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A simplified DQ Controller for Single-Phase Grid-Connected PV Inverters Abdalbaset M. Mnider, David J. Atkinson, Mohamed Dahidah, Matthew Armstrong School of Electrical and Electronic Engineering, Newcastle University, Newcastle Upon Tyne, UK [email protected] Abstract- S ynchronous dq-frame controllers are generally accepted due to their high performance compared to stationary a Jl -frame ones, as they operate on dc quantities, achieving zero steady-state error. In single-phase systems, however, PI-based dq controllers cannot be directly applied due to the reduced number of input signals available compared to three-phase systems. The common approach in single-phase systems is to create a synthesized phase signal orthogonal to the fundamental signal of the single-phase system so as to obtain dc quantities by means of (o.p to dq) transformation. The orthogonal components in conventional approaches are usually obtained by phase shifting the real signals by a quarter of the fundamental period. The introduction of such delay in the system deteriorates the dynamic response, which becomes slower and oscillatory. In this paper, an alternative dq dq controller scheme which referred as to simplified controller is proposed. The proposed scheme does not require orthogonal quantities to be generated, making it easier to be implemented. The simplified dq control method is experimentally compared to the conventional delay-based shown to improve the poor dynamics dq control method and of the conventional approach while not adding excessive complexity to the controller structure. A single-phase five-level diode-clamped grid-connected PV inverter is considered as an example in this paper. Keywords-Diode-clamped Multilevel inverter, single-phase dq controller, orthogonal system generation (OSG). I. INTRODUCTION Single-phase voltage-source converters (VSCs) are widely utilized as an interface between renewable energy sources and the utility grid, especially in small scale photovoltaic applications[l]. Current control of VSCs has seen great attention from researches in the recent years, and various approaches have been proposed, including, hysteresis, predictive, proportional-integral (PI) and proportional resonant (PR)-based control strategies [2]-[5]. Simple PI-based a� controllers are the most conventional approach. However, because of the time-varying nature of converter, it is difficult to achieve, a zero steady-state error. On the other hand, PI-based dq control approaches have been efficiently used to control three-phase grid connected inverters, due to their great advantages of presenting infmite control gain at the steady-state operating point, leading to zero steady-state error [6]-[8]. However, in single-phase grid-connected inverters, these controllers cannot be applied directly, where a second quantity in quadrature with the existing physical one should be created such that the physical and the synthesized components together form the stationary a� frame. In the technical literature, many attempts have been reported to obtain 978-1-4673-9768-1I16/$3l.00 ©2016 IEEE the required orthogonal signal [9]-[11]. In [9], the authors proposed to phase shifting the circuit variables by a quarter cycle of the fundamental period. However, phase shifting the real components to create the orthogonal signals deteriorates the dynamic response of the system, as the real and fictive axes do not run concurrently. In [10], the authors used differentiation to create the imaginary circuit to avoid the delay. But, the perfonnance of the differentiation approach can be deteriorated significantly under distorted grid voltage conditions. In [11], the authors developed a fictive-axis emulation technique to create the imaginary circuit with a fictive axis running concurrently with the real circuit, which helps improving the poor dynamics of the conventional approaches. In this paper, an alternative dq current-control scheme is proposed based on the use of the deference between the desired and real measured currents in the stationary reference frame to generate the steady state error signals in the dq frame which are required for the dq controller. Contrary to the conventional dq controller, the proposed method does not require producing any orthogonal signal, making it easy to be implemented with superior dynamic performance compared to that of the conventional delay-based approach. The rest of this paper is organized as follows. In Section II, a description of the utilized