Control of Grid-Connected Voltage-Source Converters: The Relationship Between Direct-Power Control and Vector-Current Control

This article discusses the mathematical relationship between gridvoltage-modulated (GVM) direct-power control (DPC) and vector-current control (VCC) for three-phase voltage-source converters (VSCs). GVM-DPC is equivalent to VCC at the steady state, and it presents a superior transient performance by removing the need for the phase-locked loop (PLL). This means that GVMDPC solves the disadvantage of conventional DPC, such as poor steady-state performance. Moreover, GVM-DPC will reduce the computational burden in comparison with VCC due to the absence of Park transformation and PLL. Consequently, we can expect that the GVM-DPC method will be capable of plug-and-play for VSCs. Finally, the experimental results match the theoretical expectations closely.

The conventional control of grid-connected VSC is using vector current controller (VCC), which is designed in a synchronously rotating reference frame and a proportional integral (PI) control with a decoupling term is applied to control d-q axes currents independently [8]. The key point is that it uses coordinate transformation to transform the AC components to the DC ones and thus the linear PI controller can be used. Moreover, the VSC system is changed to a linear time invariant (LTI) system in the rotating reference frame, which means that the system can be readily designed and analyzed through linear control techniques [9]- [11]. The main disadvantage of the conventional VCC is that it suffers from a slow transient response, since it uses a phaselocked loop (PLL) system for the coordinate transformation. In addition, the interaction between the PLL system and the current loop control system causes harmonic problem in a weak grid, even destabilizes the system [12]- [15].
An alternative control strategy has emerged for induction machine drives, which is the direct torque control (DTC) [16], [17]. It has a simple structure in comparison with the VCC. To achieve the constant switching frequency, a modified DTC strategy based on space vector modulation was developed [18], [19]. Based on the DTC strategy concept, the direct power control (DPC) was developed for grid-connected VSCs [20]- [22]. In [20] and [21], a look-up-table (LUT)-DPC was developed, where the proper switching states are selected from a predefined optimal switching table based on the instantaneous errors of active and reactive powers and the angular position of the VSC terminal voltage. However, the variable switching frequency is resulted with a broadband harmonic spectra, which complicates the design of line filters. To solve such a problem, various DPC algorithms were developed for a constant switching frequency [23], [24]. In addition, for a robust control, the sliding mode control (SMC) based DPC [25] and passivity-based control (PBC)-DPC [26] have been reported to obtain a faster transient response than that of the PI controller and a better robustness to parameter uncertainties than that of the LUT-DPC. However, there are still large ripples in both active and reactive powers. Another control strategy, i.e. model predictive control (MPC)-DPC performs a good closed loop behavior with consideration of the system constraints [27], [28]. However, an incorrect voltage sequence selection could affect its performance [29]. Recently, a grid voltage modulated-DPC (GVM-DPC) was introduced in [30] to design a robust but simple control law for not only the convergence rate of the instantaneous active and reactive powers, but also the steady-state performance of VSC, especially reducing the power ripples and total harmonics distortion (THD) of the output current in comparison with the SMC-DPC and PBC-DPC. Another advantage of the GVM-DPC is that it converts the original nonlinear system into an LTI one, which can easily be analyzed and designed by using the conventional linear control techniques [31]. In [32], Gui et al. proposed a novel vector current control method based on the concept of the GVM-DPC.
Although the GVM-DPC shows a better steady-state performance compared with the SMC-DPC and PBC-DPC, there is no analysis to explain how and why it can obtain a better performance in detail. This article is firstly to present a main feature between the GVM-DPC and the conventional VCC designed in d-q frame for three-phase VSC in detail. We mathematically prove that the DPC model of VSC is equal to the current model in the d-q frame, which reveals that the GVM-DPC is equivalent to the VCC at the steady state, yet presents superior transient performance by removing the need of the PLL. That means, the GVM-DPC method could achieve the same property of steady-state performance as the VCC but better tracking performance, since there is no PLL. In addition, the GVM-DPC method will reduce computational burden in comparison with the VCC since there is no Park transformation or a PLL system. Consequently, it can be expected that the GVM-DPC method could be applied to various applications and be modified to solve various industry issues.

