Dynamic Stability of Three-Phase Grid-Connected Photovoltaic System using Zero Dynamic Design Approach

Transcription

1 Dynamic Stability o Three-Phase Grid-Connected Photovoltaic System using Zero Dynamic Design Approach M. A. Mahmud, Student Member, IEEE, H. R. Pota, and M. J. Hossain, Member, IEEE Abstract This paper presents a new approach to control the grid current and dc link voltage to maximum power point tracking (MPPT) and dynamic stability o a three-phase gridconnected photovoltaic (PV) system. To control the grid current and dc link voltage, zero dynamic design approach o eedback linearization is used which linearizes the system partially and enables to design controller or reduced order photovoltaic systems. This paper also describes the zero dynamic stability o three-phase grid-connected PV system which is a key requirement or the implementation o such controller. Simulation results on a large-scale grid-connected PV system show the eectiveness o the proposed control scheme in terms o delivering maximum power into the grid. Index Terms Dynamic stability, grid-connected PV system, maximum power point, zero dynamic controller I. INTRODUCTION Due to global concern on climate change and sustainable electrical power supply, renewable energy is increasingly getting popular in the developed countries. Among dierent sources o renewable energy, PV system is a promising energy source in the recent years as PV installations are increasing due to their environment riendly operation [1]. Grid-connected PV system has gained popularity due to the eed-in-tari and the reduction o battery cost. However, the intermittent PV generation varies with the change in atmospheric conditions. Maximum power point tacking (MPPT) techniques [2] are used to deliver maximum power into the grid. Eicient and advanced control schemes are essential to ensure maximum power output o a PV system at dierent operating conditions. Implementation o proper controllers on a grid-connected PV system maintain the stable operation under disturbances like the change in atmospheric conditions, change in load demands, or an external ault within the system. This can be done by regulating the switching signal through the inverter, i.e., i a proper controller is applied through the inverter o the system, then the desired perormances will be obtained. A number o techniques are available in the literature or designing the MPPT [3]. Perturb and observe (PO) [4], and incremental conductance method are commonly used techniques in the area o photovoltaic systems. In the PO method, the derivative M. A. Mahmud and H. R. Pota are with the School o Engineering and Inormation Technology (SEIT), The University o New South Wales at Australian Deence Force Academy (UNSW@ADFA), Canberra, ACT 26, Australia. M.Mahmud@ada.edu.au and h.pota@ada.edu.au. M. J. Hossain is with the Center o Wireless Monitoring and Applications, Griith School o Engineering, Griith University, Gold Coast Campus, Gold Coast, QD 4222, Australia. j.hossain@griith.edu.au. o power (dp) and the derivative o voltage (dv) need to be measured to determine the movement o operating point. I the ratio o dp and dv is positive, the reerence voltage is increased by a certain amount and vice versa [5]. A similar approach is ollowed or incremental conductance method by comparing incremental conductance and instantaneous conductance o the PV arrays [6]. The importance o DC voltage control or a grid-connected PV system can be ound in [7]. In [7] a model prediction based voltage controller is proposed to improve the reliability and lietime o the inverter and to enhance perormance o the MPPT through the reduction o DC link capacitance. Recently, an improved MPPT based on voltage-oriented control is proposed in [8] by using a proportional-integral (PI) controller in the outer DC link voltage loop. Current controllers are used to maintain stable operation o a grid-connected PV system as they can regulate the current to ollow the reerence current. There are several techniques to control the current such as PI controller, hysteresis controller, predictive controller, sliding mode controller, and so on. In [9], PI current control scheme is proposed to keep the output current sinusoidal and to have ast dynamic responses under rapidly changing atmospheric condition and to maintain the power actor at unity. The diiculty o using a PI controller is to tune the