test system, with mathematical model for the adopted single-phase system is provided. Section III briefly explains the conventional dq current-control strategy. Section IV introduces the proposed simplified dq control scheme with mathematical analyses. In Section V the performance evaluation of the proposed and the conventional approaches based on experimental results are presented. Finally, Section VI concludes this paper. II. SYSTEM DESCRIPTION AND MODELLING Fig. 1 shows the schematic diagram of the implemented test system in which a five-level diode-clamped PV inverter is connected to the utility grid through an LCL filter. The first stage is a dc-dc converter used for maximum power point tracking (MPPT) algorithm of the PV array [12]-[15]. For simplicity, this stage is replaced by four ideal voltage sources, neglecting the dc-side dynamics. In this case, capacitor voltages are balanced and fixed at a desired level. The parameters of the adopted test system are given in Table I. In the following, a mathematical model of the single-phase system in Fig. 1 is described, and a structural diagram of that is derived, which is adopted for the design of both conventional and simplified controllers in the dq reference frame. PV Generator Vgrid_s Five-level DC Diode-clamped Inverter Link MPPT LCL Filter - Vinv_s i1 Vdc DC/DC MPPT R1 Vinv (R1+R2) Utility Grid - L1 R2 Vcf is 1 S(L1+L2) i2 . Fig.2. Structural diagram of the test system in the αβ reference frame L2 Cf Vgrid_d Vgrid (R1+R2) Gating Signals i1 or i2 Vgrid DQ Current Controller Vinv_d - VC_d - + Fig.1. Schematic of the implemented single-phase test system id ωff (L1+L2) Table 1 Hardware experimental test system parameters -ωff (L1+L2) Symbol Description Value Unit Vgrid P ωff fsw Vdc Cdc L1 R1 Cf L2 R2 Nominal grid voltage (rms) Rated output power Nominal grid frequency Switching frequency DC-link voltage DC-link capacitance LCL-filter inverter-side inductor LCL-filter inverter -side resistor LCL-filter parallel capacitor LCL-filter grid-side inductor LCL-filter grid-side resistor 90 0.4 2.π.50 20 280 4×1000 1.2 0.113 3.3 0.5 0.103 V kW Rad/s kHz V µF mH Ω µF mH Ω (R1+R2) Vinv = R1 i1 + SL1 i1 + Vcf Vcf = R 2 i2 + SL2 i2 + Vgrid ⟦ ⟧ vcf i1 = i2 + Cf S VC_q - - 1 S(L1+L2) iq Vgrid_q Fig.3. Structural diagram of the test system in the RRF (1) Vinvd = (R1 + R 2 )id + S(L1 + L2 )id −ωff (L1 + L2 )iq + Vgrid_d ⟦ ⟧ Vinvq = (R1 + R 2 )iq + S(L1 + L2 )iq (3) +ωff (L1 + L2 )id + Vgrid_q The system in the rotating reference frame (RRF) based on (3) is diagrammatically illustrated in Fig 3, containing both, the typical compensation and the decoupling terms. III. CONVENTIONAL SINGLE-PHASE DQ CURRENT CONTROLLER where, Vinv, Vcf, Vgrid, i1, and i2 represent the inverter terminal voltage, the capacitor voltage, the utility grid voltage, the converter-side current and the grid current, respectively. By neglecting the capacitor’s influence in the current control design as its value is low, the vector control for the proposed system takes exactly the same policy as for VSC with an L filter. Thus, in the (α-β frame), the single-phase grid connected inverter equation (1) is simplified into (2). Vinv_α = (R1 + R 2 )iα + S(L1 + L2 )iα + Vgrid_α ⟧ Vinv_β = (R1 + R 2 )iβ + S(L1 + L2 )iβ + Vgrid_β + Vinv_q Based on the system in Fig. 1, the dynamic of the ac side of the test system can be described as: ⟦ 1 S(L1+L2) (2) Based on (2), a structural diagram of the system in the stationary reference frame is drawn as in Fig. 2. The subscript “s” in Fig. 2 denotes the quantities in the (αβ) stationary reference frame. Further, after applying a stationary-to-rotating transformation to (2) according to xdq = xαβ e−jwt , the dynamic of the ac-side variables in the rotating-frame (dq frame) is derived as (3) The conventional dq current control strategy is well known and widely studied in the literature [6-8]. Adopting (3), in order to achieve a decoupled control of id and iq, the inverter terminal voltage should be controlled as follows: Vinv_d = VC_d − ωff (L1 + L2 )iq + Vgrid_d ⟦ ⟧ Vinv_q = VC_q + ωff (L1 + L2 )id + Vgrid_q (4) where, VC_d and VC_q represent the control signals of the d and q axes in the RRF respectively, while ωff is the nominal grid frequency (rad/s). Substituting Vinv_d and Vinv_q from (4), in (3), the following decoupled system is deduced: VC_d = (R1 + R 2 )id + S(L1 + L2 )id ⟦ ⟧ VC_q = (R1 + R 2 )iq + S(L1 + L2 )iq (5) Therefore, the transfer function of the decoupled system described in (5) can be derived as Vgrid_d i*d id - ɛd PI VC_d + Power