Grid-Connected Inverter Control Via Linear Parameter-Varying System Approach

Depósito de investigación de la Universidad de Sevilla https://idus.us.es/ This is an Accepted Manuscript of an papers published IECON 2022 – 48th Annual Conference of the IEEE Industrial Electronics Society available at: 10.1109/IECON49645.2022.9968687 “© 2022 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other uses, in any current or future media, including reprinting/republishing this material for advertising or promotional purposes, creating new collective works, for resale or redistribution to servers or lists, or reuse of any copyrighted component of this work in other Works” Grid-Connected Inverter Control Via Linear Parameter-Varying System Approach Wensheng Luo Sch. of Electr. Engine. and Auto. Harbin Institute of Technology Harbin, China [email protected] Shuhao Li Sch. of Electr. Engine. and Auto. Harbin Institute of Technology Harbin, China [email protected] Sergio Vazquez Electr. Engine. Depart., Sch. of Engine. Universidad de Sevilla Seville, Spain ser[email protected] Jinqian Du Sch. of Astronautics Harbin Institute of Technology Harbin, China [email protected] Ligang Wu Sch. of Astronautics Harbin Institute of Technology Harbin, China [email protected] Leopoldo G. Franquelo Electr. Engine. Depart., Sch. of Engine. Universidad de Sevilla Seville, Spain [email protected]g Abstract—This paper is concerned with the controller design for grid-connected inverter facing parameter variation and stochastic perturbation. Considering these two factors, a stochastically perturbed linear parameter varying (LPV) system for the inverter is formulated, upon which the stability analysis and controller synthesis have been conducted. Parameter dependent sufficient conditions have been obtained to guarantee the asymptotical and exponential mean square stability, with which the asymptotical and exponential controllers are designed respectively. A two-level three-phase inverter is used to verify the effectiveness of proposed theories. Simulation results show that both the controllers are effective under parameter variation and stochastic perturbation, and the exponential controller performs better than the asymptotical controller. Index Terms—linear parameter varying system, stochastic perturbation, grid-connected inverter I. INTRODUCTION Grid-connected inverters are commonly used to connect the dc sources (such as photo-voltaic system, energy storage system) to the utility grid. To control the inverters, many control strategies have been proposed, including PID control, sliding mode control, model predictive control, etc. [1], [2]. For these methods, the inverter is mostly modelled as timeinvariant model. However, the inverter parameter such as the filtering inductance and its internal equivalent resistance may vary with time. Also, the grid consists in various devices and can bring stochastic perturbations to the inverter [3], [4]. Linear parameter-varying (LPV) systems are those systems dependent on unknown but measurable time-varying parame- This work was supported in part by the National Key R&D Program of China under Grant 2019YFB1312001, the National Natural Science Foundation of China, under grants 62003114 and 62033005, the China Postdoctoral Science Foundation, under grant 2020M681097, the Heilongjiang Postdoctoral Fund, under grant LBH-Z20134, the Heilongjiang Touyan Program, and the Consejeria de Economia, Conocimiento, Empresas, y Universidad, Secretaria General de Universidades, Investigacion y Tecnologia, under project P18-RT-1340. Sergio Vazquez gratefully acknowledge the financial support provided by the project PID2020-115561RB-C31 funded by MCIN/AEI/10.13039/501100011033. ters, where the measurement of the time-varying parameters provides real-time information on the variations of the plant’s characteristics. It