gain with the change in atmospheric conditions. The hysteresis controller as proposed in [1] has ast response by varying switching requency. The predictive controller [11] overcomes the limitation o hysteresis controller as it has constant switching requency but the main problem associated with this controller is that it cannot properly match with the existing problems related to the change in environmental actors. Grid-connected PV systems are highly nonlinear systems where most o the nonlinearities occur due to the intermittency o the sunlight, the switching unctions o the converters and inverters. The dynamic stability o a grid-connected PV system is analyzed in [12], [13] based on the eigenvalue analysis. In [12], [13], no controller is proposed to enhance the stability o PV system. However, the controllers proposed in [8-12] are mainly designed based on the linearized model o photovoltaic system which can provide satisactory operation or a predeined set o changes. To ensure the operation o gridconnected photovoltaic systems over a wide range o operating points, the design and implementation o a nonlinear controller is important. A sliding mode current controller or a three-phase grid-

2 connected PV system is proposed in [14], [15] along with a new MPPT technique to provide robust tracking against the uncertainties within the system. In [14], [15], the controller is designed based on a time-varying sliding surace. However, the selection o a time-varying surace is diicult without prior knowledge o the system. Exact linearization which is a straightorward way to design nonlinear controllers, transorms a nonlinear system into a ully linear system [16], [17] by canceling the inherent nonlinearities within the system. This linearization technique is independent o the operating point. Exact eedback linearization o grid-connected PV system is proposed in [18], [19] to enhance the dynamic stability. The control scheme proposed in [18], [19] is independent o environmental changes. However, in these papers [18], [19], the input DC current to the inverter is expressed in terms o the input and output power relation o the inverter to make the system exactly linearizable rather than using the nonlinear switching unctions. Moreover, the inverter is considered as ideal and the resistances within the lines connected to the grid are neglected in [18], [19] and the calculation o reerence values or the linear controllers are not discussed. But i the dynamics o the DC link capacitor are expressed in terms o switching unction o the inverter, more accurate results can be obtained. In this case, the gridconnected photovoltaic system will be partially linearized rather than ully linearized. When the system is partially linearized, exact linearization is no more applicable. Zero dynamic design approach o eedback linearization algebraically transorms nonlinear system dynamics into a partly linear one which is a reduced order linear system independent o operating points. Finally, linear controller can be designed or the reduced order system. Zero dynamic design approach also introduces the internal dynamics o the photovoltaic system which should be stable. Thereore, it is essential to ensure the stability o the internal dynamics beore implementing zero dynamic design approach on a gridconnected photovoltaic system. The aim o this paper is to design a new controller through zero dynamic design approach to control the DC link voltage and grid current using similar control algorithm. The partial linearizability o grid-connected photovoltaic systems are discussed along with the stability o internal dynamics o the systems. Finally, the stability o the system with the proposed control scheme is investigated by considering the environmental changes. The rest o the paper is organized as ollows. The mathematical model o a three-phase grid-connected PV system is shown in Section II. Section III presents the partial linearizability o PV system to prove the suitability o the proposed model or zero dynamic design approach. The controller design or a three-phase grid-connected photovoltaic system is shown in Section IV and relation o MPPT with the proposed control scheme is shown in Section V. Section VI shows the simulation results with the proposed controller under dierent circumstances. Finally, the paper is concluded with uture trends and urther recommendations in Section VII. II. PV