} k_{n2}e_{n2}-\frac{\hat{z}_{n1}}{\lambda _n}e_{n3} \end{aligned}$$

(C10b)

$$\begin{aligned} \dot{e}_{n3}&= \gamma _n \frac{k_{n1}}{\lambda _n}\hat{z}_{n1}e_{n1}+\gamma _n k_{n2}\hat{z}_{n1}e_{n2}. \end{aligned}$$

(C10c)

Consider the candidate Lyapunov function to establish the asymptotic stability to the origin of (C10), as

$$\begin{aligned} V_n(e_n)= \dfrac{1}{2}\Big (e_{n1}^2+\lambda _n^2\,k_{n2}\,e_{n2}^2+\dfrac{\lambda _n}{\gamma _n}e_{n3}^2\Big ). \end{aligned}$$

(C11)

From (C10a), with positive constants $k_{n1}$ and $k_{n2}$, it is immediate to see that $\,e_{n1} \rightarrow 0\,$ as $\,t \rightarrow \infty \,$, then by taking this fact in (C10b)–(C10c), the time derivative of (C11) becomes

$$\begin{aligned} \begin{aligned} \dot{V}_{n}(e_{n})&= -\frac{k_{n1}}{k_{n2}}e_{n1}^{2}+\lambda _{n}^{2}k_{n2}e_{n2}\left( -\xi _{n}k_{n2}e_{n2}-\frac{\hat{z}_{n1}}{\lambda _{n}}e_{n3} \right) \\&\quad +\frac{\gamma _{n}}{\lambda _{n}}e_{n3}\left( \gamma _{n}k_{n2}\hat{z}_{n1}e_{n2} \right) \\&= -\frac{k_{n1}}{k_{n2}}e_{n1}^{2}-\lambda _{n}^{2}k_{n2}^{2}\xi _{n}e_{n2}^{2}. \end{aligned} \end{aligned}$$

(C12)

Therefore, $\dot{V}_n(e_n)$ is negative semidefinite, and by using the LaSalle’s theorem [42], function $\dot{V}_n(e_n)=0$ for $\,e_{n1} = 0\,$ and $\,e_{n2} = 0\,$, consequently $\dot{e}_{n2}=0$ and $\dot{e}_{n2} = 0$, and from (C10b) one obtains $0=-\dfrac{\hat{z}_{n1}}{\lambda _n}e_{n3}$, then the trivial solution $e_{n3} = 0$ is fulfilled. Hence, the estimation error (C10) is asymptotically stable [42].

Recognize that (C10) can be rewritten as

$$\begin{aligned} \dot{e}_{n}&=\underbrace{\left[ \begin{array}{ccc} -\frac{k_{n1}}{k_{n2}}&{}0&{}0\\ -\frac{1}{\lambda _{n}}(z_{n3}+\xi _{n}k_{n1})&{}- \xi _{n}k_{n2}&{}-\frac{\hat{z}_{n1}}{\lambda _{n}}\\ \gamma _{n}\frac{k_{n1}}{\lambda _{n}}\hat{z}_{n1}&{}\gamma _{n}k_{n2}\hat{z}_{n1}&{}0 \end{array} \right] }_{\mathcal {A}_{1}(z_{n3},\hat{z}_{n1})} \left[ \begin{array}{c}e_{n1}\\ e_{n2}\\ e_{n3} \end{array} \right] \end{aligned}$$

(C13a)

$$\begin{aligned}&=\mathcal {A}_{n}(z_{n3},\hat{z}_{n1})e_{n}. \end{aligned}$$

(C13b)

Because the asymptotic stability of system (C10) has been demonstrated through (C11), it is assumed that there exists a positive definite Lyapunov function of the form $V(e_{n})$ such that its time derivative can be represented as $\dot{V}(e_{n})= -\eta _{n} \left\| e_{n}\right\| ^{2}$, where $\eta _{n}$ becomes a stability degree [42], defining its amount of negativeness.

