See discussions, stats, and author profiles for this publication at: https://www.researchgate.net/publication/224213848 Single-phase grid-tie inverter control using DQ transform for active and reactive load power compensation Conference Paper · January 2011 DOI: 10.1109/PECON.2010.5697632 · Source: IEEE Xplore CITATIONS READS 75 6,808 4 authors, including: Ehab El-Saadany L. El Chaar University of Waterloo General Electric 456 PUBLICATIONS 18,350 CITATIONS 39 PUBLICATIONS 838 CITATIONS SEE PROFILE L. A. Lamont Mott MacDonald Group 32 PUBLICATIONS 774 CITATIONS SEE PROFILE Some of the authors of this publication are also working on these related projects: Solving the nonconvex economic dispatch problem View project Short Term Load Forecasting View project All content following this page was uploaded by Ehab El-Saadany on 14 September 2015. The user has requested enhancement of the downloaded file. SEE PROFILE 2010 IEEE International Conference on Power and Energy (PECon2010), Nov 29 - Dec 1, 2010, Kuala Lumpur, Malaysia Single-Phase Grid-Tie Inverter Control Using DQ Transform for Active and Reactive Load Power Compensation B. Crowhurst, E.F. El-Saadany, L. El Chaar and L.A. Lamont Abstract—This paper presents a current control for single phase grid connected inverters. The method allows for inverter active and reactive power control. The method uses the Direct-Quadrature (DQ) synchronous reference frame transformation for single-phase converters. This method transforms an orthogonal pair consisting of the inverter output current and a time shifted version of this current from a stationary frame to a rotating frame synchronous to the fundamental output frequency. Alternatives to using the time shifted current are discussed. The steady state current components in this rotating DQ frame are DC values and thus PI control methods can be used with zero error. A household scale grid-tie inverter is used as an example application of this method. A Simulink simulation model and results where the inverter output is controlled to match a local load's active and reactive power demand is presented. Index Terms— Pulse width modulated inverters, Reactive power control, Power electronics. I. INTRODUCTION D istributed Generation (DG) systems offer secure and implemented for the interconnection of certain renewable energy resources to the grid [3]. Many DG systems are installed at residential or rural locations which are often only served by a single phase of the electrical distribution system. In these cases a single-phase inverter is required. The FACTS systems previously mentioned are typically larger scale three-phase systems, therefore if we wish to implement equivalent power electronic converter control algorithms in single-phase inverters they must either be modified to suit that application or new algorithms must be developed. Some three-phase compensation algorithms also suffer from the inability to generate sinusoidal current waveforms in the utility side when utility voltages and/or currents are not balanced [4] which could be a motive to further investigate single-phase compensation techniques. In the past, sine wave inverters used open loop feed- forward control with output RMS voltage feedback to regulate the magnitude for standalone applications. Grid-tie systems require a more instantaneous control system to achieve lower Total Harmonic Distortion, and improved disturbance rejection [5]. The application of time-invariant system theory to control AC values is problematic and typically causes significant steady state error in both amplitude and phase [6] referred to as “following error” [5] which must be reduced by operating the power converter at a very high switching frequency. A common theory employed in active filters and reactive power compensators that addresses this problem is Synchronous Reference Frame Theory (DQ). The advantage of this theory is that the fundamental frequency components of an AC signal are mapped to DC values. It has been shown to be effective in the presence of harmonic distortion. One of the inherent DQ transformation properties is that it can only be applied to three-phase systems [7]. In single phase systems, the instantaneous power contains sinusoidal components at twice the grid frequency which complicates the grid connected system [8]. This paper investigates a new method that overcomes the problem listed above and applies the DQ transform to singlephase systems for the purpose of controlling active and reactive power injection of a single phase inverter. diversified energy options, increase generation and transmission efficiency, reduce greenhouse gas emissions, improve power quality and system stability, cut energy costs and capital expenditure, and alleviate the bottleneck caused by distribution lines [1]. DG systems that are based on renewable energy sources are intermittent source of energy and power electronics converters are needed as interface to connect them to the grid. Hence, Power electronic converters or inverters play an