was firstly brought up by Shamma in 1991 [5]. Since then, researches on this field have been growing greatly, both in theoretical research and engineering practice. On theoretical field, to name a few, a linear matrix inequality (LMI) method was brought up to analyze the LPV systems in [6], the L2synthesis problem for LPV systems via parameterdependent Lyapunov functions was addressed in [7], the H∞ and L2-L∞filtering problem of time-delayed LPV systems was dealt with in [8]. On the other hand, LPV control are very popular in practical engineering, for example, the LPV control was applied in hypersonic air-breathing vehicles [9], the robust control of DC-DC power converters was realized via LPV method [10], also the doubly fed induction machines was controlled with LPV method [11]. Stochastic systems attracts much attention due to its use in many branches of science and engineering applications. For example, the robust stability and stabilization problem was dealt with by employing the LMI approach [12], the stochastic stabilization was addressed for bilinear continuous delayed systems [13], the robust H∞synthesis was conducted for statedelayed stochastic systems [14], a sliding mode controller was designed for delayed stochastic systems [15], the H∞output feedback controller was synthesized for stochastic systems with time-varying delays [16], a sliding mode controller was designed for a switched stochastic system [17]. To authors’ knowledge, though there are many researches concerning with the LPV systems due to its importance in theory and popularity in practical engineering, these systems disturbed by stochastic noises receive little attention. Considering that stochastic phenomenon is very common in various kinds of systems, investigation of LPV systems under stochastic perturbation would be meaningful. In the meantime, considering the parameter variations and stochastic perturbations faced by the grid-connected inverter, it is proper to model it as a LPV system with stochastic perturbation. The rest of this paper is organized as follows. The system description and preliminaries are put in Sections 2. Section 3 presents main theoretical results. Simulation results are given in Section 4 and Section 5 concludes this paper. II. SYSTEM DESCRIPTION AND PRELIMINARIES Itˆ o stochastic systems have drawn great attention in the past decades due to its wide applications in branches of science and industry, which includes chemical process control, random errors in computer operation, economics, etc. [18]. In this paper, a parameter varying system under stochastic perturbation is considered: dx(t)=[A(ρ(t))x(t) + B(ρ(t))u(t)]dt +Bω(ρ(t))x(t)dω(t) (1) where x(t)∈Rnis the state vector; u(t)∈Rmis the control input; ω(t)is a one-dimensional (1-D) Brownian motion satisfying E{dω(t)}= 0 and E{dω2(t)}=dt; ρ(t)∈Rsis vector-valued parameter evolving continuously over time and its range is limited to a compact subset, and ρ is used instead of ρ(t)for simplicity in the rest of this paper. The value of ρis unknown, but can be measured in real time. Matrix B is assumed to be of full column rank. The autonomous system of (1) can be formulated as dx(t) = A(ρ)x(t)dt +Bω(ρ)x(t)dω(t)(2) Before presenting the main results, first introduce the following lemmas and definitions which will play important roles in deriving the theorems. Definition 1: System (1) is called asymptotically mean square stable if lim x→∞ Ekx(t)k2= 0 for any initial state x(0) ∈Rn. Definition 2: System (1) is called asymptotically mean square exponentially stable if Enkx(t)k2o≤ηkx(0)k2e−λt,∀t≥0(3) where η≥1, λ > 0for any initial state x(0) ∈Rn. Lemma 1: ([19]. Itˆ o’s formula) Let x(t)be an ndimensional Itˆ o’s process on t≥0with the stochastic differential dx(t) = f(t)dt +g(t)dω(t)(4) where f(t)∈Rnand g(t)∈Rn×m. Let V(x(t), t)∈ C2,1(Rn×R+;R+). Then,V(x(t), t)is