SYSTEM MODE The schematic diagram o a three-phase grid-connected PV system which is main ocus o this paper is shown in Fig. 1. The considered PV system consists o a PV array, a DC link capacitor C, a three-phase inverter, a ilter inductor and connected to the grid with voltage e a, e b, e c. In this paper, the main target is to control the voltage v dc across the capacitor C and to make the input current in phase with grid voltage or unity power actor by means o appropriate control signals through the switches o the inverter. The mathematical model o the system is presented in the next subsections. A. PV Cell and Array Modeling PV cell is a simple p-n junction diode which converts the irradiation into electricity. Fig. 2 shows an equivalent circuit diagram o a PV cell which consists o a light generated current source I, a parallel diode, shunt resistance R sh and series resistance R s. In Fig. 2, I ON is the diode current which can be written as: I ON = I s [exp[α( +R s i pv )] 1] (1) q where α = AkT C, k = JK 1 is the Boltzmann s constant, q = C is the charge o electron, T C is the cell s absolute working temperature in Kelvin, A is the p-n junction ideality actor whose value is between 1 and 5, I s is the saturation current, and is the output voltage o PV array which in this case is the voltage across C, i.e., v dc. Now, by applying Kirchho s Current aw (KC) in Fig. 1, the output current (i pv ) generated by PV cell can be written as, i pv = I I s [exp[α( +R s i pv )] 1] +R s i pv (2) R sh The light generated current I depends on the solar irradiation which can be related by the ollowing equation: s I = [I sc +k i (T C T re )] (3) 1 where, I sc is the short circuit current,s is the solar irradiation, k i is the cell s short circuit current coeicient and T re is the reerence temperature o the cell. The cell s saturation current I s varies with the temperature according to the ollowing equation [19]: [ ] 3 [ ( TC qeg 1 I s = I RS exp 1 )] (4) T re Ak T re T C where, E g is the band-gap energy o the semiconductor used in the cell and I RS is the reverse saturation current o the cell at reerence temperature and solar irradiation. Since the output voltage o PV cell is very low, a number o PV cells are connected together in series in order to obtain higher voltages. A number o PV cells are put together and encapsulated with glass, plastic, and other transparent materials to protect rom harsh environment, to orm a PV module. To obtain the required voltage and power, a number o modules are connected in parallel to orm a PV array. Fig. 3 shows an electrical equivalent circuit diagram o a PV array

3 Fig. 1. Three-phase grid-connected PV system [15] Fig. 2. Equivalent circuit diagram o PV cell [15] Fig. 3. Equivalent circuit diagram o PV array [15] wheren s is the number o cells in series andn p is the number o modules in parallel. In this case, the the array i pv can be written as [ ( )] ] vpv i pv = N p I N p I s [exp α 1 N p R sh ( vpv N s + R si pv N p + R si pv N s N ) p (5) B. Three-Phase Grid-Connected PV System Modeling For a control scheme to be eective or three-phase gridconnected PV system, some details o the system is essential. The details o any system can best be described by the mathematical model. In state-space orm, Fig. 1 can be represented by the ollowing equations [19], [2]: i a = R i a 1 e a + 3 (2K a K b K c ) i b = R i b 1 e b + 3 ( K a +2K b K c ) (6) i c = R i c 1 e c + 3 ( K a K b +2K c ) where, K a, K b, and K c are the input switching signals. Now, by applying KC at the node where DC link is connected, we get = 1 C (i pv i dc ) (7) But the input current o the inverter i dc can be written as [15], [2] which yields i dc = i a K a +i b K b +i c K c (8) = 1 C i pv 1 C (i ak a +i b K b +i c K c ) (9) The complete model o a three-phase grid-connected PV can be presented by equation (6) and (9) which are nonlinear and time variant. This time variant model can be converted into time invariant model by applying dq transormation [15] using the angular requency ω o the grid, rotating reerence rame synchronized with grid where the d component o the grid voltage E d is zero [15], [19]. By using dq transormation, equation (6) and (9) can be written as I d = R I d +ωi q E d + K d I q = ωi d R I q E q + K q = 1 C i pv 1 C I dk d 1 C I qk q (1) Equation (1) represents the complete mathematical model o the three-phase grid-connected PV system which is nonlinear due to the switching unctions and diode current. In equation (1), K d and K q are the control inputs and the output variables are I q and. Since E d =, the real power delivered to the grid can be written as P = 3 2 E qi q (11)

