A Robust Active Damping Control Strategy for an LCL

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS

A Robust Active Damping Control Strategy for an LCL-based Grid-connected DG Unit Mahdieh S. Sadabadi, Aboutaleb Haddadi, Member, IEEE, Houshang Karimi, Senior Member, IEEE, and Alireza Karimi, Member, IEEE

Abstract—The connection of a DG unit to a weak power system is challenging due to stability issues resulted from dynamic interactions between the DG unit and the grid. LCL-based DG unit is a particularly challenging case due to the presence of a high resonant peak in its frequency response. This paper proposes a robust control strategy to overcome the stability issues of an LCL-based DG unit connected to a weak grid. The main advantage of the proposed control strategy is that it guarantees stability and satisfactory transient performance against the variations of grid impedance. Moreover, it is able to decouple the d- and q channels of the control system which enables independent regulation of the real and reactive output power of the DG unit. Real-time simulations and experimental tests illustrate the effectiveness of the proposed controller in terms of improved transient performance, robust stability, and satisfactory controller set-point tracking. Index Terms—Active damping control, grid-connected microgrids, grid impedance uncertainty, LCL filters, robust current control, voltage-sourced converter.

I. I NTRODUCTION

E

CONOMIC, technical, and environmental incentives are deriving electric power systems towards an era where a large number of distributed generation (DG) units will considerably contribute to the production of electrical energy. Such large-scale integration of DGs can adversely affect system stability, particularly in small or isolated power systems. The interconnection of a DG unit to such a weak system is challenging due to stability issues resulted from intermittent nature of renewable energy sources [1], [2]. Even in a relatively stiff grid, grid impedance may be subject to significant variations during operations such as faults, tripping of lines, or load variations. This paper addresses the control design challenges Manuscript received September, 2016; revised December 2016 and March 2017; accepted March 2017. This work was supported by the Swiss National Science Foundation under Grant No. 200020-130528, the Commission for Technology and Innovation (CH), and the Swiss Competence Center for Energy Research-Future Swiss Electrical Infrastructure. M. S. Sadabadi is with the Computational and Biological Learning (CBL) Group, Department of Engineering, University of Cambridge, Cambridge, United Kingdom (e-mail: [email protected]). A. Haddadi and H. Karimi are with Polytechnique Montreal, Montreal, QC, Canada (e-mails: [email protected] and [email protected]). A. Karimi is with the Automatic Control Laboratory at Ecole ´ erale ´ Polytechnique Fed de Lausanne (EPFL), Lausanne, Switzerland (Corresponding author, phone: +41 21 693 5925; e-mail: [email protected]).

of an electronically-interfaced DG unit connected to a weak grid through an LCL filter. The objective is to develop a robust control strategy to cope with the variations of grid impedance (grid weakness) and the stability problems associated with the LCL filter dynamic. A DG unit is commonly interfaced to the host grid through a voltage-sourced converter (VSC) and an L or LCL filter. The interface filter attenuates high-frequency harmonics injected by the VSC. The LCL filter is more advantageous due to its cost-effectiveness in terms of size and weight as well as the efficient attenuation of switching harmonics [3]. Nevertheless, the use of LCL filters results in stability issues as an LCL filter introduces high resonant peak in the frequency response of the DG unit. To address this challenge, a number of passive (e.g. [4], [5]) and active damping approaches have been proposed in the literature (e.g. [3], [6]–[17]). Passive damping methods often use a resistance in series with the capacitor of the filter. Although this strategy is simple and reliable, the damping resistance adversely affects the highfrequency harmonic attenuation property of the LCL filter and increases power losses. Active damping methods, on the other hand, reshape the current control system of the DG unit to guarantee robust stability. One of the most widely used approaches is vector current control which is based on the control of two independent d- axis and q-axis current components in rotating reference frame while the synchronization is done via a phase-locked loop (PLL) [18]. The vector current control approaches for L-type VSCs usually utilize conventional PI controllers [19] or modified PIs [20], [21] in order to compensate for the only dominant pole of each axis by the zero of the PI controllers. However, the PI controllers are not capable of compensating for all poles of LCL-filter-based VSCs; therefore, damping strategies are required to attenuate the effects of the uncompensated poles of the system. A common strategy is based on use of additional feedbacks or state feedback controllers, e.g. [6], [9], [11], [15]–[17]. Nevertheless, the multi-loop and state feedback controllers require more sensors leading to increased overall cost and reduced system reliability. To solve such a problem, dynamic output feedback controllers (filter-based methods) are usually employed for the purpose of active damping control of VSCs with LCL-type filters [8], [12]– [14]. A shortcoming of the existing output feedback control strategies, such as those proposed by [8], [12]–[14], is that they do not guarantee stability against the variations of grid


V dc

P,Q

LCL filter

vt,abc

ic,abc

rc

rg

Lc

Vs Lg1

C Gating pulses

Lg 2 PCC

6

Gating signal generator

uabc

Utility grid

ig,abc

vc,abc

VSC

vg,abc

Control circuit

dq/abc

udq

dq current controller

ig,dq

abc/dq

ig,abc PLL

θ (t) θ (t)

yref

vg,abc

Fig. 1. Schematic diagram of an LCL-based grid-connected VSC.