important role in these systems particularly in cases where power is generated in a format incompatible with the distribution system such as the DC output of batteries, photovoltaic modules and fuel-cells or the variable frequency AC output of variable speed wind turbines. Moreover, such interface must be outfitted with control strategies to maximize the power extracted from the source and transferred to the grid without violating the grid standards of quality [2]. Power electronic converters are also used for Active Power Filtration and Power Factor Correction in Flexible AC Transmission System (FACTS) systems such as the Static II. ALTERNATIVE CONTROL ALGORITHMS Compensator (STATCOM) and the Unified Power Flow Controller (UPFC). DG systems, which employ the same A few alternative power electronic converter control converters, are thus also capable of performing these functions. methods that were reviewed but not implemented in this FACTS technologies are extremely beneficial when study are briefly described here. B. Crowhurst and E.F. El-Saadany are with the Electrical and Computer Engineering department, the University of Waterloo, Waterloo, Ontario Canada. (e-mail: ehab@ uwaterloo.ca). L. El-Chaar and L. A. lamont are with the petroleum Institute, Abu Dhabi, UAE. (e-mail: lelchaar@pi.ac.ae; llamont@pi.ac.ae). 978-1-4244-8946-6/10/$26.00 ©2010 IEEE A. Space Vector Transformation The space vector transformation is another method used in controlling power electronic converters. This method overcomes a weakness of the DQ transform that there is a 489 strong coupling between the transformed coordinates (dq) [9]. IV. POWER GRID AND LOAD MODELS B. Fuzzy Logic Control Fuzzy logic controllers are another method of control for power electronic converters. Fuzzy logic control is non-linear and adaptive in nature, has robust performance under parameter variation and load disturbance, and its implementation does not require a sophisticated microcontroller such as a DSP [10]. External connections to the inverter include a simple model of the connection to the power grid and a similarly simple load model to represent a load at the site of the DG to be compensated. III. THE SINGLE PHASE INVERTER The single-phase inverter considered in this paper consists of an IGBT based H-bridge and an LCL output filter. The combination of the H-Bridge and LCL filter implemented as a Simulink model is shown in Fig. 1. The inverter output voltage and current measurement “sensors” are also shown in this model. These signals are fed back to the control system which generates the pulses or IGBT gate control signals. The signals are generated such that in each leg of the bridge the bottom transistor gate control is the inverse of the top to avoid creating a low impedance path between the DC source terminals. A. Power Grid Model An infinite bus model of the power grid was used as shown in Figure 2. This model consists of an AC voltage source and a small RL branch component. The inclusion of the RL branch allows for voltage variation at the point of common coupling (PCC) dependent on load and inverter power injection. Fig. 2. Power Grid Model B. Load Model The local load was modeled as two separate loads shown in Fig. 3, one fixed and the other switched. This allows for the observation of inverter response to changes in load. The constant load is purely resistive whereas the switched load is partially inductive. The switching of this load thus changes both the active and reactive power demand of the total load. The switch used is the ideal switch with infinite snubber capacitance. Fig. 1. Inverter circuit with H-Bridge, and LCL filter A. H-Bridge The H-bridge is also known as a full bridge. This is a simple circuit topology with low component count which leads to low cost and high efficiency [1]. This topology was chosen over a half bridge which consists of only one pair of switches because a lower DC voltage source is required. Half bridge inverters require twice the DC voltage and two capacitors in series are required to provide the neutral output. The H-Bridge is modeled using the Universal Bridge Simulink model block with two legs. The gates of the H-bridge are driven by a Pulse Width Modulation (PWM) generator discussed in section V. The average amplitude of the voltage output of the H-Bridge (VAB in figure 1) over a switching cycle is directly proportional to the commanded duty cycle of the inverter (±100%) and the amplitude of the DC bus [5]. B. Output Filter Inverters require an output filter to limit the high frequency current ripple. There is a tradeoff between filter component size and switching frequency. The former increase inverter size and cost whereas the later increases switching power loss. The output filter selected for simulation is the LCL filter. This filter has gained popularity due to its smaller size however it presents potential stability problems in control [11]. Fig. 3. Switched load used in simulation V. DQ SINGLE-PHASE TRANSFORM A. DQ transform As previously mentioned, the single-phase DQ transform is an important element in the control circuit of the inverter system. The DQ transform is an example of a state space transform. The state of a system at any instant t0 is the smallest set of variables which is sufficient to determine the behavior of the system for all time t > t0 when the inputs to the system are known [12]. This system state can be specified in many different ways which means the state variable are not unique. Alternative state representations can be obtained via linear transformation. The three phase DQ transform can be implemented directly from the ABC time varying signal space to the DQ space with the linear transformation in (1) [3]. 