Stage Vdc Vinv_d + i Five-Level Diode Clamped Inverter (R1+R2) (L1+L2) Vgrid Vinv ωff (L1+L2) SPWM -ωff (L1+L2) iq PI VC_q Vinv_q + iα α-β - iq dq iβ α-β Vinv_β (6) in which, the time constant TS = (L1 + L2 )⁄(R1 + R 2 ) and K S = 1⁄(R1 + R 2 ) It should be noted that, since id and iq respond to VC_d and VC_q through a simple first-order transfer function, the control rule of (4) is completed by defining feedback loops with simple first order PI controllers. Based on (4), the structural diagram of the current regulator based on PI controllers is shown in Fig. 4 in which the voltage feedforward and the coupling terms are shown. The structural diagram of the test system along with the conventional single-phase dq controller is shown in Fig. 5, in which the orthogonal imaginary current component (iβ) is obtained by phase shifting the real component (iα) by a quarter of the fundamental period. The measured and the shifted current components are then employed in a (αβ–dq) transformation, and a conventional dq current controller shown in Fig. 4, with decoupling strategy is implemented. The resulting output control signals in the dq-frame (Vinv_d) & (Vinv_q) are transferred back to the α-β frame to obtain the corresponding ac control signals. Usually, the α component (Vinv_α) of the control signal is employed and fed into the pulsewidth modulation (PWM) modulator, while the β component (Vinv_β) is discarded. Note that, a single-phase phase-locked loop (PLL) based on second-order generalized integrator (SOGI) [16] is used, to generate the reference phase angle (θPLL) for the (αβ–dq) and the (dq-αβ) transformation as shown in Fig. 5. This approach is relatively simple and straightforward; however, phase shifting the current to create the required orthogonal signal tends to deteriorate the dynamic response of the system, as the real and fictive components do not run in the same time. Therefore, any transient in the real physical component is also experienced in the fictive orthogonal component a quarter of fundamental period later. Since the reference current is subject to frequent step changes, delaying the current deteriorates the dynamics of the system and makes it slower and oscillatory. To avoid this shortcoming, the simplified dq current control strategy is proposed in the coming section. Vinv_α sT 4 e ωPLL ƟPLL dq i*q Vgrid_q Fig. 4 Structural diagram of the conventional dq current controller 1 KS GDe (S) = = (R1 + R 2 ) + S(L1 + L2 ) 1 + STS ɛq Vinv_d ɛq + Vinv_q - - id ɛd dq Current controller SOGI PLL i*q i*d Vgrid Control Circuit Fig. 5 Test system along with the conventional dq current controller IV. SIMPLIFIED SINGLE-PHASE DQ CURRENT CONTROL The control strategy of the previous section necessitates a αβ to dq transformation, which, in single-phase systems, is not viable because there is only one phase variable available, while this transformation needs at least two orthogonal variables. Therefore, to make the aforementioned current-control strategy applicable to single-phase systems, fictitious component orthogonal to the existing physical one should be created by phase shifting the measured real signal such that the physical and fictitious signals together form the stationary or αβ frame. The introducing of such delay in the system deteriorates the dynamic of the system, which becomes slower and oscillatory. To overcome the aforementioned shortcoming, a simplified current controller scheme as depicted in Fig 6, is proposed where the difference between the reference and the measured currents iα∗ and iα in the stationary reference frame is used to generate the steady state error signals in the dq frame ƹd and ƹq .It is worth noting that the proposed scheme does not require any artificial orthogonal generator by simply assuming that i∗β = iβ at the steady state [17]. The relationship between stationary and rotating frames is given by (7) which defines the transformation from stationary to rotating frame, and (8) from rotating frame to stationary frame cos() sin() (7) [ ]=[ ].[ ] −sin() cos() cos() −sin() [ ] = [ ].