4 From the above equation, it is clear that by controlling I q the active power P can be controlled. III. PARTIA FEEDBACK INEARIZATION AND PARTIA INEARIZABIITY OF PV SYSTEM The mathematical model o three-phase grid-connected PV system can be expressed as the ollowing orm o multi-input multi-output (MIMO) nonlinear system: where and ẋ = (x)+g 1 (x)u 1 +g 2 (x)u 2 y 1 = h 1 (x) y 2 = h 2 (x) (x) = x = [ I d I q ] T R I d +ωi q E d ωi d R I q Eq 1 g(x) = C i pv I d C u = [ K d y = [ I q Iq C K q ] T ] T (12) Consider, the ollowing nonlinear coordinate transormation z = [h 1 h 1 r1 1 h 1 h 2 h 2 r2 1 h 2 ] T (13) where, h 1 (x) = h1 x (x) and h 2 (x) = h2 x (x) are the ie derivative [16] o h 1 (x) and h 2 (x) along (x), respectively. The above transormation transorms the nonlinear system (12) with state vector x into a linear dynamic system with state vector z, described as provided that the ollowing conditions ż = Az +Bv (14) g1 k h i (x) = g2 k h i (x) = where,i = 1, 2, k < r i 1 and g1 ri 1 h i (x) g2 ri 1 h i (x) (15) (16) are satisied and n = r 1 +r 2 where r 1 and r 2 are the relative degree [16] o the system corresponding to output unction h 1 (x) and h 2 (x), respectively. I these conditions are satisied, the linear controller can be design or the linearized system (14) by using any linear controller design technique. When (r 1 +r 2 ) < n, then we can do only partial linearization and this case, the transormed states z can be written as z = φ(x) = [ z ẑ] T (17) where, z represents the states obtained rom nonlinear coordinate transormation up to the order r 1 +r 2 and ẑ represents the states related to the remaining n (r 1 + r 2 ) order. The dynamics o ẑ is called the internal dynamics whose stability need to be ensured beore designing the linear controller or the ollowing partially linearized system. z = Ã z + Bṽ (18) Now, the partial linearizability o the PV system represented in the orm (12) can be obtained by calculating relative degree corresponding to the output unctions. The relative degree corresponding to h 1 (x) = I q can be calculated as g h 1 (x) = g 1 1 h 1 (x) = K q (19) where r 1 = 1. Similarly, the relative degree corresponding to h 2 (x) = can be calculated as ollows: g h 2 (x) = 1 C (I dk d +I q K q ) (2) which indicates r 2 = 1. Thereore, r 1 +r 2 = 2 which means that (r 1 +r 2 ) < n as n = 3. Thereore, the system is partially linearized and exact linearizing controller is not possible or such system. In this case, the only way to implement eedback linearizing control is to implement the zero dynamic design approach provided that the zero dynamic o the system is stable which is shown in the ollowing section. IV. CONTROER DESIGN This section presents the controller design algorithm or a three-phase grid-connected PV system using the zero dynamic design approach as the system is partially linearized. The ollowing steps needs to be ollowed to obtain the control law through zero dynamic design approach: Step 1: Nonlinear coordinate transormation and ormulation o partial linear system rom state x to z A nonlinear coordinate transormation can be written as: z = φ(x) (21) where, φ is the unction o x. For a three-phase grid-connected photovoltaic system, we choose and z 1 = φ 1 (x) = h 1 (x) = I q (22) z 2 = φ 2 (x) = h 2 (x) = (23) Using the above transormation, the partially linearized system can be obtained as ollows: z 1 = h 1(x) ẋ = h 1 (x)+ g1 h 1 (x)u 1 + g2 h 1 (x)u 2 x z 2 = h (24) 2(x) ẋ = h 2 (x)+ g1 h 2 (x)u 1 + g2 h 2 (x)u 2 x For the photovoltaic system considered in this research, equation (24) can be rewritten as: z 1 = ωi d R I q E q + K q 1 z 2 = C i pv 1 C I dk d 1 (25) C I qk q