Then, the estimator stability analysis can be extended to the case when n signals (harmonics and interharmonics) are involved in a signal s(t). In this case, the interconnection terms appear in the analysis by considering that a signal $s=\sum _{n=1}^N s_n=\sum _{n=1}^N \left( \frac{k_{n1}}{\lambda _{n}}z_{n1}+k_{n2} z_{n2}\right)$. Particularly, for the error $e_1$ one obtains the interconnection described as

$$\begin{aligned} \begin{aligned} \dot{e}_{11}&= -\frac{k_{11}}{k_{12}}e_{11}+b_{11}e\\ \dot{e}_{12}&= -\frac{1}{\lambda _{1}}\left( z_{13}+\xi _{1}k_{11} \right) e_{11}-\xi _{1}k_{12}e_{12}-\frac{\hat{z}_{11}}{\lambda _{1}}e_{13}+b_{12}e\\ \dot{e}_{13}&= \gamma _{1}\frac{k_{11}}{\lambda _{1}}\hat{z}_{11}e_{11}+\gamma _{1}k_{12}\hat{z}_{11}e_{12}+b_{13}e. \end{aligned} \end{aligned}$$

(C14)

where

$$\begin{aligned} \begin{aligned} b_{11}&= \left[ 0,0,0,-\frac{\lambda _{1}}{\lambda _{2}}\frac{k_{22}}{k_{12}},0,\ldots , \frac{\lambda _{1}}{\lambda _{n}}\frac{k_{n1}}{k_{12}}, -\lambda _{1}\frac{k_{n1}}{k_{12}},0 \right] \\ b_{12}&= \left[ 0,0,0,-\xi _{1}\frac{k_{21}}{\lambda _{2}},-\xi k_{22},0,\ldots , -\xi _{1} \frac{k_{n1}}{\lambda _{n}}, -\xi _{1} k_{n2}, 0 \right] \\ b_{13}&= \Big [0,0,0,\gamma _{1}\hat{z}_{11}\frac{k_{21}}{\lambda _{2}}, \gamma _{1}\hat{z}_{11}k_{22},0,\ldots ,\\&\quad \gamma _{1}\hat{z}_{11}\frac{k_{n1}}{\lambda _{n}}, \gamma _{1}\hat{z}_{11}k_{n2},0 \Big ]. \end{aligned} \end{aligned}$$

Notice that system (C14) can be rewritten into a compact form as

$$\begin{aligned} \dot{e}_{1}=\mathcal {A}_{1}(z_{1},\hat{z}_{1})e_{1}+b_{1}e \end{aligned}$$

(C15)

where

$$\begin{aligned} \mathcal {A}_{1}(z_{1},\,\hat{z}_{1})&= \left[ \begin{array}{ccc} -\frac{k_{11}}{k_{12}}&{}0&{}0 \\ -\frac{1}{\lambda _{1}}\left( z_{13}+\xi _{1}k_{_{11}}\right) &{} -\xi _{1}k_{12} &{} -\frac{\hat{z}_{11}}{\lambda _{1}} \\ \gamma _{1}\frac{k_{11}}{\lambda _{1}}\hat{z}_{11}&{} \gamma _{1}k_{12}\hat{z}_{11} &{} 0 \end{array} \right] \\ b_1&=[b_{11} \,\,\,b_{12} \,\,\,b_{13}]^T \end{aligned}$$

with $e_1=[e_{11} \,\,\,e_{12} \,\,\,e_{13}]^T$ and $e=[e_{11} \,\,\,e_{12}\,\,\,e_{13}\,\,\,\ e_{21} \,\,\,e_{22}\,\,\,e_{23} \,\,\,\ldots \,\,\,\ e_{n1} \,\,\,e_{n2}\,\,\,e_{n3}]^T$. In a similar way, the error $e_2$ can be analyzed, resulting in

$$\begin{aligned} \begin{aligned} \dot{e}_{21}&= -\frac{k_{21}}{k_{22}}e_{11}+b_{21}e \\ \dot{e}_{22}&=-\frac{1}{\lambda _{2}}(z_{23}+\xi _{2}k_{21})e_{21}-\xi _{2}k_{22}e_{22}-\frac{\hat{z}_{21}}{\lambda _{2}}e_{23}+b_{22}e \\ \dot{e}_{23}&=\gamma _{2}\frac{k_{21}}{\lambda _{2}}\hat{z}_{21}e_{21}+\gamma _{2}k_{22}\hat{z}_{21}e_{22}+b_{23}e. \\ \end{aligned} \end{aligned}$$