[ ] sin() cos() (8) In single-phase systems, however, this transformation cannot be applied directly because there is only one variable available iα . This drawback however, can be solved without creating a fictitious current iβ as following Using the estimated phase angle generated by the PLL (θPLL), two orthogonal reference currents can be generated as ∗ i∗α = i∗ = Im cos(θPLL ) (9) Power Stage Five-Level Diode Clamped Inverter Vdc Vgrid_d i ɛd (R1+R2) (L1+L2) id iq iα dq Current controller ɛα - Vinv_d ωff (L1+L2) -ωff (L1+L2) i*q i*α i*d - ɛq PI cos Vinv_d Vinv_q ɛq -sin + + SPWM ɛd VC_d i*d - Vgrid Vinv PI ƟPLL dq SOGI PLL α-β VC_q + + Vinv_q Vgrid_q Vgrid Fig. 7 Structural diagram of the simplified dq current controller Vinv_α Vinv_β Control Circuit Fig. 6 Test system along with the simplified dq current controller ∗ i∗β = Im sin(θPLL ) (10) Applying the dq transformation in (7) with (θ=θPLL), two reference currents in the dq frame can be found as (11) and (12). ∗ ( 2 ∗ (11) i∗d = Im θPLL + 2 θPLL ) = Im ∗ i∗q = Im (−cos θPLL sin θPLL + cos θPLL sin θPLL ) = 0 The measured real current iα is defined as in (13) (13) iα = i = Im cos(θPLL ) While, the synthesized current iβ is defined as the reference current i∗β as in (14) ∗ iβ = i∗β = Im sin(θPLL ) (14) Applying the assumption that i∗β = iβ at the steady state, a simplification will result in the current control loop system as follows The steady state errors in the dq frame ƹd and ƹq are given by (15) and (16) ƹd = id∗ − id ƹd = [i∗α cos(θPLL ) + i∗ sin(θPLL )] − [iα cos(θPLL ) + iβ sin(θPLL )] ƹd = (i∗α −iα )cos(θPLL ) + (i∗ −iβ )sin(θPLL ) ƹd = (i∗α −iα )cos(θPLL ) ⟦ ⟧ (15) ƹq = i∗q − iq ƹq = ⟦ [−i∗α sin(θPLL ) + i∗ cos(θPLL )] − [−iα sin(θPLL ) + iβ cos(θPLL )] ƹq = −(i∗α −iα )sin(θPLL ) + (i∗ −iβ )cos(θPLL ) ƹq = −(i∗α −iα )sin(θPLL ) ⟧ Fig. 8. Experimental setup (12) (16) From (15) and (16), both ƹd and ƹq errors were calculated based on the real current error ƹα and not on the knowledge of dq currents as in the conventional dq current controller. The dq current controller based on (15) and (16) which used in the simplified approach is shown in Fig 7. V. PERFORMANCE EVALUATION The purpose of this section is to experimentally evaluate the performance of the simplified-dq current control scheme and also to compare it with that of the conventional delay-based control strategy. The presented test results show that the proposed simplified dq controller has the following features: 1) It is capable of tracking the reference signals with a zero steady-state error within few milliseconds; 2) has fast dynamics; and 3) contrary to the conventional approach, provides practically decoupled d and q current axes. Adopting the test system in Fig. 1 and the corresponding parameters in Table I, the single-phase experimental setup shown in Fig. 8 is implemented based on that. To implement the control strategies, a Texas Instrument TMS320F28335 digital signal processor (DSP) is used. The control algorithms are written in C-code and are developed using Code Composer Studio CCS5.5 software. The sampling frequency has been fixed to 20 kHz with a dead-time period of 1 μs. The PI controller gains of the control algorithms are set as follows: kp =1300 and ki=20000 which are tuned using trial-error method to obtain the optimal dynamic performance. To evaluate the performance of both current regulation schemes experimentally, a reference tracking test is conducted for each control strategy by stepping –up and stepping-down in the d-axis reference value while that of the q-axis is kept constant at zero. 1.5 1) Conventional Delay-Based Controller Adopting the control strategy of Fig. 5, the inverter initially injects zero current. At time instant t 0.1s, the reference value of the d-axis steps up to 0.5 p.u. Moreover, at time instant t O.2s, the reference value of d-axis steps up to Ip.u, and at t 0.3s it steps down to zero, while the reference value of q-axis is kept constant at zero p.u during the whole process. As shown in Fig. 9(a), upon each step change in the d-axis references, the controller tries to regulate the a current at the desired value; however, due to excessive transients, there is an overshoot in the regulated current. Moreover, as shown in Fig. 9(b), the d axis of the current experiences non-negligible transients for about a cycle due to the delay used in the controller. Therefore, it takes about 20ms for the d component of the current to track the reference value with a zero steady-state error. The conducted study demonstrates that the delay-based control scheme suffers from excessive transients subsequent to any step change in its d -axis. In Fig. 9(c), although the reference value of the q-axis is kept constant, however, subsequent to each step change in the d-axis, the q-axis also experiences a non-negligible transient, which verifies that the conventional current control strategy suffers from coupled axes. Note that in the steady state, the controller can regulate the current with zero steady-state error. Moreover, the measured total harmonic distortion (THD) of the grid current in the steady-state is 3.67% as shown in Fig 10. = = = �= I�1111I� 0 .