2012 Australian Control Conference
15-16 November 2012, Sydney, Australia
Nonlinear Controller Design for Single-Phase Grid-Connected Photovoltaic Systems Using Partial Feedback Linearization M.A. Mahmud, H.R. Pota, M.J. Hossain
Abstract—This paper deals with the design of a nonlinear controller for single-phase grid-connected photovoltaic (PV) systems to maintain the current injected into the grid in phase with grid voltage and to regulate the DC link voltage for achieving maximum power point tracking (MPPT). The controller is designed based on the partial feedback linearization which transforms the nonlinear system into a reduced order linear system and an autonomous system whose dynamics are known as internal dynamics of the system. This paper also deals with the stability of internal dynamics of PV systems which is a basic requirement to design partial feedback linearizing controller. The performance of the proposed controller is evaluated in terms of delivering maximum power and synchronization of grid current with voltage under changes in atmospheric conditions. Index Terms—DC link voltage, grid current, grid-connected PV system, maximum power point tracking, partial feedback linearizing controller.
I. I NTRODUCTION The integration of renewable energy sources is increasingly being pursued all over the world as a supplement and an alternative of conventional fossil fuel generation to meet the increased energy demand and keep the environment free from fuel exhaustion. The PV system is the most employed renewable energy source among different sources. Residential PV installations are increasing significantly due to their small relative size, noiseless operation, and feed-in tariff [1], [2]. The major concerns of integrating PV into the grid are stochastic behaviors of solar irradiations and interfacing of inverters with the grid. The intermittent PV generation varies with changes in atmospheric conditions. Due to the high initial investment and reduced life time of PV system as compared to traditional energy sources, it is essential to extract maximum power from PV systems [3]. Controllers on grid-connected PV systems are applied to achieve the desired performance under disturbances like changes in atmospheric conditions, changes in load demands, or external faults within the system. In a grid-connected PV system, control objectives are obtained by means of a strategy based on two cascaded control loops using a pulse width modulation (PWM) scheme [4]. The two M. A. Mahmud and H. R. Pota are with the School of Engineering and Information Technology (SEIT), The University of New South Wales at Australian Defence Force Academy (UNSW@ADFA), Canberra, ACT 2600, Australia. E-mail:
[email protected] and
[email protected]. M. J. Hossain is with the Center of Wireless Monitoring and Applications, Griffith School of Engineering, Griffith University, Gold Coast Campus, Gold Coast, QLD 4222, Australia. E-mail: j.hossain@griffith.edu.au.
ISBN 978-1-922107-63-3
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cascaded control loops consist of an outer voltage control loop to settle the PV array at MPP and an inner current control loop to perform the task of establishing the duty ratio for the generation of a sinusoidal output current that is in phase with the grid voltage [4]. The importance of DC link voltage control for a gridconnected PV system can be seen in [5] where a model prediction based voltage controller is proposed to improve the reliability and lifetime of the inverter and to enhance performance of the MPPT through the reduction of DC link capacitance. In [6], a PV-system-sampled data linear model based on the system energy-balance approach is used to control the voltage by using conventional linear discrete control technique. Recently, an improved mppt method has been proposed in [7], based on voltage-oriented control, using a PI controller in outer DC link voltage loop. There are several techniques to control the current such as PI controller, hysteresis controller, predictive controller, sliding mode controller, and so on. In [8], PI current control scheme is proposed to keep the output current sinusoidal and to have fast dynamic responses under rapidly changing atmospheric condition and to maintain the power factor at unity. The difficulty of using a PI controller is the necessity to tune the gain with changes in atmospheric conditions. The hysteresis controller as proposed in [9] has fast response but a varying switching frequency. The predictive controller [10] overcomes the limitation of hysteresis controller as it has constant switching frequency. Grid-connected PV systems suffer from nonlinear behaviors where most of the nonlinearities occur due to the variation of solar irradiance