Shicheng Zheng*,Zhuangzhuang Li, and Menglin Cao

(School of Electrical and information Engineering, Anhui University of Technology, Ma’anshan 243002 China)

Abstract: In the case of asymmetric loads of power grid, load currents are composed of four components: positive sequence active and reactive components and negative sequence active and reactive components that can pollute the power grid with harmonics and reactive power and interrupt the normal operation of power grid. Therefore, increasing numbers of static var generators (SVGs) are used to stabilize the power grid. In this study, a new type of current compensation control strategy of a three-phase three-wire SVG under an asymmetric load is proposed. According to the symmetric component method and dual dq synchronous transformation theory, the load currents are transformed into four components under dual dq coordinates. Each component is compensated separately by the SVG. Simultaneously, the proportional-limiter method is adopted to minimize the harmonics from the SVG by optimizing the waveform of the modulation wave and limiting its amplitude. Finally, the validity and feasibility of the control strategy are verified by simulation and experimental results.

Keywords: Dual-dq transformation, proportional-limiter, positive and negative components, active and reactive components

1 Introduction1

Recently, more loads with fluctuating and impacting characters are being widely used in industries. Thus, the power grid has become vulnerable to harmonics and reactive power and become unstable[1-3]. To stabilize the power grid,many types of handling equipment must be installed to clean the power grid at the customer’s end. Among these equipment, static var generator (SVG) has the most application in improving the power quality by advantages such as continuous compensation, fast regulation, wide operating range, and fixed repetition of harmonic compensation[4-7].

Generally, SVG which connected to the power grid is designed on accordance with nominal compensation capacity. Because of the unpredictable load variation,the output currents of SVG need to be limited according to the designed capacity. Otherwise, load negative sequence currents are required by asymmetric load, the power devices will collapse[8]. When using SVG to compensate asymmetrical three-phase load currents, the currents show characteristics such as different amplitudes, negative sequence currents, and unbalanced voltage of point of common coupling (PCC)[9-11]. After dual-dq transformation of load currents, the direct current (DC) signals of positive sequence reactive and negative sequence currents can be obtained and their amplitudes must be limited, which are applied as instruction signals of SVG. Because the positive and negative sequence currents are not orthogonal, the synthetic signal is very difficult to be directly calculated.Therefore, the output currents of SVG cannot be effectively ensured without overload.

To solve this problem, the traditional method(truncation-limiter) of the amplitude of three-phase modulation wave signals is required. Though the peak values of SVG output currents are less than the maximum permissible value of insulated gate bipolar transistor(IGBT), when load current exceeds SVG capacity, the output currents of SVG are peak-leveled sine waves that introduce harmonics and threaten the power grid.

To balance the voltage of PCC, three-layer DC side voltage control is proposed in [12]. The first layer is the overall DC side voltage control, second layer is balancing control between the three phases, and third layer is balancing control between modules of each phase. However, the negative sequence current is not considered. In paper [13], the theory of instantaneous symmetrical components for generating instantaneous reference current waveforms is utilized to obtain the negative sequence components. This means that the reference waveforms are alternating current (AC)components. Hence, the control strategy to balance the asymmetric currents is difficult to achieve by proportional-integral (PI) controller.

In this paper, the case of asymmetric load is observed, where after measuring the asymmetric load currents and decomposing them into AC components by instantaneous symmetrical component method, the obtained positive and negative sequence currents are transformed as reference DC signals by dual-dq synchronous transformation. Then, through PI decoupling control, the complete decoupling of positive sequence active and reactive currents and negative sequence active and reactive currents is achieved. The positive and negative sequence reference voltage vectors are also obtained. Finally,pulse-width modulation (PWM) drive signals, used to control power devices, are generated by space vector pulse width modulation (SVPWM) modulation.

2 SVG mathematical model under asymmetric load

2.1 SVG system structure

SVG system (shown in Fig. 1) consists of a threephase bridge converter, a signal detection and conditioning circuit, a main control circuit based on TMS320F2812DSP, a driving circuit, and a protection circuit[14-15]. The three-phase grid voltages, load currents,SVG output currents, and SVG DC side capacitor voltage are detected by signal detection circuit.