a real-valued Itˆ o’s process with its stochastic differential given by dV (x(t), t) = LV(x(t), t)dt +Vx(x(t), t)g(t)dω(t)(5) LV(x(t), t) = Vt(x(t), t) + Vx(x(t), t)f(t) +1 2trace(gT(t)Vxx(x(t), t)g(t)) (6) where C2,1(Rn×R+;R+)denotes the family of all real-valued functions V(x(t), t)such that they are continuously twice differentiable in xand t. If V(x(t), t)∈ C2,1(Rn×R+;R+), set Vt(x(t), t) = ∂V (x(t), t) ∂t Vx(x(t), t) = ∂V (x(t), t) ∂x1 ,...,∂V (x(t), t) ∂xn Vxx(x(t), t) = ∂2V(x(t), t) ∂xixjn×n Itˆ o’s formula plays very important role in the stability analysis of stochastic systems. Lemma 2: [14] For system (1), if there exists a Lyapunov function V(x(t), t), which satisfies LV(x(t), t)<0(7) Then the system is asymptotically mean square stable. III. STABILITY ANALYSIS AND STABILIZATION In this section, the stability analysis and stabilization of system (2) will be carried out. Theorem 1: If there exists a family of parameter-dependent continuous differentiable symmetric positive matrices P(ρ), such that for all the parameter trajectories, it satisfies ˙ P(ρ) + sym (P(ρ)A(ρ)) BωT(ρ)P(ρ) ∗ −P(ρ)<0(8) Then the system (2) is asymptotically mean square stable. Proof. Choose Lyapunov function candidate as V(x, ρ) = xT(t)P(ρ)x(t)(9) According to Itˆ o’s formula, dV (x, ρ) = LV(x, ρ)dt + 2xT(t)P(ρ)Bω(ρ)x(t)dω (t) (10) where LV(x, ρ) =xT(t)˙ P(ρ)x(t)+2xT(t)P(ρ)A(ρ)x(t) +xT(t)BωT(ρ)P(ρ)Bω(ρ)x(t) =xT(t) [ ˙ P(ρ) + sym (P(ρ)A(ρ)) +BωT(ρ)P(ρ)Bω(ρ)]x(t)(11) If the following inequality holds, ˙ P(ρ) + sym (P(ρ)A(ρ)) + BωT(ρ)P(ρ)Bω(ρ)<0(12) Then LV (x, ρ)<0. And according to Lemma 2, system (2) is asymptotically mean square stable. By Schur complement, inequality (12) equals the inequality (8). Proof completed.  Remark 1: Theorem 1 introduces a parameter dependent Lyapunov function matrix P(ρ), which greatly reduces the conservativeness comparing with employing a common matrix Pfor all the parameter trajectories. This would enormously increase the computation burden, but improves the feasibility of finding the Lyapunov function matrix. Theorem 2: If there exists a family of parameter-dependent continuous differentiable symmetric positive matrices R(ρ) and matrices F(ρ), such that for all the parameter trajectories, the following inequality holds: ΠBωT(ρ)R(ρ) ∗ −R(ρ)<0(13) where Π = sym {A(ρ)R(ρ) + B(ρ)F(ρ)} − s X i=1 τi ∂R (ρ) ρi (14) Then the system (1) is mean square stable under u= K(ρ)x(t)with K(ρ) = F(ρ)R−1(ρ). Proof. Consider system (1) with u(t) = K(ρ)x(t), in which K(ρ)is a parameter-dependent state feedback controller, then the closed-loop system can be written as dx(t)=(A(ρ) + B(ρ)K(ρ))x(t)dt +Bω(ρ)x(t)dω(t)(15) Multiply (8) by P−1(ρ)both on left and right side, then it is obtained that P−1(ρ)˙ P(ρ)P−1(ρ) + A(ρ)P−1(ρ) + P−1(ρ)AT(ρ) +P−1(ρ)BT ω(ρ)P(ρ)Bω(ρ)P−1(ρ)<0 (16) where ˙ P=dP dρ ·dρ dt = s X i=1 τi ∂R(ρ) ∂ρi , τi=dρi dt (17) Let matrix R(ρ) = P−1(ρ), then it is obtained that ∂R(ρ) ∂ρi =−R(ρ)∂P (ρ) ∂ρi R(ρ)(18) Thus (16) can be rewritten as − s X i=1 τi ∂R(ρ) ∂ρi + sym (A(ρ)R(ρ)) +R(ρ)BT ω(ρ)P(ρ)Bω(ρ)R(ρ)<0(19) By Schur complement, (19) equals    sym (A(ρ)R(ρ)) − s X i=1 τi ∂R(ρ) ∂ρi R(ρ)BT ω(ρ) ∗ −R(ρ)   <0(20) For closed-loop system with state feedback u(t) = K(ρ)x(t), (20) changes into Π1R(ρ)BT ω(ρ) ∗ −R(ρ)<0(21) where Π1= sym (A(ρ) + B(ρ)K(ρ))R(ρ)) − s X i=1 τi ∂R(ρ) ∂ρi Let F(ρ) = K(ρ)R(ρ), then (13) cab be obtained. Proof completed.  