impedance; thus, a change in the grid impedance may reduce the stability/performance of the designed controller. Another shortcoming of the existing active damping methods is that they do not provide decoupled control of real and reactive power, i.e., a change in the set-point of the output real power leads to a change in the output reactive power and vice versa. This paper proposes an active damping method for an LCLbased DG unit which guarantees robust stability and robust performance against a (pre-specified) wide range of variation of grid impedance. The proposed controller is a robust MIMO low-/fixed-order dynamic output feedback controller which is designed to meet the performance specifications such as fast transient response, small overshoot, high closed-loop bandwidth, and low interactions between the control channels. The control problem is formulated as an H∞ control design problem and is formulated as a non-convex optimization problem. The non-convexity of the set of low-/fixed-order controllers makes the design problem theoretically challenging [22]. In this paper, we propose an inner convex approximation of the problem where the non-convexity is obviated by introducing several slack matrices determined by a linear matrix inequality (LMI)-based algorithm. The performance of the proposed active damping controller is tested using real-time simulations carried out in OPAL-RT simulator. To show the feasibility of hardware implementation and validate the performance against real-life implementation issues, the proposed controller has been implemented in an experimental setup including an LCLbased grid-connected DG unit. The real-time and experimental test results illustrate the effectiveness of the proposed current control approach in terms of dq reference current signal tracking and robust stability against grid impedance uncertainties. The proposed control strategy is also applicable to L-filterbased VSCs. However, the results are not included in this paper as the main focus is on the more challenging case of LCL-type filter. Throughout the paper, matrices I and 0 are the identity matrix and the zero matrix of appropriate dimensions, respectively. The symbols T and ? indicate the matrix transpose and symmetric blocks, respectively. For symmetric matrices, P > 0 (P < 0) shows the positive-definiteness (negative-definiteness). II. S YSTEM U NDER S TUDY The system under study in this paper is a grid-connected LCL-based electronically-interfaced DG unit, Fig. 1. As

TABLE I PARAMETERS OF THE TEST SYSTEM OF F IG . 1. DG unit parameters Lc = 1.5mH VSC-side inductance of LCL filter rc = 0.1Ω Internal resistance of Lc Lg1 = 1mH Grid-side inductance of LCL filter rg = 0.1Ω Series resistance of Lg1 C = 15µF LCL filter capacitance Vdc = 440V DC bus voltage fs = 10020Hz Controller sampling frequency fsw = 5010Hz PWM switching frequency Sbase = 16kVA VSC rated power 5.27 ≤ SCR ≤ 7.75 Short circuit ratio Grid parameters Lg2 ∈ [0 0.5mH] Grid inductance Vs = 220V Grid voltage (line-to-line rms) f0 = 60Hz System nominal frequency

shown, the power circuit of the DG unit consists of a conditioned prime energy source which is modeled by a dc voltage source, a VSC, and a three-phase LCL filter. The per-phase inductances and capacitance of the filter are denoted by Lc , Lg1 , and C, respectively. The resistances rc models the ohmic power loss of Lc and the VSC power losses, and rg models the ohmic loss of Lg1 . The DG is connected to the grid at the point of common coupling (PCC) whose voltage is represented by vg,abc . The grid is modeled by a voltage source Vs in series connection with an inductance represented by Lg2 . The DG unit employs a current control system to regulate real/reactive powers. Fig. 1 indicates that the current control task is performed in a rotating dq frame. The dq frame is defined such that its d-axis makes an angle θ (t) with respect to the horizontal stationary axis. The variable θ (t) is obtained by means of a unified three-phase signal processor (UTSP) which is an enhanced PLL system [23]. The input of UTSP is the voltage signal at the PCC. The mathematical model and structure of UTSP are given in Equation (9) of [23] and due to space limitations are not presented here. The parameters of the UTSP system are µ1 = 67, µ2 = 67, µ3 = 67, µ4 = 10000, µ5 = 130, µ6 = 433, and µ7 = 1333. The dynamics of the PLL are neglected because the grid under study is not assumed to be very weak, i.e. short-circuit capacity ratio is one. The current control loop regulates the DG terminal current ig,abc by adjusting the VSC terminal voltage vt,abc . The instantaneous values of DG terminal current ig,abc are first transformed into dq frame signals ig,dq which is then supplied to the dq current controller. The controller compares the measured currents with their corresponding set-points represented by yre f (provided by an external control mechanism referred to as secondary controller) and generates the control command signal u. This control command is then applied to the VSC through a gating signal generator block. The VSC employs a Pulse Width Modulation (PWM) scheme to generate the desired voltage vt,abc at its terminals. Table I presents the parameters of the system under study. As mentioned in Section I, the grid impedance Lg2 is subject to variation which may negatively affect the transient performance and stability of the overall system. This can be demonstrated by investigating the resonance frequency of the LCL filter of the DG unit of Fig. 1. One can show that the


resonance frequency of the filter is given by s Lg + Lc 1 fres = 2π Lc LgC

(1)

where Lg = Lg1 +Lg2 . Equation (1) indicates that the resonance frequency depends on the grid inductance, i.e., the higher the grid inductance, the lower is the resonance frequency of the filter. The frequency response of the LCL filter exhibits a peak at the resonance frequency. In a stiff grid, Lg2 is small and, provided a properly designed filter, the resonant frequency of the filter is sufficiently larger than the pass-band of the current controller. However, in a weak grid where Lg2 is relatively large, the resonance frequency decreases and the resonance peak may enter the pass-band of the current controller which in turn results in stability issues. The short circuit ratio (SCR), defined as the ratio of the short-circuit capacity of the hosting grid at the PCC to the rated power of the VSC, is usually used to characterize the grid stiffness/weakness. It is mathematically defined as follows [1], [24]: Vs2

SCR = q /Sbase ω02 Lg2 + rg2

(2)

where ω0 = 2π f0 , Vs is the nominal RMS line-to-line voltage of the grid and Sbase is the rated power of the VSC. The grid is considered as weak when the SCR is less than 3. The objective of this paper is to design a current controller so that a satisfactory transient performance and stability is guaranteed under a pre-specified range of variations of the parameter Lg2 . In this paper, it is assumed that Lg2 varies in the interval [Lg2min Lg2max ] given in Table I. To design such a controller, the first step is to develop a mathematical model for the DG system of Fig. 1.

x˙g (t) = Ag xg (t) + Bg vt (t) + Bv vs (t)