Fig. 1 SVG system structure

When SVG is applied to compensate the reactive power of asymmetric load, the output currents of SVG should be equal to the vector sum of negative sequence active and reactive currents and positive sequence reactive currents of load to ensure that grid currents are in phase with grid voltages. To logically deduce the SVG control strategy, the mathematical model of SVG under steady state and asymmetric conditions should be established based on the symmetric component method and dual-dq synchronous transformation theory. Then, according to the instantaneous reactive power theory, the SVG control strategy under asymmetric load is studied.

2.2 Symmetric component method and dual-dq synchronous transformation

From Fig. 1, the three-phase load currents in the asymmetric load of system are unbalanced. The unbalanced components can be decomposed into balanced components that include positive, negative,and zero sequence components. For each AC component (xa(t), xb(t), xc(t)) in an unbalanced system,instantaneous components can be expressed as

Where, X+, X-, X0are peak values of positive,negative, and zero sequence components, respectively and φ+,φ-,φ0are initial phase angles of each component, respectively.

Owing to the three-phase three-wire SVG, positive and negative sequence components except zero sequence components exist in the three-phase AC components.

The phasor calculation form is applied in traditional symmetrical component method to obtain the phase and amplitude of AC components. However,phasor calculation is only applied to the steady state analysis of the asymmetric components. Hence,transient asymmetric component transformation must adopt instantaneous symmetrical component method that can transform three-phase instantaneous currents to positive and negative sequence components.According to the definition of symmetrical component method, the transformation formula is given as

In SVG system, instantaneous grid voltages, load currents, and SVG compensation currents can be transformed into positive and negative sequence symmetrical components under three-phase stationary coordinates by equation (2) or (3). Then, to analyze the dynamic mathematical model of SVG, the three-phase stationary AC components must be transformed into DC components in two-phase synchronous rotating coordinates.

According to phase-lock information, three-phase symmetry positive and negative sequence components are transformed into positive synchronous rotating reference d+q+coordinates and negative synchronous reference rotating d-q-coordinates by dual-dq transformation. The synchronous speed is ω and -ω,respectively. The principle is shown in Fig. 2.

Fig. 2. Dual-dq rotating reference coordinates

Taking asymmetrical load current as an example,positive sequence dq transformation matrix, equation(4) is deduced from Fig. 2. From equation (4), two DC components Ld i+ and Lq i+ are obtained by transforming positive sequence components from three-phase stationary coordinates to positive synchronous rotating coordinates. However, negative sequence components are transformed into two double frequency AC components 'Ld i - and 'Lq i - under positive synchronous rotating coordinates. Because PI controller is a regulator without steady error only for DC components, steady error for AC components exists.

To control negative sequence components by PI controller, the negative sequence dq transformation must be applied. The transformation matrix in equation(5) is introduced, from which negative sequence components can be transformed into two DC components,andSimilarly, positive sequence components are transformed into double frequency AC components by negative dq transformation. From the above analysis, the PI controller based on unidirectional synchronous rotating dq coordinates is only valid for the same direction components. If the PI controller is applied to regulate double frequency AC components,the proportional gain must be increased, which will result in unstable system or even divergent. Hence,positive and negative sequence components should be transformed under different dq coordinates to obtain DC components.

Traditionally, positive and negative sequence transformations are applied to transform three-phase asymmetry components and the low-pass filter is adopted to filter the double frequency components to obtain DC components[16-17]. But in the actual working conditions, low-pass filter can cause signal distortion and delay. In this paper, symmetrical component method is applied to separate the positive and negative sequence components from three-phase AC components and these components are transformed into DC components by dual-dq transformation. This method has two merits:

(1) The outputs only contain DC components.

(2) There are no low-pass filters.

Grid voltage Usand SVG output currents are also transformed into positive and negative sequence components by the above method. After the asymmetrical three-phase AC components are transformed into DC components and regulated by PI controller, positive sequence active and reactive components and negative sequence active and reactive components need to be inversely transformed into three-phase AC components. Then, modulation waves of SVPWM containing the value of reactive power needed by SVG to output or absorb are obtained by superposing positive and negative AC components. Hence, dual-dq inverse transformation formula can be deduced as

2.3 Steady-state mathematical model of SVG

The steady-state mathematical model of SVG is studied to determine the intrinsic relationship and dynamic characteristics of each physical parameter in the system. Without compromising the accuracy of the study, the following assumptions are made for the convenience of model:

(1) Three-phase main circuit of SVG is completely symmetrical.