490 ⎡vd ⎤ ⎡sin ωt sin (θ − 2π 3) sin (θ + 2π 3) ⎤ ⎡va ⎤ ⎢ ⎥ 2⎢ ⎥⎢ ⎥ = v ⎢ q ⎥ 3 ⎢cos ωt cos(θ − 2π 3) cos(θ + 2π 3) ⎥ ⎢vb ⎥ ⎢v ⎥ ⎢⎣1 2 ⎥⎦ ⎢⎣vc ⎥⎦ 12 12 ⎣ 0⎦ where θ=ωt, and ω = electric system frequency (1) This transform can also be accomplished by first transforming to the αβ space. The αβ space is used in PQ theory and three phase voltages and currents are represented as a single vector which rotates about the stationary orthogonal axes thus α and β projections of sinusoidal abc values are themselves sinusoidal. The transformation from αβ space to DQ can then be achieved by effectively rotating the αβ frame at the fundamental frequency as shown in Figure 4. In this rotating frame the resulting D and Q vectors will be constant for sinusoidal signals at the fundamental frequency. What this indicates is that the transformation to the synchronous frame requires two orthogonal components, the equivalent of the αβ components of three-phase systems. In other words, to achieve an orthogonal plane, the projections of two or more variables are necessary [3]. This is a problem in single-phase systems. Fig. 4. Rotating DQ Frame In order to get around this requirement several methods have been suggested to generate the missing orthogonal vector. One method involves using the capacitor current of an output LC filter as in [5]. This current is 90 degrees out of phase with the output voltage. Another is to use a quarter period time delayed version of the signal as the orthogonal vector as in [7]. This is the method that is examined here. If we consider a current signal to be transformed to the DQ frame using the time delay method we can refer to the actual current signal as the real current, ir, and the time delayed version the imaginary current, ii, where the real current corresponds to the α and the imaginary to the β. Equations for ideal sinusoidal versions of these currents are given in (2.a) and (2.b) i r = A sin (ωt + δ ) (2.a) ii = A sin (ωt + δ − π 2 ) = − A cos (ωt + δ ) (2.b) The linear transform corresponding to the rotation of the αβ frame is given in (3). Note that this transform is different if cos(ωt) is considered the reference (i.e. replace sin(ωt) with cos(ωt) and cos(ωt) with -sin(ωt)). ⎡I d ⎤ ⎡sin ωt I dq = ⎢ ⎥ = TI ri = ⎢ ⎣cos ωt ⎣I q ⎦ − cos ωt ⎤ ⎡ I r ⎤ ⎥⎢ ⎥ sin ωt ⎦ ⎣ I i ⎦ (3) Equations (4) and (5) gives the resulting id and iq values for this ideal sinusoidal case. These values are constant DC values for fixed amplitude and phase input as previously described. i d = A sin (ωt + δ ) sin (ωt ) + A sin (ωt + δ − π 2)(− cos(ωt )) = A cos δ i q = A sin (ωt + δ ) cos(ωt ) + A sin (ωt + δ − π 2)(sin (ωt )) = A sin δ (4) (5) It should be noted here that DC components of the original signal are mapped to AC signals at the fundamental frequency in the rotating DQ frame. Thus the presence of a DC component will result in control system oscillation and make zero steady state error impossible. In order to transform back to the real and imaginary frame the inverse transform is applied as shown in (6). Again, if cos(ωt) is considered the reference this transform must be modified (i.e. replace sin(ωt) with cos(ωt) and cos(ωt) with-sin(ωt)). ⎡I r ⎤ ⎡sin ωt I ri = ⎢ ⎥ = T −1 I dq = ⎢ I ⎣cos ωt ⎣ i⎦ cos ωt ⎤ ⎡ I d ⎤ ⎥⎢ ⎥ − sin ωt ⎦ ⎣ I q ⎦ (6) B. Simplified technique Other techniques for implementing the single-phase DQ transform avoid the need for a second orthogonal variable by using notch filters tuned at twice the line frequency [13][6]. In this case the transform is applied with i q = 0. The resulting derivation is shown in equation series (7) and (8). It can be seen that similar results to the original are obtained however a double line frequency sinusoid is added. It is this double frequency component that must be filtered in order to obtain the DC quantities desired for zero steady state error. This simplified technique was simulated without the use of notch filters and because of the DC component in the D and Q components, zero steady state error was possible. This was the method that was described for the in-class presentation. The results of this simulation are not presented in this report as it was decided to implement the time delay version instead. i d = A sin (ωt + δ ) sin (ωt ) [ ] = A (cos δ ) sin 2 (ωt ) + (sin δ )(sin ωt ) cos(ωt ) (7) A ( cos δ ) − A [sin δ sin (2ωt ) − (cos δ ) cos(2ωt )] 2 2 i q = A sin (ωt + δ )(− cos(ωt )) = [ ] = − A (sin δ ) cos 2 (ωt ) + (cos δ )(sin ωt ) cos(ωt ) −A = (sin δ ) − A [sin δ sin (2ωt ) + (cos δ ) cos(2ωt )] 2 2 (8) C. DQ Transform Circuit The simulation model used to implement the single-phase DQ transform is shown in figure 5. The generation of the sine and cosine signals involved is discussed in the next section. 