(3)

y(t) = Cg xg (t)

where xg (t) = [ic,d ic,q vc,d vc,q ig,d ig,q ]T is the state vector, vt (t) = [vt,d vt,q ]T is the input, vs (t) = [Vsd Vsq ]T is the disturbance signal, and y(t) = [ig,d ig,q ]T is the output vector. Further, the state-space matrices are as follows: − Lrcc

ω0

− L1c

0

0

0



 −ω  0  1   C Ag =   0   0  0 " 1

− Lrcc

0

− L1c

0

0

0

0

ω0

− C1

0 − C1

         

Bg = " Bv = Cg =

1 C

−ω0

0

0

0

1 Lg

0

− Lgg

ω0

0

0

1 Lg

−ω0

− Lgg

0

0

0

0

0

0

1 Lc

0

0

0

0

0

0

0

− L1g

0

0

0

0

0

0

0 0

0 0

0 0

0 0

− L1g

1 0

0 1

r

#T

0

It is assumed that the grid inductance value is not precisely known but belongs to a given interval, i.e., Lg2 ∈ [Lg2min Lg2max ]. Therefore, the state space matrices Ag and Bv have parameter uncertainty. In order to mathematically describe the uncertainty, we adopt polytopic uncertainty representation modeled via a convex hull of two given vertices as follows: Ag (λ ) = λ Ag1 + (1 − λ )Ag2

(5)

Bv (λ ) = λ Bv1 + (1 − λ )Bv2

where 0 ≤ λ ≤ 1. Vetrices Ag1 , Ag2 , Bv1 , and Bv2 are obtained by substituting the maximum and minimum values of the grid inductance Lg2 . The state space model given in (3) is transformed to discrete-time using the zero-order hold (ZOH) method [25] with the sampling time Ts = f1s as follows: xg (k + 1) = Agd (λ )xg (k) + Bgd vt (k) + Bvd (λ )vs (k) y(k) = Cgd xg (k)

(6)

where Agd = eAg Ts , Bgd = 0Ts eAg τ Bg dτ, Bvd = 0Ts eAg τ Bv dτ, and Cgd = Cg . It is assumed that there exists one sample delay between the converter voltage command u(k) and the VSC terminal voltage vt (k), i.e. vt (k) = u(k − 1) [11]. Therefore, by considering the delay, the following augmented model G is derived. R

R

y(k) = Cgaug xgaug (k)

Under balanced conditions, the dynamics of the DG unit of Fig. 1 are described by the following state-space equations:

Lc

B. Grid Impedance Uncertainty

xgaug (k + 1) = Agaug (λ )xgaug (k) + Bgaug u(k) + Bvaug (λ )vs (k)

A. Mathematical Model of the DG Unit



It should be mentioned that ic,dq , ig,dq , vc,dq , vt,dq , and Vs are the dq-components of the converter-side current, the grid-side current, the capacitance voltage, the VSC terminal voltage, and the grid voltage, respectively.

r

(7) where xgaug (k) = [xgT (k) vtT (k)]T and " # " # Agd (λ ) Bgd (λ ) 0 Agaug (λ ) = ; Bgaug = 0 0 I T Bvaug (λ ) = Bvd (λ ) 0 ; Cgaug = Cgd 0

(8)

Since the sampling time Ts is small enough, it can be assumed that the discrete-model of (7) and (8) represents a polytope with q = 2 vertices. III. C ONTROLLER D ESIGN M ETHOD This section proposes a robust fixed-structure control strategy for the current controller of the grid-connected DG system of Fig. 1.

(4)

A. Controller Design Requirements #T

It is desired that a current controller for the grid-connected DG unit described by (7) and (8) with the grid inductance uncertainty meet the following performance criteria: • The closed-loop system is asymptotically stable for all values of Lg2 in the given interval.


z1

A common choice for W [26] and Td in continuous-time case is given as follows:

W(z)



+ωB∗ s+ωB∗ ε

s Mw

W (s) = 

yref +

e

K(z)

u

G(z)

_



_

Td (s) = 

yr Td(z)

•

• • •

The closed-loop polytopic system should be able to track all step current reference signals with zero steady state error. The closed-loop response should have small rise time and overshoot for all values of the grid inductance within the pre-specified uncertainty interval. The closed-loop system should eliminate the impact of the disturbance signal vs . High control bandwidth is required to reject low frequency harmonics generated by the grid voltage. The coupling between the d and q output channels should be minimized.

To satisfy the aforementioned criteria, a current controller K with the following structure is proposed: u(k) = Ck xk (k) + Dk (yre f (k) − y(k))

(9)

where Ak ∈ Rm×m and Bk , Ck , and Dk are of appropriate dimensions. The controller in (9) is a solution of the following optimization problem: min K(z)

α1 µ1 + α2 µ2

subject to kW S(λ )k2∞ < µ1

0

0 ωB∗ s+ωB∗



(12)

 

(13)

The weighting filter W and the model reference Td are discretized using the ZOH method. We assume that the statespace equations of these transfer functions are given by: xw (k + 1) = Aw xw (k) + Bw (yre f (k) − y(k)) z1 (k) = Cw xw (k) + Dw (yre f (k) − y(k)) xd (k + 1) = Ad xd (k) + Bd yre f (k) yr (k) = Cd xd (k) + Dd yre f (k)

(14) (15)

To obtain the state-space representation of W S, first the dynamic equations of the plant in (8) and the weighting filter W are augmented as follows: xˆg (k + 1) = Aˆ g (λ )xˆg (k) + Bˆ g (λ )u(k) + Bˆ v (λ )vd (k) + Bˆ w w(k) y(k) ˆ = Cˆg xˆg (k) + Dˆ yw w(k)

B. Structure of the Proposed Current Controller

xk (k + 1) = Ak xk (k) + Bk (yre f (k) − y(k))

ωB∗ s+ωB∗

+ωB∗ s+ωB∗ ε

s Mw

where ωB∗ is the desired closed-loop bandwidth, ε is the maximum tracking steady-state error, and Mw is the maximum peak value of S typically set to 2.