(2) The fundamental component is considered in modeling and harmonic components are not considered.

(3) The internal power loss of SVG reduces to connection reactor.

First of all, Sa, Sb, and Scare defined as the switching states of the three-phase bridge arms;switching function Skis shown as

Where, udcis the capacitor voltage of DC, Usa,b,care three-phase grid voltages, and ica,b,care output currents of SVG.

From Kirchhoff’s voltage law and Fig. 1,equation (8) can be deduced. The mathematical model described above is a description of the switching function that is more intuitive in three-phase stationary coordinates. However, the controller of this model is not suitable for the dynamic status of AC. So, equation(8) can be transformed into dq coordinates of equation(9) by rotating coordinate transformation.

Where, icdand icqare output currents of SVG that are transformed to dq coordinates.

2.4 Mathematical model of SVG under unbalanced conditions

Herein, to study the SVG control strategy under unbalanced conditions, the dynamic process of SVG is logically analyzed by constructing a mathematical model. The load currents will be asymmetric if the system has unbalanced load, which unbalances the voltage of PCC. The mathematical model of SVG under unbalanced condition is deduced and shown as

Compared with the balanced condition, the negative sequence components are generated at the unbalanced condition by (10). Positive components have no direct mathematical relationship with negative components, so, they can be controlled. However, the independent relation between positive and negative sequence components is verified at mathematical level.Then, the relationship of both components is verified further by physical significance. Under two-phase stationary αβ coordinates, the equation (11) is given as

Equation (13) shows that the output positive and negative currents of SVG are related to positive and negative voltages, respectively. Considering the laws of physics, positive and negative currents can be controlled independently. Owing to the variation of load, the negative current of SVG may increase, which might cause over-current of SVG and destroy the safe operation of power devices.

3 Principle of proportional-limiter

With the purpose of limiting over-current, the traditional control strategy introduces truncation limiter that transforms modulation waves into peak-leveled sine waves. This method could generate significant harmonic and harm the safety of grid. To ensure that the compensation current of SVG is lower than the maximum permitted current (Icmax), the proportionallimiter method, which will not only protect the normal operation of power equipment but decrease the harmonics and reactive power, is proposed in this paper.

Fig. 3 The calculation principle of proportional-limiter factor

The calculation method of proportional-limiter factor (A), calculated by root mean square (RMS) of asymmetric load current and maximum permitted current value of SVG, is shown in Fig. 3. Because the positive sequence active power current of load ( Ld) is supplied by the power grid and other load currents are compensated by SVG, the value of Ld i+ need not be limited. Therefore, Ldequals 0 and the remaining components have the original value as inputs of dq inverse transformation.

The positive and negative sequence three-phase AC current values (andas shown in Fig.3) are obtained by dual-dq inverse transformation and all compensation load currents are calculated after superposingand. Then, to calculate the RMS (ILa,b,c) of each phase current and select the maximum Imax, the proportional-limiter factor A is obtained through Icmaxdivided by Imax.

Fig. 4 The working principle of proportional-limiter

Fig.4 shows the working principle of proportionallimiter. After calculating the proportional-limiter factor A, the working principle of proportional-limit in this paper should be discussed in detail.

(1) A≥1 indicates that the maximum RMS of load current is lesser than the permitted maximum value of compensation current of SVG. SVG is not only operating in safe area but also normally compensating the reactive power at the moment.Hence, the proportional-limit does not act and the instruction currents of SVG are positive sequence reactive power current and negative sequence active and reactive power current of load.

(2) A＜1 indicates that the compensation load current exceeds the compensation range of SVG.Therefore,are controlled as instruction values of SVG after multiplying A and SVG is working at maximum load status.

4 Control strategy

To effectively control the positive and negative currents of SVG, two sets of controllers are adopted in this research, whose purpose is to suppress asymmetric load disturbance. One is a positive sequence PI controller that regulates the output positive current of SVG. The other is the negative sequence PI controller that is designed to regulate the output negative current of SVG. The unbalanced voltage of PCC caused by asymmetric load can be regulated to be balanced based on the following controllers.