and nonlinear switching functions of inverters. Linear controllers for nonlinear PV systems as presented in [5]–[10], provide satisfactory operation over a fixed set of operating points as the system is linearized at an equilibrium point. The restrictions of operating points can be solved by implementing nonlinear controllers for nonlinear PV systems. A sliding mode current controller for singlephase grid-connected PV system is proposed in [11] along with a new MPPT technique to provide robust tracking against the uncertainties within the system. In [11], the controller is designed based on a time-varying sliding surface. However, the selection of a time-varying surface is a difficult task and the system stays confined to the sliding surface. Feedback linearization is a straightforward way to design nonlinear controllers as it transforms a nonlinear system into a fully linear or a partly linear system by canceling the
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inherent nonlinearities within the system and linear controller design techniques can be employed to design the controller for the linearized system [12], [13]. When feedback linearization transforms a nonlinear system into fully linear system, the approach is called exact feedback linearization and if the system is transformed into partially linearized system, the approach is known as partial feedback linearization [12]. Since feedback linearization cancels nonlinearities by introducing nonlinear term, the feedback linearized system is independent of operating points. The research on the implementation of feedback linearizing controller for PV systems application is in primary stage. In [14], [15], exact feedback linearizing controller of grid-connected PV systems is proposed. But a PV system can be partially linearized rather than fully linearized and in this case, exact linearization is no more applicable. The main focus of this paper is to design a partial feedback linearizing controller which is able to control both the DC link voltage and the current injected into the grid. Partial feedback linearization not only transforms the nonlinear system into a reduced order linear system but also an autonomous system whose dynamics are known as internal dynamics. The stability of these internal dynamics needs to be ensured before implementing partial feedback linearizing controller. This paper also addresses the issue related to the stability of internal dynamics of single-phase grid-connected PV systems. The performance of the proposed controller is investigated by considering the environmental changes and comparing with the conventional PI controller. The rest of the paper is organized as follows. The mathematical model of a single-phase grid-connected PV system is shown in Section II. Section III presents a brief overview of partial linearization and partial linearizability of PV system to prove the suitability of the proposed model for implementing partial feedback linearizing controller. The controller design for a single-phase grid-connected photovoltaic system is shown in Section IV. Section V presents the simulation results with the proposed controller under different operating conditions and comparison of performances with PI controllers. The paper concludes with comments on future trends and recommendations for research in Section VI.
Schematic diagram of a single-phase grid-connected PV system which is the main focus of this paper is shown in Fig. 1. The considered PV system consists of a PV array, a DC link capacitor C, a single-phase inverter, a filter inductor, and connected to the grid with voltage e. In Fig. 1, the PV array represents the equivalent model as presented in [16], v is the output of DC link which is to be adjusted at a level through MPPT to make it suitable for the inverter, S1 , S2 , S2 , and S4 are the four switches of the inverter, R is the line resistance, L is combination of filter and line inductance, i = Im sin ωt is the current injected into the grid, and e(t) = Vm sin ωt is the grid voltage where Vm is the maximum value of the grid voltage, ω = 2πf is the angular frequency and f is the grid frequency.
31
Fig. 1. Equivalent circuit diagram of single-phase grid-connected PV system
Now, by applying KCL at the node where DC link is connected as shown in Fig. 1, we get = idc + iC
ipv
(1)