Fig. 2. Closed-loop block diagram with weighting functions.

•

0

z2

y +



0

(10)

z1 (k) = Cˆz xˆg (k) + Dˆ zw w(k) (16) T where xˆg (k) = xgTaug (k) xwT (k) , w(k) = yre f (k), y(k) ˆ = yre f (k) − y(k), and " # " # Agaug (λ ) 0 Bgaug Aˆ g (λ ) = ; Bˆ g = −BwCgaug Aw 0 T T (17) Bˆ v (λ ) = Bvaug (λ ) 0 ; Bˆ w = 0 Bw ˆ ˆ Cg = −Cgaug 0 ; Dyw = I Cw ; Dˆ zw = Dw Cˆz = −DwCgaug

kT (λ ) − Td k2∞ < µ2 where S = (I + GK)−1 , T = GK(I + GK)−1 , W , and Td are the sensitivity function, the complementary sensitivity function, the weighting filter, and the desired closed-loop model, respectively. The positive scalars α1 and α2 characterize the emphasis on the H∞ norm of the weighted sensitivity transfer function and the model matching problem kT (λ ) − Td k∞ . The weighting filter W shapes the sensitivity function S. Essentially, the minimization of kW S(λ )k∞ provides the desirable performance characteristics of the closed-loop system while the minimization of kT (λ ) − Td k∞ decouples the two output channels. Fig. 2 depicts the structure of the closed-loop system including performance weights. The signals z1 and z2 are defined as follows: z1 (k) = W (yre f (k) − y(k)) z2 (k) = y(k) − Td yre f (k)

Then, by augmenting the dynamic equations of the augmented plant in (16)-(17) and the controller in (9), the statespace representation of the system W S is obtained as follows: x1 (k + 1) = A1 (λ )x1 (k) + B1 yre f (k) z1 (k) = C1 x1 (k) + D1 yre f (k) where x1 (k) = [xˆg (k) xk (k)]T and " # Aˆ g (λ ) + Bˆ g DkCˆg Bˆ gCk A1 (λ ) = BkCˆg Ak " # Bˆ g Dk Dˆ yw + Bˆ w B1 = Bk Dˆ yw C1 = Cˆg 0 ; D1 = Dˆ zw

(18)

(19)

(11) In a similar fashion, the dynamic equations of the system


T − Td are given by: x2 (k + 1) = A2 (λ )x2 (k) + B2 yre f (k) z2 (k) = C2 x2 (k) + D2 yre f (k)

(20)

where x2T (k) = [xgTaug (k) xkT (k) xdT (k)], z2 (k) = y(k) − yr (k), and   Agaug (λ ) − Bgaug DkCgaug Bgaug Ck 0   −BkCgaug Ak 0  A2 (λ ) =  0

0

B2 =

Bgaug Dk Bk Bd

C2 =

Cgaug

0

Ad

T

−Cd

;

D2 = −Dd (21)

It has been assumed that specific structural constraints on the controller matrices can be imposed. These constraints can be in the form of fixed-order dynamic output feedback where the order of the controller is independent of the plant order and it is fixed a priori and fixed-structure matrix Ak which must have two poles at z = 1. The constraints of Ak come from that reason that the controller K(z) must contain integrators to track step references and reject the disturbance signal vs .

Lemma 1. The following set of inequalities is equivalent to (22) [27]:   PT ji − A ji T PT ji A ji ? ? ?  P A +M −X A 2X j − PT ji ? ?  Tj j ji  T ji ji   >0 T T T  B ji MT j − B ji X j A ji B ji X j I ?  0

C ji

D ji

µ jI

(23) for i = 1, 2 and j = 1, 2, where PT ji = T −T Pji T −1 , MT j = T j−T M j T j−1 , and X j = T j−T T j−1 . It should be noted that the choice of slack matrices M j and T j , j = 1, 2 affects the conservativeness of the proposed controller design method. Therefore, these matrices should be determined in an appropriate way. In the following, a heuristic approach for choosing the slack matrices is given. A solution for choosing the slack matrices M j and T j for j = 1, 2 is to design a fixed-structure H∞ controller for one vertex of the polytope, e.g. vertex l, using available approaches, e.g. [28]–[31]. Then, a new polytope with following vertices is built in the iteration h of an iterative algorithm as explained in the next subsection. [h]

Agi = α [h] Agi + (1 − α [h] )Agl [h]

C. Controller Design Method

Bvi = α [h] Bvi + (1 − α [h] )Bvl

The aforementioned control design requirements described in (7)-(8) can be satisfied by a set of LMI-based conditions as presented in the following.

[h] Bgi

Theorem 1. For given slack matrices M1 , M2 and nonsingular matrices T1 , T2 , if there exist symmetric positive-definite matrices P11 , P12 , P21 , and P22 such that:   Pji − M Tj Pji M j ? ? ?  P M − M + T −1 A T 2I − Pji ? ?  j ji j  ji j  j  >0 −1 T  0 (T j B ji ) I ?  C ji T j

0

D ji

µ jI

(22) for i = 1, 2 and j = 1, 2, then the controller in (9) with a fixed structure guarantees:

= Bg ,

[h] Cgi

(24)

= Cg

for i = 1, 2, where α [h] (0 ≤ α [h−1] ≤ α [h] ≤ 1) is a scaling factor for the original polytope. Note that α [h] = 0 and α [h] = 1 mean that the l th vertex and the original polytope are covered, respectively. Moreover, it can be easily shown that the new polytopic system in the iteration h encompasses the old one (for α [h−1] ). The objective is to create a polytopic system with maximum scaling factor 0 ≤ α ≤ 1 in which the stability as well as the performance criteria are satisfied. In the next subsection, a procedure to design a fixed-structure H∞ controller with the maximum polytopic uncertainty domain is presented. D. Controller Design Procedure