(1) Positive sequence controller: According to instantaneous reactive power theory, reactive power is only exchanged between three phases and the fluctuation of capacitor voltage is caused by active power. To stabilize the voltage of DC side, the gridconnected active power P is controlled by regulating the positive sequence active current icd+. The relationship betweenand grid active power is shown as

From (10), the grid voltage is used as feed forward control parameter and the positive sequence active components should be decoupled with the reactive ones. The positive sequence active current controller is obtained by

An SVG control strategy to compensate the positive reactive current of load () is proposed in(16) by combining the mathematical model of SVG under unbalanced condition given by (10).stands for input value of positive sequence reactive voltage of positive dual-dq inverse transformation.

(2) Negative sequence controller: To compensate the negative sequence currents of load, negative sequence controllers are applied to ensure that the phase of grid current is the same as grid voltage. From(10), the load current components

Fig. 5 The control strategy of the system

From the above discussion, the complete control strategy can be obtained by equations (15)～(18), as shown in Fig. 5. The proportional-limiter is applied to improve the modulation wave to avoid exceeding the compensation range of SVG in complex working conditions. The controlled load current components are extracted as the reference compensation components of SVG after multiplying the proportional-limiter factor A. The error value between capacitor voltage value of DC side and set value is controlled by automation voltage regulator (AVR) and its output is the positive active current reference value of SVG.After obtaining the four error values (between the sets and each sequence components of SVG), the closedloop feedback control is applied with automation current regulator (ACR). Then, both decoupling control and grid voltage feedforward control are put into use.Finally, the DC voltage signals (,,and) are obtained by superposing the three output signals as ACRs, grid voltage feedforward, and decoupling controllers, respectively. After the dual-dq inverse transformation of DC voltage signals, the three-phase positive and negative sequence voltage signals are calculated and used as modulation waves by superposing these signals.

5 Simulation and experiment verification

5.1 Simulation analysis

An SVG prototype with 220 V (per phase RMS)/50 Hz was built to test and verify the above control strategy. As shown in Fig. 1, the test system consisted of a three-phase SVG and three-phase symmetry load. At 0.2 s, the asymmetry resistor-inductance load was thrown into the system. System parameters are given in Tab. 1.The dynamic performance of control strategy was verified by a Matlab/Simulink simulation of the system.

Tab. 1 Test system parameters

Fig. 6a shows the waveforms of load current. Its asymmetry load is suddenly thrown into the system at 0.2 s. The symmetry positive sequence current (shown in Fig. 6b) and negative sequence current (shown in Fig. 6c) were obtained by instantaneous symmetry component method after a transient fluctuation. From Fig. 6c, it is noted that the value of negative components equals zero before 0.2 s, which represents the accuracy of the method.

Fig. 6 The verification of instantaneous symmetrycomponent method

After obtaining the symmetry components, the DC components of each sequence of load currents were obtained by dual-dq transformations and were treated as the set values of each PI controller. According to equations (5) and (6), the dual-dq transformation can be realized as shown in Figs. 7 and 8.

Fig. 7 Load current transformation under positive sequence dq coordinates

Fig. 8 Load currents under negative sequence dq coordinates

Fig. 7 shows that the positive sequence components of load currents were transformed into DC components, whereas, the negative sequence components of load currents were transformed into double frequency AC components under positive sequence dq coordinates. Similarly, the negative sequence components of load currents were transformed into DC components and the positive sequence components of load currents were transformed into double frequency AC components under negative dq coordinates, which are shown in Fig.7. Therefore, the DC components were obtained by instantaneous symmetry component method and dual-dq transformation. Compared with the traditional method to calculate DC components, the filter was left out in the above method that reduced the delay time.

Figs. 9 and 10 show the sudden increase of load currents and compensation currents of SVG. Fig. 9a shows that the positive sequence active current of load was provided by power grid. Although it was suddenly increased at 0.2 s, the positive sequence active current of SVG was equal to 0 after small variations. Fig. 9b shows that with the positive sequence reactive current of load increasing to 50 A, the positive sequence reactive current of SVG increased rapidly to compensate. Fig. 10a shows that the negative sequence active current of load was stabilized after a transient fluctuation and the negative sequence active current of SVG increased rapidly to compensate. Similarly, Figs.10a and 10b show that the negative sequence reactive current of load could be rapidly compensated by SVG. As discussed above, the SVG compensation control strategy based on instantaneous symmetric component method and dual-dq transformation theory are effective and feasible.