where idc is the input current of the inverter and iC = C dv dt . After doing some simplification equation (1) can be written in the following form: 1 dv = (ipv − idc ) (2) dt C Since idc is the input current of the inverter, in terms output current i of the inverter it can be written as follows: → − → mi (3) idc = − → where − m represents the space-phasor corresponding to the PWM modulating signals which are normalized to the peak → − value of the triangular carrier signal, i is the phasor representation of i, and idc = |m||i|. Finally, equation (2) can be written as 1 dv → − → = ipv − − mi (4) dt C Applying KVL at the AC side of the inverter as shown in Fig. 1, it can be written that → − di → − → → − +− e (5) v out = R i + L dt where − → represents the phasor quantity of the corresponding variables. In terms of input DC voltage v of the inverter the → output AC voltage − v out can be expressed as follows: vout
II. PV S YSTEM M ODEL
→ = − mv
(6)
Using the above relation, equation (5) can be simplified as follows: → − di 1 − → → − → = mv − R i − − e (7) dt L For analysis and control purposes, space-phasor variables in PV systems model can be projected on a dq-frame which can be achieved by replacing each space-phasor by its dq-frame equivalent as [17] − → x =
(Xd + jXq ) ejωt
− If → x represents a state variable, then → dXq dXd d− x = +j ejωt + ω (Xd + jXq ) ejωt dt dt dt
(8)
(9)
where ω is the dq-frame angular frequency. The space-phasor to dq-frame can be obtained by using the following relation: − x e−jωt (Xd + jXq ) = →
(10)
Using the relations presented by equations (8)-(10), PV system models represented by equations (4) and (7) can be transformed into the following dq-frame: v R Ed + Md I˙d = − Id + ωIq − L L L Eq v R ˙ (11) Iq = −ωId − Iq − + Mq L L L 1 1 1 v˙ pv = ipv − Id Md − Iq Mq C C C Equation (11) represents the complete mathematical model of the single-phase grid-connected PV system which is nonlinear due to the switching functions and diode current. In equation (11), Md and Mq are the control inputs and the output variables are Iq and v. III. PARTIAL F EEDBACK L INEARIZATION AND PARTIAL L INEARIZABILITY OF PV S YSTEM The mathematical model of a single-phase grid-connected PV system can be expressed a multi-input multi-output (MIMO) nonlinear system as follows: x˙ = f (x) + g1 (x)u1 + g2 (x)u2 y1 = h1 (x)
(12)
y2 = h2 (x) where
x = Id
Iq
v
T
⎡
Ed −R L Id + ωIq − L E R ⎣ f (x) = −ωId − L Iq − Lq 1 C ipv ⎡ v ⎤ 0 L v ⎦ g(x) = ⎣ 0 L Iq Id −C −C
and
u = Md
Mq
y = Iq
T
v
⎤
z = [h1
h2
where, i = 1, 2; k < ri − 1 and Lg1 Lrfi −1 hi (x) = 0
(15)
Lg2 Lrfi −1 hi (x) = 0
where, Lg Lf hi (x) is the Lie derivative of Lf hi (x) along g(x). Integers r1 and r2 are known as the relative degree of the system corresponding to output function h1 (x) and h2 (x), respectively [12]. If these conditions are satisfied, a linear controller can be design for the linearized system (16). z˙ = Az + Bv
(16)
where, A is the system matrix for feedback linearized system, B is the input matrix for feedback linearized system, and v is the new control input for feedback linearized system. When (r1 + r2 ) < n, we can do only partial linearization and this case, the transformed states z can be written as z = φ(x) = [ z
zˆ]T
(17)
where, z represents the states obtained from nonlinear coordinate transformation up to the order r1 + r2 and zˆ represents the states related to the remaining n − (r1 + r2 ) order. The dynamics of zˆ is called the internal dynamics and its stability needs to be ensured before designing the linear controller for the following partially linearized system (18).
(18)
is the system matrix for partially linearized system, where, A B is the input matrix for partially linearized system, and v is the new control input for partially linearized system. The partial linearizability of the PV system represented by equation (12) can be obtained by calculating relative degree corresponding to the output functions. The relative degree corresponding to h1 (x) = Iq can be calculated as
T
Lf h2 · · · Lrf2 −1 h2 ]T
(14)
Lg2 Lkf hi (x) = 0
z + B v z ˙ = A
⎦
Lg h1 (x) = Lg L1−1 h1 (x) = f
Nonlinear system in (12) can be often linearized using feedback linearization. Consider, the following nonlinear coordinate transformation Lf h1 · · · Lrf1 −1 h1
Lg1 Lkf hi (x) = 0
(13)
where, r1 < n and r2 < n are integers, Lf hi (x) = ∂hi ∂x f (x), i = 1, 2 are the Lie derivative of hi (x) along f (x) [12]. This transforms the nonlinear system (12) with state vector x into a linear dynamic system with state vector z provided that the following conditions are satisfied for n = r1 + r2 :
32
v L
(19)
where r1 = 1. Similarly, the relative degree corresponding to h2 (x) = v can be calculated as follows: Lg h2 (x) = −
1 (Id + Iq ) = 0 C
(20)
which indicates r2 = 1. Therefore, r1 + r2 = 2 which means that (r1 + r2 ) < n as n = 3. Therefore, the system is partially linearizable for the chosen output functions. To implement feedback linearizing control for this system the partial feedback linearization technique needs to be used provided that internal dynamics of the system is stable. The design process is covered in the following section.