1) closed-loop robust stability for all values of the grid inductance in the given interval 2) kT (λ ) − Td k2∞ < µ2 3) kW S(λ )k2∞ < µ1 4) zero steady state error for tracking of the step grid current references and grid voltage disturbance rejection

This section presents an algorithm for the current controller design. To facilitate the presentation of the algorithm, the inequalities in (22) and (23) are respectively defined as follows:

Proof. According to [27], LMI conditions given in (22) for j = 2 ensures the robust stability of the closed-loop system with linearly parameter-dependent Lyapunov matrix P2 (λ ) = λ P21 + (1 − λ )P22 where 0 ≤ λ ≤ 1. Moreover, they satisfy kT (λ ) − Td k2∞ < µ2 which ensures decoupling between the d and q components of the output signal. The set of LMI conditions (22) for j = 1 guarantees that desired robust closed-loop performance, i.e. kW S(λ )k2∞ < µ1 with linearly parameterdependent Lyapunov matrix P1 (λ ) = λ P11 + (1 − λ )P12 where 0 ≤ λ ≤ 1 [27]. Property 4) is also ensured due to the two poles of matrix Ak at z = 1.

for i = 1, 2 and j = 1, 2. The sign | in the arguments of F ji and H ji separates the decision variables and the known parameters in the LMIs. The controller design procedure contains the following steps: 1) Select the values α1 , α2 , the weighting filter W , and the desired transfer function Td . Then determine their state-space representations in (12) and (13). 2) Obtain the augmented plant of (16) and (17). 3) Choose the order of controller. 4) Use the following steps to design a fixed-structure H∞ controller.

F ji (Pji , K, µ j | M j , T j ) < 0

(25)

H ji (PT ji , MT j , X j , µ j

(26)

| K) < 0


From: In(1)

40

We also set α1 = 0.2 and α2 = 1. The objective is to design a robust dq-based current controller for a VSC with an LCL filter described by (7). It has been assumed that the grid inductance Lg2 belongs to [0, 0.5mH]. Following the control design procedure in Subsection III-D, the final H∞ controller is obtained after 5 iterations as follows:

From: In(2)

To: Out(1)

20 0 -20

Magnitude (dB)

-40

 -60 40

       Ak =       

To: Out(2)

20 0 -20 -40 -60

10 2

10 3

10 4 10 2 Frequency (rad/s)

10 3

10 4



Fig. 3. Frequency response of proposed active damping controller.

Step 1 (Initialization): Put the iteration number h = 1. Design a fixed-structure H∞ controller for the l th vertex of the polytopic set. [h] [h] Step 2: Matrices MT j and X j can be considered as a feasible solution of the the following convex optimization problem: α [h] = max α [h]

[h]

s.t. [h]

H ji (PT j , MT j , X j , µ j , α | K [h−1] ) < 0; i

i = 1, 2, j = 1, 2

(27)

where α represents the uncertainty bounding set. Compute the [h] [h] instrumental matrices M j and T j as follows: [h] T

[h]

M j = Tj

[h] [h]

MT j T j

[h]

[h]

T j = (chol(X j ))−1 ;

(28) j = 1, 2

where chol denotes Cholesky factorization. Step 3: Solve the following set of LMIs to obtain a fixedstructure H∞ controller K [h] for the current polytope α [h] : [h]

[h]

[h]

F ji (Pji , K [h] , µ j | M j , T j , α [h] ) < 0;

i = 1, 2, j = 1, 2 (29)

Step 4: If either ∆α = α [h+1] − α [h] < ε or α [h] = 1, go to Step 5, else use the obtained controller in Step 3 as an initial controller and return to Step 2 with h ← h + 1. Step 5: The obtained controller in Step 4 can be employed as an initial controller. IV. R OBUST C URRENT C ONTROLLER In order to design a fixed-structure controller for the voltagesourced converter of Fig. 1, the weighting filter W and desired closed-loop transfer function Td are chosen as follows:  s  3 +1000 0  W (s) =  s+0.01 (30) s 3 +1000 0 s+0.01 " # 1000 0 s+1000 Td (s) = (31) 1000 0 s+1000

1 0

0 1

−0.058 −0.028

0 0 0 0 0 0

0 0 0 0 0 0

0.008 0.024 0.036 0.003 0.009 0.011

3.045 0.003 −0.5 −1.002 −2.006 −0.0011

       Bk =        −0.0014 −0.0017 3.1729 Dk = 0.1488

0.046 0.046

−0.020 −0.023

−0.005 −0.150

−0.011 0.173

0.493 0.003 0.002 0.0004 −0.018 0.507 0.0003 0.004 −0.029 0.012 0.003 0.007 −0.005 0.002 0.014 0.482 −0.013 0.005 0.035 −0.038 −0.02 0.01 0.062 −0.073   T −0.001 0.005 0.025   4.614  0.004   −0.016      2.204 0.0002  −0.218       0.001  0.352   −5.347   ; Ck =    4.254 0.0004  −0.119       −0.5  2.163   0.197      −0.337 −5.274  −1.0005  −2.002 0.124 4.212 −0.1467 3.1662

 0.016 −0.088    −0.002   −0.005    −0.011   0.009    0.519  0.037

(32)