491 The transport delay achieves the π/2 phase shift. This transport delay can easily be implemented using a digital controller shifted queue [6]. Fig. 5. Time Delay Single Phase DQ Transform The inverse transform is implemented at the load following control level discussed in the next section. Only the “real” signal needs to be generated in this case. VI. LOAD FOLLOWING CONTROL A. Control Circuit The load following control circuit is shown in Figure 6. This circuit includes sine and cosine reference generation, two DQ transform blocks, D and Q PI controllers and the inverse DQ transformation. commanded current and the inverter output current is subtracted to generate D and Q error signals which are then fed into PI controllers. The PI controllers adjust their output so as to eliminate this error. The PI controller output is then transformed back to the stationary frame using only the “real” portion of the inverse transform shown in (6). This output is used to command the PWM generator which controls the output voltage of the Hbridge. Because the simulated example inverter control attempts to match both the active and reactive components of the load current the error signal could be generated in the stationary frame before the DQ transform and thus only one DQ transform block would be required. This topology was not used because the ability to control active power independent of load is a more likely scenario for a grid-tie inverter. In this case only the Q component of the load current would be required to achieve reactive power compensation. D. Pulse Width Modulation Pulse Generator The control loop is completed with the PWM pulse generator which can be seen in the top level simulation schematic shown in Figure 7. The output of the load following control circuit is used to modulate the PWM gate signals. A modulation process example is shown in Figure 8. By changing the amplitude and phase of the command signal to the PWM generator the magnitude and phase of the inverter output voltage can be controlled. The load following current controller thus adjusts the inverter output voltage such that the inverter output current matches the load current in both phase and magnitude. The Simulink PWM generator block was used to generate the gate pulses as it is easily configured to generate the appropriate output for an H-bridge. Fig. 6. Load Following Control Circuit B. Reference current generation This circuit uses a Phase Lock Loop (PLL) to generate the sine and cosine references from the PCC voltage. The load current is determined by subtracting the output current from the grid current. This technique was used because measuring load current directly in a residential situation may be difficult as grid-tie inverter typically connect directly to the main household breaker panel making measurement of load current independent of inverter current infeasible. This indirect measurement technique could cause controller stability problems particularly under zero load conditions and thus a different connection location for the inverter output that would allow for direct measurement of the load current would be preferred. C. PI Control in the DQ Reference Frame Both the inverter output current and the load current are transformed to the DQ frame. The load current is used as the 492 Fig. 7. Top level simulation circuit Fig. 8. Pulse Width Modulation VII. SIMULATION RESULTS A time domain simulation of the circuit shown in Figure 7 and previously described was carried out in the MATLAB Simulink environment. The total simulation time was 0.45 seconds and the inductive load was switched in at 0.15 seconds and out at 0.3 seconds. The simulation circuit parameters are shown in Table I. reactive power are increased. This can be seen as the current magnitude increases and phase shift to compensate. After the second switching event the current returns to its original magnitude and phase. A. H-Bridge Output Voltage The output voltage and current of the H-bridge while operating at unity power factor are shown in Figure 9. Note that the voltage and current are out of phase due to the fact that the inverter must compensate for the output filter in order to achieve unity power factor at the filter output. C. Active and Reactive Power TABLE I SIMULATION CIRCUIT PARAMETERS PWM Switching frequency Voltage Universal Bridge Parameters Inductance 1 Inductance 2 Capacitance Resistance Inductance AC peak voltage Constant load active power Constant load reactive power Switched load active power Switched load reactive power Proportional Control gain Integral Control gain DQ transform Transport delay Switched load turn on time Switched