5 The above system can be written as the ollowing linearized orm: z 1 = v 1 z 2 = v 2 (26) where v 1 and v 2 are the linear control inputs expressed as v 1 = ωi d R I q E q + K q 1 v 2 = C i pv 1 C I dk d 1 (27) C I qk q and which can be obtained by using a linear control techniques or the system (26). But beore designing and implementing controller through zero dynamic design approach, it is essential to check the stability o zero dynamics which is presented in the next step. Step 2: Zero dynamic stability o Grid-Connected PV system In the previous step, the third-order PV system is transormed into a second order system which represents the external dynamics o the system. These external dynamics need both stability and good perormance which can be obtained through the designed controller. To do this, the control law needs to be chosen in such a way that lim h i(x) t which implies [z 1 z 2 z r ] T = [16]. For the PV system considered in this work, this means that z 1 = z 2 = which indicates z 1 = z 2 = (28) The remaining irst order system needs to be determined to obtain the zero dynamics o the system or which it is essential to ind out the rest state. et, the remaining state is ẑ 3 = ˆφ(x). This needs to be selected in such a way that it must satisy the ollowing conditions [21]: g1ˆφ(x) = g2ˆφ(x) = (29) For a three-phase grid-connected PV system, equation (29) will be satisied i we chose ˆφ(x) = ẑ 3 = 1 2 I2 d I2 q Cv2 pv (3) By usingz 1 = I q andz 2 =, equation (3) can be written as: I 2 d = 2 ẑ3 z 2 1 C z2 2 (31) Thus, the remaining dynamics o the system can be obtained as ollows: For z 1 = z 2 =, equation (32) can be simpliied as: ( ż 3 = I d R I d +ωi q E ) d (33) SinceI q = z 1 = ande d =, equation (33) can be written as: ẑ 3 = RI 2 d (34) Substituting the value o Id 2 rom (31) into (34), it can be written as: ( 2 ẑ 3 = R z 2 ẑ3 1 C ) z2 2 (35) Again, i we use z 1 = z 2 =, equation (35) can be written as ẑ 3 = 2R ẑ3 (36) Equation (36) is the zero dynamic equation o the threephase grid-connected photovoltaic system which is stable. Thereore, zero dynamic design approach can be implemented or three-phase grid-connected photovoltaic system. It is clear that the zero dynamic design approach divides the dynamics o a nonlinear system into two parts: one is external dynamics which needs both stability and good perormance; and the other is internal dynamics which needs stability only. The derivation o the proposed control law is shown in the ollowing steps. Step 3: Derivation o Control aw or a Grid-Connected PV system Using equation (25) and (26), the control law can be obtained as ollows: K d = ( v 1 +ωi d + R I q + E q K q = C I q [ v 2 + i pv C I d C ) ( v 1 +ωi d + R I q + E q )] (37) Equation (37) is the inal control law or a grid connected photovoltaic system. At this point, the only thing which needs to be designed is the new linear control input v 1 and v 2 by using any linear controller design technique. In this paper, proportional-integral (PI) controllers are used to track the reerence outputs I qre and re. Following two PI controllers are considered to track the output, t v 1 = k 1p (I qre I q )+k 1i (I qre I q )dt t v 2 = k 2p (re )+k 2i (re )dt (38) Here, the gains need to be selected in such a way that the output ollows the reerence current to minimize the error. In this paper, the gains are considered as ollows: ẑ 3 = ˆφ(x) = Id 1 +z 1 2 +Cz 2 3 (32) k 1p = 2I qre, k 1i = I 2 qre

6 Fig. 4. Control block diagram. and k 2p = 2re, k 2i = v 2 pvre In this case, the gains o the PI controllers depend on the reerence values o voltage and current. These reerence values are calculated rom MPPT as shown in the next section. Thus, the gain o PI controllers will be updated automatically with the change in atmospheric conditions. V. MPPT AGORITHM MPPT technique adjust the PV array voltage in order to extract available maximum power under any atmospheric changes. MPPT uses and i pv to detect the slope and generatesp re to track the maximum power [15]. In this paper, incremental conductance method as presented in [19] is used to obtain the maximum power. At maximum power point, dp pv d = (39) where,p pv = i pv and i we use this relation, then rom (35) = i pv (4) Here, is the incremental conductance, and ipv is the instantaneous conductance. Maximum power point (MPP) can be obtained by considering the ollowing conditions: Condition 1: At MPP = i pv Condition 2: At let o MPP > i pv Condition 3: At right o MPP < i pv I a PV system satisies condition 1, the voltage is ascertained at MPP voltage and ixed at this voltage until the MPPT encounters a change due to the change in atmospheric conditions. I the atmospheric conditions change in such a way that the PV system holds condition 2, then it is essential to increase the reerence voltage to achieve MPPT and the opposite phenomena is true or condition 3. At MPP, the reerence output power generated by PV system is P re = i pv (41) Since P re is the maximum power which supplied to grid, thereore, P re = P = 3 2 E qi qre (42)