Remark 2: τidenotes the varying rate of parameter i which is assumed to be measurable in real time. If τiis not measurable, but its bound is known a prior, i.e. |τi| ≤ υi, then Ps i=1 τi∂X(ρ) ∂ρican be approximated by Ps i=1 ±υi∂X(ρ) ∂ρi to obtain a new stability condition. Ps i=1 ±(·)represents the sum of every combination of +(·)and −(·), which contains a total of 2scombinations. Theorem 3: For a given scalar λ > 0, if there exist parameter dependent positive definite matrices P(ρ), such that for all parameter changing trajectories, the following inequality holds, ˙ P(ρ) + sym (P(ρ)A(ρ)) + λP (ρ)BωT(ρ)P(ρ) ∗ −P(ρ)<0 (22) Then the system (2) is exponentially mean square stable with u(t) = 0. Moreover, the decaying rate is Enkx(t)k2o≤ηkx(0)k2e−λt, where λ= max λmax (P(ρ)) /min λmin (P(ρ)). Proof. Choose Lyapunov function candidate as V(x, ρ) = xT(t)P(ρ)x(t), then (10) and (11) can be obtained. According to the Schur complement, the inequality (22) implies ˙ P(ρ)+sym (P(ρ)A(ρ))+BωT(ρ)P(ρ)Bω(ρ)<−λP (ρ) (23) then, dV (x, ρ)<−λxT(t)P(ρ)x(t) + 2xT(t)P(ρ)Bω(ρ)x(t)dω (t) =−λV (x, ρ)+2xT(t)P(ρ)Bω(ρ)x(t)dω (t) (24) Consider, deλtV(x, ρ) =λeλtV(x, ρ)dt +eλtdV (x, ρ) <eλt[λV (x, ρ)dt −λV (x, ρ)dt + 2xT(t)P(ρ)Bω(ρ)x(t)dω (t)] =2eλtxT(t)P(ρ)Bω(ρ)x(t)dω (t)(25) Integrate the both sides of (25) from 0 to t, and take expectation, then EeλtV(x, ρ)−V(x(0) , ρ (0))<0(26) let m= max all ρ λmax (P(ρ)) , n = min all ρ λmin (P(ρ)) (27) then E{V(x, ρ)} ≥ nE nkx(t)k2o, E{V(x(0) , ρ (0))} ≤ mkx(0)k2(28) then Enkx(t)k2o≤1 nE{V(x, ρ)} ≤1 ne−λtV(x(0) , ρ (0)) ≤m ne−λtkx(0)k2(29) let η=m n, then Enkx(t)k2o≤ηe−λtkx(0)k2(30) Then R(ρ) = 3 X j=1 fj(ρ)Rj, F(ρ) = 3 X j=1 fj(ρ)Fj(34) Then the problem becomes finding matrices Rj(j= 1 ∼3) and Fj(j= 1 ∼3), to satisfy R(ρ)>0and (13) and (31). Gridding the parameter space uniformly using a 20 ×20 grid, and approximating P2 i=1 τi∂X(ρ) ∂ρiby P2 i=1 ±υi∂X(ρ) ∂ρi, a total of 1600 = 202×(22)LMIs can be obtained. C. Asymptotical Mean Square Stabilization By solving the conditions in (13) in Theorem (2), it can be obtained that R0=0.0088 −0.0109 −0.0109 0.0292  R1= 10−3−0.1248 −0.2344 −0.2344 −0.2473  R2= 10−4−0.3002 0.0195 0.0195 −0.4529  F0=0.0001 −0.0021  F1= 10−40.2155 −0.2893  F2= 10−50.4061 −0.4361  And the obtained controller is as follows: K0=−0.1576 −0.1315  K1=0.0603 −0.0598  K2=0.1287 0.1018  Applying the obtained controller to the inverter LPV model, and perturbing the system with noise of Brownian motion, the resulted state response is shown in Fig. 2. Fig. 2.(a) shows the state response trajectory under one random perturbation, which is asymptotically stable and converges to the equilibrium very fast. Fig. 2.(b) shows the state response trajectories under ten random perturbations, in which all the trajectories converge to the equilibrium. This shows that the system is asymptotically stable in the face of parameter variations and stochastic perturbations. D. Exponential Mean Square Stabilization Let λ= 20, by solving the conditions in (31) in Theorem (4), it can be obtained that, R0=0.0046 −0.0110 −0.0110 0.0374  R1= 10−30.2483 −0.4546 −0.4546 0.7436  R2= 10−4−0.6224 0.1375 0.1375 0.5201  F0=0−0.0029  F1= 10−40.3962 −0.6655  F2= 10−5−0.4328 0.5035  K0=−0.6200 −0.2583  K1=0.0361 −0.0674  K2=0.0859 0.0741  The obtained state response with the exponential controller is shown in Fig. 3. Fig. 3.(a) shows the state response trajectory under one random perturbation, and Fig. 3.(b) shows the state response trajectories under ten random perturbations, in which all the trajectories converge to the equilibrium fast. Therefore the exponential controller is effective under the parameter variations and stochastic perturbations. Moreover, comparing Fig. 2.(a) and Fig. 3.(a), it can be observed that the state displacement is smaller when using the exponential controller. This difference is more obvious when comparing Fig. 2.(b) and Fig. 3.(b). This result demonstrates that the exponential controller obtains better performance than the asymptotical controller. V. CONCLUSION To address the parameter variation and stochastic perturbation problems faced by the grid-connected inverter, a linear parameter varying system (LPV) method has been proposed. The inverter is modelled by LPV system with stochastic perturbation. 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