The resulting controller ensures the robust stability as well as the robust performance criteria kW S(λ )k∞ < 1.4532 and kT (λ ) − Td k∞ < 1.5068 for the whole uncertainty range of Lg2 . Frequency response of the proposed active damping current controller is plotted in Fig. 3. Fig. 4 also depicts the frequency response of open-loop and closed-loop system with controller given in (32) for Lg = 1mH. The results confirm that the proposed active damping controller is able to damp the resonance frequency of the open-loop system. Moreover, the frequency response of the off-diagonal elements of the closed-loop system has a very low gain confirming the axisdecoupling of the proposed control strategy. Remarks: To design the H∞ controller given in (32), at the first iteration, an initial controller with integrators is designed using Frequency-Domain Robust Controller (FDRC) Toolbox [31] for the first vertex of the polytopic corresponding to Lg2 = 0. The LMI-based optimization problem is solved using YALMIP [32] and SDPT3 [33] as the interface and the solver, respectively. V. C OMPARISON WITH A M ULTIVARIABLE -PI V ECTOR C URRENT C ONTROLLER The main purpose of this section is to illustrate incapability of the multivariable PI-based controller in damping the closedloop system around the resonance frequency as well as the superiority of the proposed active damping current controller given in (32). The structure of the multivariable-PI vector current controller is as follows [20]:


From: In(1)

50

Closed-loop system Open-loop system

From: In(2)

From: In(1)

50

From: In(2)

0 To: Out(1)

To: Out(1)

0 -50

-50 Magnitude (dB)

Magnitude (dB)

-100 -150 50

To: Out(2)

To: Out(2)

0

-100 50

-50

0

-50 -100 -150

10 2

10 3


10 3

-100

10 4

Fig. 4. Frequency response of open-loop (red) and closed-loop system (black) for Lg = 1mH.

10 2

10 3


10 3

10 4

Fig. 5. Frequency response of open-loop (red) and closed-loop system with multivariable PI-based vector current controller (black) for Lg = 1mH.

where Ti = 0.05 and Tn = 0.015. To illustrate the performance of both controllers in terms of robust stability, robust performance, and reference tracking, a number of case studies are presented. To that end, the test system of Fig. 1 has been implemented in the OPALRT real-time environment. The real-time target computer is equipped with 12×2.7GHz cores and running on the Red Hat Enterprise Linux real-time operating system. The test cases study the transient response of the DG unit to a step change in the set-points of d- and q-axis currents, assuming an ideal grid with Lg2 = 0 mH. Frequency response of the open-loop and closed-loop system with the multivariable PI-based vector current controller is plotted in Fig. 5. The results show that the multivariable PI-based current controller cannot damp the resonance frequency of the open-loop system, and it is not able to decouple dq current axes. Fig. 6 shows the results of this case study for both proposed controller and the multivariable-PI vector current controller. As shown, the system starts from a steady-state where the DG unit is connected to the grid for a sufficiently long time and the set-points of the d- and q-axis currents are zero. At t = 5 s and t = 6 s, the set-points of the d- and q-axis components of the current are increased to 50 A and 25 A, respectively. Both controllers track the set-points; the transient response under the proposed control is well damped with negligible overshoot while the responses of the d- and qaxis current under the multivariable-PI-based controller exhibit resonances. As demonstrated, the proposed active damping controller provides a superior transient response as compared to the multivariable-PI controller in an ideal grid. To further demonstrate the features of the proposed con-

50 25 0

(b)

idq (A)

1 + sTn ω0 Tn (ig,d − ig,dre f ) − (ig,q − ig,qre f ) +Vsd sTi Ti (33) 1 + sTn ω0 Tn vt,q = (ig,q − ig,qre f ) + (ig,d − ig,dre f ) +Vsq sTi Ti

vt,d =

idq (A)

(a)

idmulti iqmulti

50

idprop iqprop

25 0 5

5.5 time (s)

6

6.5

Fig. 6. Real-time simulation of the transient response of the test system of Fig. 1 with Lg2 = 0mH under (a) multivariable-PI vector current control proposed robust controller and (b) proposed active damping controller.

troller, the grid inductance is increased to Lg2 = 3mH, equivalent to SCR = 2. Fig. 7 shows the result of this case study. As shown, the proposed controller provides the same superior transient performance under the increased grid impedance which illustrates its robustness against the variation of grid inductance. However, the transient response under the multivariable-PI current controller exhibits resonances and ripples. The switching frequency ripples are visibly smaller in Fig. 7 as compared to those of Fig. 6 due to the larger damping provided by the increased overall inductance. It should be noted that in AC weak systems, perfect tracking of current reference signals does not always imply that the desired power is dispatched as the voltage signals are also affected [34]. Fig. 8 depicts the frequency response of the off-diagonal elements of the closed-loop system with both proposed and multivariable PI current controllers for SCR = 2. Although both controllers are not able to damp the larger resonant frequency, the proposed active damping controller has superiority in damping the cross coupling effects.


50

(a)

25

(b) 50

idprop iqprop

0 To: Out(1)

0

idq (A)

From: In(1) system with proposed active damping From: In(2) control Closed-loop current Open-loop system

50 Magnitude (dB)

idq (A)

(a)

idmulti iqmulti

-50 -100 -150

25

10 2

10 3

0 6.5

10 4

vg,d vg,q

100

0 5.5 time (s)

6

6.5

Fig. 7. Real-time simulation of current signals of the test system of Fig. 1 with Lg2 = 3mH under (a) multivariable-PI vector current control proposed robust controller, (b) proposed active damping controller, and (c) voltage signals under proposed active current controller.