- 0.5 -0 II -1 .5 = It 0.1 o a ill "( 0.2 0.4 0.3 (a) -� i/ -� id 1.5 � I) s .e, ._u 1\ 0.5 � o if 0.2 0.1 0.4 0.3 (b) 0.6 0.4 0.2 1\ 0 Iv'r<! � � r-'!'PTI 'f 'f 'I -0.2 0.2 0.1 0.4 0.3 (c) Time(s) = = )11 i -1 cr- In order to fairly compare the performance of both methods, exactly the same test as that of the previous section is conducted for the simplified-based approach, and the same PI controllers with the same gains are utilized. The inverter of Fig. 6 initially injects zero current. Keeping the q-axis reference value constant at zero, at time instant t O.1s, the reference value of the d-axis steps up to 0.5 p.u. Moreover, at time instant t O.2s, the reference value of the d- axis is changed to 1p.u. and finally at t 0.3s, the d-axis reference value is set back to zero p.u. Subsequent to each change in the d axis reference value, the controller regulates the current at the desired level in almost 2ms with zero steady-state error as shown in Fig. Il(a). Moreover, as shown in Fig. I I(b), the d axis of the currents tracks the reference value in almost 2 ms with zero steady-state error. However, contrary to the conventional controller, upon each step change in the d-axis, the q-component of the current experiences very short and negligible transients as shown in Fig. ll(c) . It should be noted that similar to the delay-based controller, the controller is capable of regulating the current with zero steady-state error. Moreover, the measured THD of the grid current in the steady state is 3.76% as shown in Fig 12, which verifies that the performances of the controllers in the steady state are quite similar. However, the conducted studies demonstrate that the proposed simplified control scheme has superior dynamic performance compared with that of the conventional delay based approach. 0.5 :::J ci. S -e, 2) Simplified-Based Controller -� -- .- Fig.9. Experimental results for the transient response of the conventional dq controller during step changes in d-axis. (a) The grid current and its emulated orthogonal component. (b) The d-axis of the currents. (c) The q-axis of the currents. Fundamental (50Hz) = 5.0!9 , 1HD= 3.67% 2 o � 0 II I 10 ,1. 1 I ,11.1 . . . . h, . 11 . . 30 20 . Harmoruc order ,I. ,,II •• 40 .. 50 Fig.!O. Harmonic spectrum of the grid current using the conventional scheme 1.5 1 i i *, i (p.u) 0.5 0 -0.5 based on experiments, and it was compared with that of conventional approach. The conducted studies conclude that the proposed method is characterized by the following. 1) It can maintain the stability of the system and track reference values with zero steady error. 2) It is much faster than the conventional approach. 3) It has superior axis decoupling capability compared to the conventional approach. ACKNOWLEDGMENT A. M. Mnider would like to acknowledge the Libyan Ministry of Higher Education and Scientific Research, and the University of Elmergib, for sponsoring his Ph.D. project. i * -1 -1.5 0 0.1 0.2 (a) 0.3 0.4 1.5 i * d i id*, i d (p.u) 1 REFERENCES d [1] 0.5 [2] 0 -0.5 0 0.1 0.2 (b) 0.3 [3] 0.4 0.6 [4] iq (p.u) 0.4 0.2 [5] 0 -0.2 -0.4 [6] 0 0.05 0.1 0.15 0.2 0.25 (c) Time (s) 0.3 0.35 0.4 [7] Fig.11. Experimental results for the transient response of the simplified dq controller during step changes in d-axis. (a) The grid current and its reference. (b) The d-axis of the current. (c) The q-axis of the current. Mag (% of Fundamental) Fundamental (50Hz) = 5.019 , THD= 3.76% [8] [9] 2 [10] 1.5 [11] 1 0.5 [12] 0 0 10 20 30 Harmonic order 40 50 [13] Fig.12. Harmonic spectrum of the grid current using the simplified scheme VI. CONCLUSION A simplified dq current control strategy for single-phase VSCs has been presented. Similar to the conventional current control strategy, the proposed method can regulate the dq components of the current at desired reference levels while it does not impose excessive structural complexity compared to the conventional approach and can be easily implemented in digital environments. In addition, the proposed strategy does not require orthogonal-current component to be generated, which results in fast and non-oscillatory dynamics. The performance of the proposed control strategy was evaluated [14] [15] [16] [17] S.B. Kjaer, J.K. Pedersen, and F. Blaabjerg, ‘‘A review of single-phase grid-connected inverters for photovoltaic modules,’’ IEEE Trans. Ind. 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