IV. C ONTROLLER D ESIGN This section presents the essential steps to design the controller for single-phase grid-connected PV systems using partial feedback linearization as the system is partially linearizable. • Step 1: Nonlinear coordinate transformation and partial linearization A nonlinear coordinate transformation can be written as: z =
φ(x)
which implies [ z1 z 2 · · · z r ]T = 0 [12]. For PV system considered in this work, this means that z 1 = z 2 = 0 which indicates z ˙ 1 = 0 (28) z ˙ 2 = 0 ˆ This needs to be Let the remaining state be zˆ3 = φ(x). selected in such a way that it must satisfy the following conditions [18]:
(21)
ˆ = Lg1 φ(x) ˆ Lg φ(x) =
where, φ is the function of x. For a single-phase gridconnected photovoltaic system, we choose z 1 = φ 1 (x) = h1 (x) = Iq
(22)
z 2 = φ 2 (x) = h2 (x) = v
(23)
and
Using the above transformation, the partially linearized system can be obtained as follows:
2
0
(29)
0
For a single-phase grid-connected PV system, equation (29) will be satisfied if we chose 1 1 1 ˆ (30) φ(x) = zˆ3 = LId2 + LIq2 + Cv 2 2 2 2 By using z 1 = Iq and z 2 = vpv , equation (30) can be written as:
2 C (31) Id2 = zˆ3 − z 12 − z 22 (x) ∂h L L 1 x˙ = Lf h1 (x) + Lg1 h1 (x)u1 + Lg2 h1 (x)u2 z ˙ 1 = ∂x (24) Thus, the remaining dynamics of the system can be obtained as follows: ˙z 2 = ∂h2 (x) x˙ = Lf h2 (x) + Lg h2 (x)u1 + Lg h2 (x)u2 1 2 ∂x ˆ (32) z1 f2 + C z 2 f3 = LId f1 + L zˆ˙3 = Lf φ(x) For the photovoltaic system (11), For z 1 = z 2 = 0, equation (32) can be simplified as: ˙z 1 = −ωId − R Iq − Eq + v Mq ˙zˆ3 = LId − R Id + ωIq − Ed L L L (33) (25) L L 1 1 1 ˙z 2 = ipv − Id Md − Iq Mq C C C Since Iq = z 1 = 0 and Ed = 0, equation (33) can be written as: The above system can be written as the following linearized zˆ˙3 = −RId2
form: z ˙ 1 = v1 z ˙ 2 = v2
(26)
Substituting the value of Id2 from (31) into (34), it can be written as:
where v1 and v2 are the linear control inputs expressed as Eq v R Iq − + Mq L L L 1 1 1 ipv − Id Md − Iq Mq C C C
v1 = −ωId − v2 =
(34)
zˆ˙3 = −R
2 C zˆ3 − z 12 − z 22 L L
(35)
Again, if we use z 1 = z 2 = 0, equation (35) can be written
(27) as
which can be obtained by using a linear control techniques for the system (26). But before designing and implementing controller through partial feedback linearization, it is essential to check the stability of internal dynamics which is presented in the next step. • Step 2: Stability of internal dynamics of single-phase grid-connected PV system In the previous step, the third-order PV system is transformed into a second-order system which represents the external dynamics of the system. Good performance of the external dynamics can be obtained through a controller. The control law needs to be chosen in such a way that lim hi (x) → 0
t→∞
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2R (36) zˆ3 zˆ˙3 = − L Equation (36) represents internal dynamics of the single-phase grid-connected photovoltaic system which is stable. Therefore, partial feedback linearization can be implemented on singlephase grid-connected photovoltaic system and the derivation of the proposed control law is shown in the following step. • Step 3: Derivation of control law for a single-phase gridconnected PV system From equation (27), the control law can be obtained as follows: Eq L R Md = v1 + ωId + Iq + vpv L L
(37) ipv Eq LId C R − v2 + v1 + ωId + Iq + Mq = − Iq C Cvpv L L
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