VI. E XPERIMENTAL VALIDATION To illustrate the feasibility of hardware implementation of the proposed controller and validate its performance against real-life implementation issues, the test system of Fig. 1 has been implemented in an HIL experimental setup shown in Fig. 9. The hardware subsystem consists of: R a 20-kVA two-level Semikron SKHI61 inverter which is a Sixpack IGBT and MOSFET driver; • a PWM signal generator device which converts the reference signals of vt,abc into gating signals; • an LCL filter consisting of two 25-A inductors representing Lc and Lg1 and two series 480-V, 3.3-kVar capacitors representing C; • a 25-A inductor representing grid inductance Lg2 ; • a DC voltage source which produces Vdc on the DC-side of the DG unit; it has been implemented using a fullbridge diode rectifier fed from a three-phase, 24.2-kVA, 50-A autotransformer with a variable output voltage of 0 to 280 V, and a 450-V DC capacitor on the DC-side of the diode rectifier; • and two 1:1 isolating transformers interfacing the PCC and the AC side of the diode rectifier to the grid to electrically isolate the DG unit from the grid for safety reasons. The controller has been implemented on an OP5600 OpalR RT real-time digital simulator. The controller subsystem of the experimental setup is basically a simulation model consisting of a state-space system implementing the proposed control, a PLL, and abc-to-dq and dq-to-abc transformation blocks. Thus, except for the proposed controller, the PLL, and transformation blocks which have been implemented on the OP5600 simulator, the rest of the experimental setup consists exclusively of hardware. The simulator and the hardware interact through an OP8660 HIL Controller and Data Acquisition Interface, which measures ig,abc and vg,abc signals and supplies

From: In(2) Closed-loop system with multivariable PI current control Open-loop system

0 -50 -100

5

From: In(1)

50

To: Out(1)

6

Magnitude (dB)

5.5 time (s) (c)

200

vg,dq (V )

10 3

(b)

5

•


10 2

10 3


10 3

10 4

Fig. 8. Frequency response of off-diagonal elements of open-loop (red) and closed-loop system (black) with SCR=2 under (a) proposed active damping current controller and (b) multivariable PI-based vector current controller.

them to the OP5600 simulator, and three output channels of the OP5600 simulator which supply the reference signals of vt,abc to the PWM signal generator device. The parameters of the control and power circuits of the experimental setup are similar to those presented in Table I, except that the DC-side voltage has been reduced to 220 V, and the AC-side voltage has been reduced to 65V line-to-line rms for safety reasons. The results of the experimental tests are recorded at the OP8660 HIL Controller and Data Acquisition Interface. Two tests have been carried out. The first test evaluates the reference tracking performance of the proposed controller for stiff grid, Lg2 = 0. Fig. 10 shows the results of this test. The system is operating initially in a steady-state and ig,dre f and ig,qre f are both zero. At t = 2s, ig,dre f is stepped up from 0 A to 20 A by manually changing ig,dre f in the simulation model on OP5600. It is observed that the controller tracks the reference setpoint with a rise time of 0.02s. The control action results in an overshoot of about 7.5% in the q-axis current. Another step change is applied at t = 8.4s whereby ig,qre f is stepped up from 0 A to 10 A. It is observed that the controller tracks the step change after nearly 0.01 s, and the overshoot in the d-axis current is small. Fig. 10(b), (c), (d), and (e) respectively show the dq-components of the PCC voltage, the output real and reactive power of the DG unit, the three-phase grid current for the step change in ig,dre f , and the three-phase grid current for the step change in ig,qre f . The results confirm the effectiveness of the reference current tracking performance of the proposed controller. Another test has been carried out to evaluate the robustness of the proposed controller against grid impedance uncertainty. To that end, using a bypass breaker, Lg2 is stepped up from 0mH to 5mH and back to 0mH while the rest of the power and control circuit remain unchanged. It should be mentioned that the inductance of Lg2 = 5mH is larger than the uncertainty limit of 0.5 mH for which the proposed controller has been designed.


(a)

ig,dq (A)

20

ig,d ig,q

L change

10

0 (b)

vg,dq (V )

80

vg,d vg,q

40

0

Fig. 9. A photo of the HIL test setup. P (kW ), Q(kV ar)

(c)

(a)

ig,dq (A)

20 10

ig,d ig,dref ig,q ig,qref

3 2

P Q

1 0 8

0

10 Lg change

vg,dq (V )

vg,d vg,q

40

P (kW ), Q(kV ar)

Lg change

0 −20

2

P Q

0

0

10.04 time (s)

10.08

15.35

15.4 time (s)

Fig. 11. Experimental test of the robustness of the proposed controller against a step change in Lg2 : (a) dq-components of the grid current ig,dq , (b) dq-components of the PCC voltage vg,dq , (c) output real and reactive power of the DG unit, (d) three-phase grid current ig,abc for the step change in Lg2 from 0 mH to 5 mH, and (e) and three-phase grid current ig,abc for the step change in Lg2 from 5 mH to 0 mH.

(c)

1

18

−20 10

0

16 (e)

20

ig,abc (A)

ig,abc (A)

(b)

−1 −2

2

4

6 time (s)

(d) step change

8

10

12

(e) step change

20

ig,abc (A)

20

ig,abc (A)

14 time (s)

20

80

3

12

(d)

0 −20

0 −20

2

2.04 time (s)

2.08

8.4

8.44 time (s)

8.48

Fig. 10. Experimental test of the current tracking performance of the proposed controller with Lg2 = 0: (a) dq-components of the grid current ig,dq , (b) dq-components of the PCC voltage vg,dq , (c) output real and reactive power of the DG unit, (d) three-phase grid current ig,abc for the step change in ig,dre f , and (e) three-phase grid current ig,abc for the step change in ig,qre f .

Nevertheless, this value has been introduced intentionally to examine the performance of the proposed controller under to a rather severe disturbance. Fig. 11 shows the results of this test, where Lg2 is initially zero (bypassed) and at t = 10s steps up to 5mH, and again at t = 13.35s steps down to zero by the bypass breaker. As shown, except for some minor transients, the variation of Lg2 does not compromise the tracking performance of the proposed controller. The results of this test validate the robust stability/performance of the proposed active current controller.