7 which implies that Since re = v dcre, thus, I qre = 2 P re (43) 3 E q v dcre = P re i pv (44) The reerence values obtained in equation (39) and (4) are used or the reerence values o linear controllers. The perormances o the proposed controller are evaluated in the ollowing section. VI. SIMUATION RESUTS To simulate the perormance o the grid-connected PV system with zero dynamic controller, a PV array consists o 2 strings, characterized by a rated current o A, are connected in parallel. Each string is subdivided into 2 modules, characterized by a rated voltage o 43.5 V and connected in series. Thus, the total output voltage o the PV array is 87 V and output current is A. The value o DC link capacitor is 4 µf. The line resistance is considered as.1 Ω and the inductance is 1 mh. The grid voltage is 44 V and requency is 5 Hz. The block diagram representation o the zero dynamic design approach is shown in Fig. 4. From Fig. 4 it is shown that the three-phase grid voltages and currents are transormed into direct and quadrature axis components through abc dq transormation which is done to match with the proposed modeling presented in Section II. The controller is the combination o linear controller and zero dynamic controller. Finally, the control inputs are again transormed into three-phase components using dq abc transormation to implement them through the inverter switches. To make the input signals suitable or switches a PWM is used. All the simulations are done in the widely used PSCAD environment. In this case, the system is simulated at standard atmospheric condition where the value o solar irradiation is considered as 1 kw m 2 and the temperature as 298 K. At this condition, the output power o PV unit which is shown in Fig. 5 where there is some luctuation due to the nonlinear characteristics o PV system. The main purpose o the control action is to extract 5 kw power rom the PV unit through MPPT and supply this power to the grid. This can be done by regulating the inverter switches through the proper control scheme when the system is operated at unity power actor. Thereore, to evaluate the perormance o the controller, it is essential to analyze the grid voltage and current. The grid voltage (solid line) and current (dotted line) o phase A are shown in Fig. 6 rom 1.1 s to 1.15 s with the proposed control scheme. From Fig. 6, it is seen that the voltage and current are in phase which ensure the operation o grid-connected PV system at unity power actor. In a practical PV system, the atmospheric condition changes continuously or which there exists variations in cell working temperature as well as in solar irradiance. Due to the change in atmospheric condition, the output voltage, current, and power o PV unit changes signiicantly. For example, i a single Fig. 5. Output power (kw) o PV unit Voltage (V) and Current (A) Time (s) Output DC power o PV Unit at standard atmospheric condition Time (s) Fig. 6. Voltage (solid line) and current (dotted line) o phase A at standard operating condition. module o a series string is shaded partially, then its output current will be reduced which will dicate the operating point o whole string. Fig. 7 shows the perormances o the proposed zero dynamic controller with the change in atmospheric conditions. From Fig. 7, it is seen that the PV system operates at standard atmospheric conditions rom 1.5 s to 1.15 s. But the irradiance changes rom 1 W m 2 to 7 W m 2 at 1.15 s and the weather remains cloudy, i.e., the PV system is shaded until 1.25 s. At this stage, the amount o power delivered to the grid will be changed and MPPT will select a dierent MPP but the grid voltage will be the same. The change in grid current is also shown in Fig. 7. Ater 1.25 s the grid