(C16)

where

$$\begin{aligned} \begin{aligned} b_{21}&= \Big [ -\frac{\lambda _{2}}{\lambda _{1}}\frac{k_{12}}{k_{22}},-\lambda _{2}\frac{k_{12}}{k_{22}},0,-\frac{\lambda _{1}}{\lambda _{2}}\frac{k_{21}}{k_{12}}, \lambda _{1}\frac{k_{22}}{k_{12}},0,\\&\quad \ldots ,-\frac{\lambda _{2}}{\lambda _{n}}\frac{k_{n1}}{k_{22}},0 \Big ]\\ b_{22}&= \Big [ -\xi _{2}\frac{k_{11}}{\lambda _{1}},-\xi _{2}k_{12},0,-\xi _{1}\frac{k_{21}}{\lambda _{2}},\xi _{1}k_{22},0,\ldots \\&\quad -\xi _{1}\frac{k_{n1}}{\lambda _{n}},-\xi _{1}k_{n2},0\Big ]\\ b_{23}&= \Big [\gamma _{2}\hat{z}_{21}\frac{k_{11}}{\lambda _{1}},\gamma _{2}\hat{z}_{21}k_{12},0,\gamma _{1}\hat{z}_{11}\frac{k_{21}}{\lambda _{2}}, \gamma _{1}\hat{z}_{11}k_{22},0,\ldots \\&\quad \gamma _{2}\hat{z}_{21}\frac{k_{n3}}{\lambda _{3}},\gamma _{2}\hat{z}_{21}k_{n2},0\Big ]. \end{aligned} \end{aligned}$$

Therefore, generalizing for n harmonics, one can describe the whole estimation error system as

$$\begin{aligned} \dot{e}= \mathcal {A}\, e+ \mathcal {B}\,e \end{aligned}$$

(C17)

where $\mathcal {A}= \text {diag}\{\mathcal {A}_{1}(z_{1},\hat{z}_{1}), \mathcal {A}_{2}(z_{2},\hat{z}_{2}), \ldots , \mathcal {A}_{n}(z_{n},\hat{z}_{n}) \}$, $\mathcal {B}=[ b_{1},b_{2},\ldots , b_{n} ]^{T}$ and $e = [e_{1},e_{2},\ldots ,e_{n}]^{T}$. Then, it is possible to state the following general Lyapunov function

$$\begin{aligned} V(e)=e^{T}Pe,\qquad P=P^T>0 \end{aligned}$$

(C18)

with time derivative

$$\begin{aligned} \dot{V}(e)= \dot{e}^{T}Pe+e^{T}P\dot{e} \end{aligned}$$

(C19)

which can be described in terms of its nominal part (already demonstrated to be asymptotically stable through (C11)–(C12) for an n estimator, with stability degree given as $\eta _n$), and now with the inclusion of the interconnection part as

$$\begin{aligned} \dot{V}(e)&\le e^{T}(A^{T}P+PA)e +2 \left\| P\right\| \left\| \mathcal {B} \right\| \left\| e_{n}\right\| ^{2} \end{aligned}$$

(C20a)

$$\begin{aligned}&\le -\eta _{P}\left\| e_{n}\right\| ^{2}+2 \left\| P\right\| \left\| \mathcal {B} \right\| \left\| e_{n}\right\| ^{2} \end{aligned}$$

(C20b)

$$\begin{aligned}&\le -\left( \eta _{P}-2 \left\| P\right\| \left\| \mathcal {B} \right\| \right) \left\| e_{n}\right\| ^{2} \end{aligned}$$

(C20c)

where $\eta _{P}$ is the general stability degree for the nominal part. Hence, if $\eta _{P}>2 \left\| P\right\| \left\| \mathcal {B} \right\| \implies \dot{V}(e)< 0$, then the error motion in (C17) is asymptotically stable. $\square$