VII. C ONCLUSION In this paper, a dq current vector controller for gridconnected DG units with LCL-type filters under polytopic uncertainties is proposed. The uncertainty is imposed by the grid inductance which belongs to a given interval. The controller assigned with integrators results from an optimal solution of a convex optimization problem subject to some LMI conditions. The proposed controller guarantees the robust stability and robust performance of the system against the grid inductance uncertainties. Moreover, the proposed controller is able to decouple the d and q components of the current signals. To verify the performance of the proposed controller, several real-time simulations and experiments are conducted. Realtime simulations as well as experimental tests confirm that the controller is robust to the grid inductance uncertainty and is able to track reference current signals with fast transient response and small overshoot. ACKNOWLEDGMENT The authors would like to acknowledge Prof. Jean Mahseredjian from Polytechnique Montreal and Dr. LucAndré Grégoire from OPAL-RT Technologies Inc., Montreal, Canada, for their kind support in the preparation of the experimental setup.


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[23] H. Karimi, A. Yazdani, and R. Iravani, “Negative-sequence current injection for fast islanding detection of a distributed resource unit,” IEEE Trans. on Power Electronics, vol. 23, no. 1, pp. 298–307, January 2008. [24] K. R. Padiyar, HVDC Power Transmission Systems: Technology and System Interactions. John Wiley & Sons Inc, Aug. 1991. [25] K. Ogata, Discrete-time control systems. Prentice-Hall, 1995. [26] S. Skogestad and I. Postlethwaite, Multivariable Feedback ControlAnalysis and design. Wiley, 2005. [27] M. S. Sadabadi and A. Karimi, “An LMI formulation of fixed-order H∞ and H2 controller design for discrete-time systems with polytopic uncertainty,” in 52nd IEEE Conference on Decision and Control, Florence, Italy, 2013, pp. 2453–2458. [28] P. Apkarian and D. Noll, “Nonsmooth H∞ synthesis,” IEEE Trans. on Automatic Control, vol. 51, no. 1, pp. 71–86, 2006. [29] S. Gumussoy and M. L. Overton, “Fixed-order H∞ controller design via HIFOO, a specialized nonsmooth optimization package,” in IEEE American Control Conference, Seattle, USA, 2008, pp. 2750–2754. [30] A. Karimi and G. Galdos, “Fixed-order H∞ controller design for nonparametric models by convex optimization,” Automatica, vol. 46, no. 8, pp. 1388–1394, 2010. [31] A. Karimi, “Frequency-domain robust controller design: A toolbox for MATLAB,” available online: http://la.epfl.ch/FDRC-Toolbox, Automatic Control Laboratory, EPFL, Switzerland, 2012. [32] J. Löfberg, “YALMIP: A toolbox for modeling and optimization in MATLAB,” in CACSD Conference, 2004. [Online]. Available: http://control.ee.ethz.ch/∼joloef/yalmip.php [33] K. C. Toh, M. J. Todd, and R. H. Tutuncu, “SDPT3: A MATLAB software package for semidefinite programming,” Optimization Methods and Software, vol. 11, pp. 545–581, 1999. [34] L. Zhang, “Modeling and control of VSC-HVDC links connected to weak AC systems,” Ph.D. dissertation, Royal Institute of Technology (KTH), Stockholm, Sweden, 2010. M. S. Sadabadi is currently a research associate in the Computational and Biological Learning (CBL) Group at the Department of Engineering, the University of Cambridge, UK. Prior to that, she was a postdoctoral fellow in the Division of Automatic Control at the Department of Electrical Engineering, Linköping University in Sweden. She received her Ph.D. in Systems and Control Theory from Automatic Control Laboratory, Swiss Federal Institute of Technology in Lausanne (EPFL), Switzerland in February 2016. She obtained a BSc and MSc with honors in Electrical Engineering from Tehran Polytechnic. She was a visiting scholar at LAAS-CNRS in Toulouse, France and Ecole Polytechnique de Montreal in Montreal, Canada. Her research interests are centered around fixed-structure controller design, control of large-scale uncertain systems, and their applications to energy systems, microgrids/smart power grids, and computational neuroscience. A. Haddadi (M’11) received the Ph.D. degree in electrical and computer engineering from McGill University, Montreal, QC, Canada in 2015. He has been a post-doctoral fellow with Montreal Polytechnique, Montreal, QC, Canada since 2015. His research interests include distributed generation, power system simulation, and power system dynamics and control. H. Karimi (M’07-SM’12) received the B.Sc. and M.Sc. degrees from Isfahan University of Technology, Isfahan, Iran, in 1994 and 2000, respectively, and the Ph.D. degree from the University of Toronto, Toronto, ON, Canada, in 2007, all in electrical engineering. He was with the Department of Electrical Engineering, Sharif University of Technology, Tehran, Iran, from 2009 to 2012. From June 2012 to January 2013, he was a Visiting Professor in the ePower lab of the Department of Electrical and Computer Engineering, Queens University, Kingston, ON, Canada. He joined the Department of Electrical Engineering, Polytechnique Montreal, QC, Canada, in 2013, where he is currently an Assistant Professor. His research interests include control systems, microgrid control, and smart grids. A. Karimi received his B.Sc. and M.Sc. degrees in Electrical Engineering in 1987 and 1990 from Amir Kabir University (Tehran Polytechnic). After 3 years of industrial experience he joined Institut National Polytechnique de Grenoble (INPG) in France and received his DEA and Ph.D. degrees both on Automatic Control in 1994 and 1997, respectively. He was Assistant Professor at Electrical Engineering Department of Sharif University of Technology in Teheran from 1998 to 2000. He is currently Senior Scientist at the Automatic Control Laboratory of Swiss Federal Institute of Technology in Lausanne (EPFL), Switzerland. He was an Associate Editor of European Journal of Control from 2004 to 2013. His research interests include closed-loop identification, data-driven controller tuning approaches and robust control.