II. GRID VOLTAGE MODULATED DIRECT POWER CONTROL
In this section, firstly, a model of the VSC in the stationary reference frame is described.
Then, the DPC modeling of VSC is briefly introduced. For the VSC system, the GVM-DPC is designed to make it be an LTI multi-input-multi-output (MIMO) system. The DC side could be connected to renewable energy sources or energy storage systems with a capacitor C. The relationship among the VSC output voltages, the grid voltages, and the output currents can be expressed as follows:

A. Modeling of vector current control
where v ga,b,c , i abc , and u abc are the three-phase of grid voltage, input current, and VSC voltage, respectively. L and R are the filter inductance and resistance, respectively. The Clark transformation is defined as follows [33]: Based on a balanced grid voltage condition, the relationship in (1) can be expressed in the stationary reference frame by using Clark transformation in (2) as [34] where v gα and v gβ indicate the grid voltages, i α and i β indicate the output currents, and u α and u β indicate the VSC output voltages in the alpha-beta frame. Then, we give Park transformation.
where θ is a phase angle and usually calculated by using PLL. In this study, the d-axis is always coincident with the instantaneous voltage vector and the q-axis is in quadrature with it, i.e., v gd = V g and v gq = 0. By using Park transformation in (4), the current model (3) can be presented in the d-q frame as [33] where v gd and v gq indicate the grid voltages, i d and i q indicate the output currents, and u d and u q indicate the VSC output voltages in d-q frame. ω is the angular frequency of the grid voltage and ω = 2πf , and f is the frequency of the grid voltage.

B. Modeling of DPC
We define the instantaneous active and reactive powers of the VSC in the stationary reference frame as follows [34]: where P and Q are the active and reactive powers of the VSC, respectively. We can express the instantaneous active and reactive powers variation based on the grid voltages and output currents variation by differentiating (6) as follows: In this study, we consider a non-distorted grid. Thus, we can obtain the following relationship such as where V g is the amplitude of the grid voltage. Then, the instantaneous grid voltage variations can be obtained by differentiating (8) as follows: Substituting (3) and (9) into (7), we can obtain a state-space model of the active and reactive powers as follows: C. Grid voltage modulated direct power control As represented in (10), the dynamics of VSC with an L fiter is a time-varying MIMO system.
In [30], the GVM control inputs are defined to decouple the outputs from the two inputs as Based on (8), the new GVM control inputs (11) are possible to be represented in the d-q frame as follows: Based on (12), the original system (10) can be represented as follows: The GVM-DPC presents the system in the d-q frame without using the PLL. Notice that, (13) is converted into an LTI system with some coupling states. Consequently, it can be easily analyzed and designed by using conventional linear control techniques [31].

III. RELATIONSHIP BETWEEN DPC AND VCC
In this section, we show a relationship between the GVM-DPC and the conventional VCC designed in synchronous rotating frame. Then, the conventional VCC and the GVM-DPC methods are designed and compared.

A. Relationship between GVM-DPC and VCC
Since the system in (5) is defined where the d-axis is always coincident with the instantaneous voltage vector and the q-axis is in quadrature with it, v gd = V g and v gq = 0. Consequently. the active and reactive powers in the d-q frame can be defined as follows: If we multiply 2 3Vg to both side of (13), then a new system can be obtained as follows: It is obvious that the DPC model is changed into a conventional d-q currents model. Consequently, we can achieve zero steady-state error by using a PI controller which is the same as the VCC. In contrast, we can expect that the controller designed based on (13) will obtain a faster transient response because there is no need for PLL.

B. Vector current control designed in d-q frame
In this article, we only show a traditional controller consisting of feedforward and feedback to regulate d-q axes currents. At first, we define errors of d-q axes currents as follows: where i dref and i qref are d-q axes currents references, respectively. A controller consisting of feedforward and feedback is designed as follows [35]: where ν i d and ν iq are the feedback control inputs. To obtain zero steady-state error, a PI controller is applied to ν i d and ν iq as follows: where K i d ,p , K i d ,i , K iq,p , and K iq,i are the PI controller gains. For the PI controller gain design, substituting from (16) to (18) into (15), the closed-loop system can be obtained such as If we differentiate (19), then it is changed to a second-order system as Applying the Laplace transform to (20) yields  or equivalently The PI controller gains could be selected based on the traditional second-order system (22) [36].