8 results, it is clear that the controller perorms better under varying atmospheric conditions. Future work will deal with the extension o the proposed method by considering some mismatches within the PV model and implement on a large interconnected system. Fig. 7. Control Signal Grid current with the change in solar irradiation Time (s) Fig. 8. Control input (Solid line K a, dashed line K b, and Solid line with dot K c). current is similar to its previous value as the system again operates at standards atmospheric conditions. From the results it is clear that the designed controller perorms well at varying atmospheric conditions and provides robust perormance. The corresponding control signals are shown in Fig. 8. VII. CONCUSION In this paper a zero dynamic control scheme is presented to analyze the perormance o a three-phase grid-connected PV system and to enhance the dynamic stability limit with the change in atmospheric conditions. The proposed controller design approach cancels all possible nonlinearities by transorming PV system into two decoupled linear subsystems with stable internal dynamics. The DC link voltage and current injected into grid are controlled to ensure the operation o the PV system at unity power actor. From the simulation REFERENCES [1] Y. T. Tan, D. S. Kirschen, and N. Jenkins, Impact o a large penetration o photovoltaics on the power system, In Proc. o CIRED 17th International Conerence on Electricity Distribution, 23. [2] I. Houssamo, F. ocment, and M. Sechilariu, Maximum power point tracking or photovoltaic power system: development and experimental comparison o two algorithms, Renewable Energy, vol. 35, no. 1, pp , 21. [3] T. Esram and P.. Chapman, Comparison o photovoltaic array maximum power point tracking techniques, IEEE Transactions on Energy Conversion, vol. 22, no. 2, pp , 27. [4] A. Mellit, H. Rezzouk, A. Messai, and B. Medjahed, FPGA-based real time implementation o MPPT controller or photovoltaic systems, Renewable Energy, vol. 36, no. 5, pp , 211. [5]. Chun-xia and. i-qun, An improved perturbation and observation MPPT method o photovoltaic generate system, In Proc. o 4th IEEE Conerence on Industrial Electronics and Applications, 29. [6] B. iu, S. Duan, F. iu, and P. Xu, Analysis and improvement o maximum power point tracking algorithm based on incremental conductance method or photovoltaic array, In Proc. o IEEE PEDS, 27. [7] F. He, Z. Zhao, T. u, and. Yuan, Predictive DC voltage control or three-phase grid-connected pv inverters based on energy balance modeling, in 21 2nd IEEE International Symposium on Power Electronics or Distributed Generation Systems (PEDG), June 21, pp [8] R. Kadri, J. P. Gaubert, and G. Champenois, An improved maximum power point tracking or photovoltaic grid-connected inverter based on voltage-oriented control, IEEE Transactions on Industrial Electronics, vol. 58, no. 1, pp , Jan [9] J. Selvaraj and N. A. Rahim, Multilevel inverter or grid-connected PV system employing digital PI controller, IEEE Trans. on Industrial Electronics, vol. 56, no. 1, pp , 29. [1] N. A. Rahim, J. Selvaraj, and C. C. Krismadinata, Hysteresis current control and sensorless mppt or grid-connected photovoltaic systems, In Proc. o IEEE ISIE, 27. [11] A. Kotsopoulos, J.. Duarte, and M. A. M. Hendrix, A predictive control scheme or DC voltage and AC current in grid-connected photovoltaic inverters with minimum DC link capacitance, In Proc. o IEEE IECON, 21. [12]. Wang and T.-C. in, Dynamic stability and transient responses o multiple grid-connected pv systems, IEEE Trans. on Energy Conversion, vol. 19, no. 4, pp , 24. [13] C. Rodriguez and G. A. J. Amaratunga, Dynamic stability o gridconnected photovoltaic systems, In Proc. o IEEE Power Engineering Society General Meeting, 24. [14] I. Kim, Sliding mode controller or the single-phase grid-connected photovoltaic system, Applied Energy, vol. 83, pp , 26. [15], Robust maximum power point tracker using sliding mode controller or the three-phase grid-connected photovoltaic system, SolarEnergy, vol. 81, pp , 27. [16] A. Isidori, Nonlinear control systems. Springer Verlag, [17] J. J. E. Slotine and W. i, Applied Nonlinear Control. New Jersey: Prentice-Hall, [18] A. O. Zue and A. Chandra, State eedback linearization control o a grid connected photovoltaic interace with MPPT, In Proc. o 29 IEEE Electrical Power and Energy Conerence (EPEC), 29. [19] D. alili, A. Mellit, N. ourci, B. Medjahed, and E. M. Berkouk, Input output eedback linearization control and variable step size mppt algorithm o a grid-connected photovoltaic inverter, Renewable Enegry, vol. 36, no. 12, pp , 211. [2] R. Marouani and A. Mami, Voltage oriented control applied to a grid connected photovoltaic system with maximum power point tracking technique, American Journal o Applied Sciences, vol. 7, no. 8, pp , 21. [21] Q. u, Y. Sun, and S. Mei, Nonlinear Control Systems and Power System Dynamics. Boston/Dordrecht/ondon: Kluwer Academic Publishers, 21.