load turn off time Simulation length f pwm V DC Various L F1 L F2 CF R line L line V AC P load1 Q load1 P load2 Q load2 Kp Ki t delay t on t off t simulation Fig. 10. Inverter PCC Voltage and Output Current 5 kHz DC 500V Default Filter 4 mH Filter 2 mH Filter 10 ℜ F Feeder 0.1 ∂ Feeder 0.69 mH Grid 340 V 5 kW 0 kVAR 5 kW 5 kVAR 0.005 0.75 1 / 240s 0.15 s 0.30 s 0.45 s Fig. 9. H-Bridge Output voltage and current when operating at Unity PF B. Voltage and Current The inverter output voltage at the PCC and the inverter current throughout the simulation are shown in Figure 10. Before the first switching event the load is purely resistive and thus the inverter output current is in phase with the voltage. At the first switching event the load power and The inverter output active and reactive power was measured with a Simulink PQ measurement block. The output of this block is shown in Figure 11. It can be seen that the inverter output active and reactive power track the load with zero steady state error. Overshoot could be eliminated by adjusting the PI control gains however this may result in longer durations to achieve zero error. Fig. 11. Inverter Output Active and Reactive Power D. DQ Transform of Inverter Output Current The inverter output current, time shifted output current, D component of the output current, and Q component of the output current are shown over the duration of the simulation in Figure 12. The D and Q components using the time shift DQ transform are seen to be DC values in the steady state. It can be seen that using the time shifted current in the DQ transform contributes to D and Q error during magnitude and phase transitions. This is because the time shifted current is not completely orthogonal to the current during these periods. E. DQ Error and PI Control Output The D error, D PI controller output, Q error, and Q PI controller output are shown in Figure 13. The high frequency current spikes resulting from the switching of the inductive load are seen in the Q error and also in the Q PI control output. The DQ transform can also be applied in higher order harmonic cancellation as discussed in [7]. 493 VIII. CONCLUSIONS This paper presented a description of the synchronous rotating frame current control for single-phase inverters. The method uses a time delayed version of the original output current signal as the orthogonal variable used in the linear transform to the synchronous frame. An example application of this control for a household scale grid-tie inverter system was described. Simulink models of this application were presented which demonstrate local load following using switched inductive load. Time domain simulation results of this example have been presented. These results demonstrated the active and reactive load power demand compensation capabilities of this control method. IX. REFERENCES [1] Fig. 12. Inverter Output Current, Time Shifted Inverter Output Current, D component of output current, Q component of Output current. Y.Xue, L. Chang, S.B. Kjær, J. Bordonau, and Toshihisa Shimizu, “Topologies of Single-Phase Inverters for Small Distributed Power Generators: An Overview ,” IEEE Transactions on Power Electronics, vol. 19, pp. 1305-1314,.Sept. 2004 [2] F. Blaabjerg, R. Teodorescu, M. 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Dame, “A grid Simulator with Control of Single-Phase Power Converters in D-Q Rotating Frame,” Power Electronics Specialists Conference, Vol. 3, pp. 14311436, 2002. [7] M. Gonzales, V. Cardenas and F. Pazos, “DQ Transformation Development for Single-Phase Systems to Compensate Harmonic Distortion and Reactive Power,” 9th IEEE International Power Electronics Congress, pp. 177-182, 2004. [8] S. Silva, B. Loopes, B. Filho, R. Campana, and W. Bosventura, “Performance evaluation of pll algorithm for single-phase grid connected systems,” Industry Applications Conference, 2004, 39th IAS Annual Meeting, Conference Record of the 2004 IEEE, vol. 4, pp. 2259-2263, Oct. 2004 [9] J. M. Aller, A. Bueno, T. Paga, “Power system analysis using spacevector transformation,” IEEE Transactions on Power Systems., Vol. 17 , pp. 957-965, Nov. 2002. Fig. 13. D error PI control input, D PI control output, Q error PI control input, Q PI control output. [10] S. Premrudeepreechacharn, T.Poapornsawan, “Fuzzy logic control of predictive current control fro grid-connected single phase inverter,” IEEE Photovoltaic Socialists Conference, pp. 1715-1718, 2000. [11] H.R. Karshenas, H. Saghafi, “Basic Criteria in Designing LCL Filters for Grid Connected Converters,” IEEE International Symposium on Industrial Electronics, Vol. 3, pp. 1996-2000, 2006. [12] B.P. Lathi, Linear Systems and Signals, Carmichael, CA:Berkely Press, p. 595, 1992. [13] U.A. Miranda and L.G.B. Rolin, “A DQ Synchronous Reference Frame Current Control for Single-Phase Converters,” Power Electronics Specialists Conference, pp. 1377-1381, 2005. Fig. 14. Combination of D and Q PID output after the inverse DQ transform to form PWM modulation signal 494 View publication stats