C. Controller design for GVM-DPC
In this Subsection, a conventional controller including feedforward and feedback is designed to make the active and reactive powers track their references. Define errors of the active and reactive powers as follows: where P ref and Q ref are the active and reactive power references, respectively. To compare with the conventional VCC fairly, we design the same control structure as the one designed in the d-q frame. To cancel the coupling terms. we take a control law with a feedforward and feedback such as where ν P and ν Q are the feedback control inputs. To obtain zero steady-state error, PI controller is applied to ν P and ν Q as follows: ν P =K P,p e P + K P,i t 0 e P (τ )dτ, where K P,p , K P,i , K Q,p , and K Q,i are the PI controller gains. Finally, the original control inputs can be calculated based on the inverse of (11).
Notice that, if we use controller gains as positive values, the system is globally exponentially stable [37]. For the controller gain design, substituting from (23) to (25) into (13), the closed-loop system is obtained such as If we differentiate (26), then it is changed to a second-order system as Applying the Laplace transform to (27) yields or equivalently

D. Controller gains tuning
Normally, the PI controller gains of the VCC method are tuned considering the overall system dynamics, which are evaluated by the crossover frequency ω c and the phase margin (PM). With the consideration of the time delay, which consists of one sampling period (T s ) of computation delay and half sampling period (0.5T s ) of pulse-width modulation (PWM) delay [38], ω c is related to PM by [39] Based on ω c in (30), the proportional gain K p can be obtained approximately as To minimize the phase contribution of the PI regulator at ω c , its corner frequency is usually set a decade below ω c [40]. Thus, the integral gain K i can be calculated as Since the GVM-DPC gets the same closed-loop dynamics as the VCC method, its controller gains can also be calculated based on the aforementioned procedure.

IV. PERFORMANCE VALIDATION
The effectiveness of the GVM-DPC method is compared with the VCC by using a three-leg three-phase 15-kVA inverter with an L filter. The control system is implemented by using the DS1007 dSPACE system, where the switching pulses are generated by using the DS5101 digital waveform output board, and the grid voltages and currents are measured by using the DS2004 high-speed A/D board. A constant dc voltage supply is used at the dc-side. Furthermore, the acside is connected to a grid simulator, which generates 110 V, as shown in Fig. 3. The parameters of the system used in the experimental test are listed in Table I. In the test, the PM in (30)  As the first test, the reference of the active power is changed from 0 kW to 1 kW. Fig. 4 shows the time response of the VCC and the GVM-DPC methods. In this case, it should be noted that the grid voltages are measured at 'A' point in Fig. 2 (i.e., the VSC continuously measures the grid voltages even it is not injecting powers, as shown in Fig. 4). The main disadvantage of the conventional DPC method is the steady-state performance (i.e., power ripple) compared to the VCC method designed in the d-q frame [25], [37]. However, the GVM-DPC has a similar active power tracking performance compared to the VCC method, as shown in Fig. 4. Moreover, from method has a same model as that in d-q frame, as we discussed in Section II. In this case, the PLL system can continually provide the correct information of phase angle, hence, we can conclude that the results in Fig. 4 are acceptable. In addition, we test the robustness of the both methods to the grid voltage, i.e. low voltage ride through (LVRT) capability. Fig. 6 shows the time response of the VCC and GVM-DPC methods when the grid has a 100% balanced voltage sag. In this case, we set the references as the grid voltages are measured from 0 to its nominal value, can also be found in the module uninterruptible power supply (UPS) system, which has a hot-swap operation property [41]. That means when one UPS module fails, the redundant power modules have to take over immediately to guarantee the electricity supply to the loads. Hence, the redundant power modules do nothing in normal time, (i.e., the grid voltages are not measured at 'B' point in Fig. 2). It can be expected that the GVM-DPC has an enhanced capability of the plug-and-play.
Finally, we test the both methods in a weak grid, where the frequency and the phase shift step and fluctuate [42]. In this case, we use 22 mH-L and 15 µF-C to construct a grid impedance, where the short circuit ratio is 1.5. From Fig. 7(a), it is observed that the VCC method destabilizes the system due to the PLL system as discussed in [15]. However, the GVM-DPC method can stabilize the system since it eliminates the PLL system, as shown in Fig. 7(b). Consequently, we can conclude that the GVM-DPC has same property in the normal cases but it has good dynamic capabilities in some special cases, where the PLL system makes some problem.

V. CONCLUSIONS
In this article, we presented a relationship between GVM-DPC and the VCC designed in synchronous rotating reference frame for three-phase VSC. We mathematically showed that the GVM-DPC is equal to the current model in the synchronous rotating reference frame. That means the GVM-DPC could achieve the same steady-state performance as the VCC but better tracking performance because of PLL elimination in the control method implementation. Furthermore, the GVM-DPC will reduce the computational burden in comparison with the synchronous controller since there is neither d-q transformation nor PLL system. Finally, the experimental results show that the GVM-DPC has same property in the normal cases as the VCC but it has better dynamic capabilities in some special cases, where the PLL system makes the problem.
In the future, the GVM-DPC will be modified for various applications to overcome their practical issues.