Fig. 9 Positive sequence current components of SVG and load

Fig. 10 Negative sequence current components of SVG and load

Fig. 11 shows the variation of amplitude and phase of grid current of A phase as the asymmetric RL load mutates. To demonstrate the current and voltage,the amplitude of voltage is zoomed out in the scope. It is noteworthy that the grid current was suddenly distorted at 0.2 s and the phase lagged behind the grid voltage. After the compensation by SVG, the grid current had the same phase as grid voltage and the amplitude increased because the mutation load contained the positive sequence active component.

Fig. 11 Grid voltage and current waveforms of A phase

Simultaneously, to verify the merits of proportional-limiter when the compensation current value exceeded the range of SVG, the simulations of proportional-limiter and truncation-limiter were analyzed. Although the truncation-limiter can achieve the normal working of SVG, the output currents are peak-leveled sine waves that introduce significant harmonics to the power grid. Therefore, the asymmetric load current whose amplitude was greater than Icmax(set as 80 A) was applied.

Figs.12a and 12b show the total harmonic distortion (THD) of A-phase grid current adopting proportional-limiter and truncation-limiter respectively.Fig. 13 shows that the THD of grid current was decreased from 18.97% to 4.23% after the proportional-limiter was applied to the system. Fig. 12a shows that the system contains lower DC and higher frequency components than Fig. 12b, which indicates that proportional-limiter is better than truncation-limiter in improving the grid current waveform.

Fig. 12 THD of grid current at maximum load

5.2 Experimental results

To verify the control strategy, 100 kvar SVG experiment platform was built as shown in Fig. 13.

Fig. 13 Experiment platform of SVG

The experimental results such as THD, current unbalanced degree, and RMS of harmonics and waveform are shown in Figs. 14, 15, and 16,respectively. The role of SVG, as obtained from the above-mentioned figures, is to compensate the reactive power and lower harmonics of grid currents and improve the power quality. There were two sets of experiments. One was to verify the compensation effect of the control strategy under the condition of B phase containing harmonics.

Fig. 14 Parametric values of grid currents

Fig. 15 Waveforms of grid currents after compensation

Fig. 16 Output currents of SVG

Fig. 14 shows the values of grid currents. The THD of B phase grid current was decreased from 7.15% to 3.92% after compensation by SVG, which indicated the compliance of harmonics with the requirements of power quality (less than 5%).Simultaneously, the unbalanced degree of harmonics and currents in phase B and C also decreased. Fig. 15 shows that the waveform of grid current was symmetrical after compensation by SVG. Fig. 16a shows the output waveforms of SVG and its parametric values are shown in Fig. 16b.

To verify the control strategy in fault working conditions, the experiment was carried out without the load of phase A. The results are shown in Figs.17 and 18. Fig. 17a shows the waveforms of load currents and its parametric values are shown in Fig. 17b. The waveforms and parametric values of grid currents are shown in Fig. 18 after compensation by SVG. Fig. 18a shows that the three-phase grid currents are symmetrical. Simultaneously, the THD of grid currents corresponds with the standard of power grid.

Fig. 17 Load currents without A phase load

Fig. 18 Grid currents without A phase load

6 Conclusion

In the case of asymmetric load, the power factor of grid is decreased the grid can easily become unstable. Based on the instantaneous reactive power theory, a new current compensation control strategy is proposed in this paper. According to the above research, the following conclusions can be drawn:

(1) The positive sequence reactive components and negative sequence components due to asymmetric load can be compensated by the control strategy proposed in this paper.

(2) The control strategy proposed in this paper can avoid the compensation current that exceeds the compensation range of SVG, thereby improving the modulation waves, protecting the power devices, and decreasing the harmonic and reactive harm to power grid.

(3) The experiment results are in accordance with the simulation and the validity and feasibility of the proposed control strategy are verified.