State feedback control of advanced static var compensator using a five-level NPC inverter topology

ABSTRACT


INTRODUCTION
The static compensators (STATCOM)s are important reactive compensation devices that have been used in power systems to maintain bus voltage at a constant level, improve the transient stability, damp the systems and suppress voltage flicker [1]- [4].
The introduction of modern semiconductor devices in the design of power electronic converters has resulted in a solid-state Var source with a simpler structure, namely the ASVC [5]- [7].An ASVC is a fully controlled switch based converter, which is an upgrade version of The Static Var Compensators (SVCs), a thyristor-based converter.Like an SVC, the ASVC provides a controllable parallel compensation.The reactive power generation or absorption by an ASVC is the same as that of an SVC.The ASVC has advantages over a conventional SVC [8]- [11].
The ASVC use a Pulse Width Modulation (PWM) controlled DC-AC VSI with a capacitor as a DC power storage device.Recently, the multilevel PWM converter topology has drawn tremendous interest in the power industry since it can easily provide the high power required for high power applications for such uses as static VAR compensation, active power filters, and the control of large motors by high power adjustable frequency drives.The most popular structure proposed as a transformerless voltage source inverter is the diode clamped converter based on the neutral point clamped (NPC) converter proposed by Nabae [12], [13].It has the advantages that the blocking voltage of each switching device is one half of DC-link voltage, and the harmonics contents output voltages are lower than those of a two-level inverter for the same switching frequency.The NPC inverter has the drawback of the unbalance DC capacitors voltage when used as a static var compensator.Different methods were proposed to solve this issue.Some researchers have proposed to insert two inductors in one leg to balance the capacitors voltage [14], [15].Others have introduced a modified switching pattern and improved the optimized PWM technique [16], [17].
This paper presents a modeling and analysis of this new type of five-level inverter used for static VAR compensation.The control design is based on the conventional proportional-integral (PI) controller and the state feedback controller (SFC) controllers to adjust the ASVC Var flow with the AC system.The SFC controller also has the ability to balance the total DC capacitors voltage without any extra device or design of a complex circuit or optimizing control method.Finally, some simulation results under various transient conditions of the proposed ASVC model and its control are given to prove their effectiveness.

OVERVIEW AND MODELING OF THE ASVC-BASED FIVE-LEVEL INVERTER
The ASVC circuit consists of twenty four-pulse VSI with four DC capacitors and a PWM modulator.The ASVC is connected to the transmission line is via a coupling transformer where Rs and Ls are the coupling transformer active losses and leakage respectively, as shown in Figure 1 [18]- [22].

PWM control strategy of the five-level NPC-VSI inverter
In this work, the PWM control strategy for a five-level inverter uses a single reference and four carriers signals as depicted in Figure 2(a) where a sinusoidal reference signal is continuously compared to four (N-1 in general) triangle waveforms where N is the number of level of the inverter.One of advantages of this technique is that the significant harmonics are concentrated at the carrier frequency [23]- [26].
The frequency modulation index and the amplitude modulation index are given by (1) where f c : Frequency of carries.A c : Peak-to-peak amplitude of carries.The mathematical model of the converter is developed using switch connection function F ki , (k=a,b,c) : names of arms and (i=0,1,2,3,4) : number of the function.The switching function, is either 1 or 0 corresponding to on and off states of switch S ki , (k=a,b,c) : names of arms and (i=1,2,3,4,5,6,7,8) : number of the switches of one arm.Table 1 lists the switch connection function of one leg and output voltage.Table 1.A possible switch combination of one phase leg for the NPC-based five-level inverter Using the upper arm of the phase-leg 'a' as an example shown by Figure 2(b).Note that the switch states S a1 , S a2 , S a3 and S a4 are complementary to S a5 , S a6 , S a7 and S a8 respectively [27], [28].

Mathematical model of the ASVC system
The simplified three-phase equivalent circuit of the ASVC connected to a transmission line is shown in Figure 3.The ASVC supplies reactive power to the AC transmission system if the magnitude of the output inverter voltage is greater than the AC terminal voltage and absorb reactive power from the AC transmission system if the magnitude of the AC terminal voltage is greater than the output inverter voltage.The Var exchange is zero when the two voltages are equal [29], [30].It is assumed that the AC side is a balanced sinusoidal three-phase voltage supply.Using matrix form, the mathematical model of the ASVC system is given by (2): The output inverter voltages relative to the point n and the DC side currents of the inverter using the connection functions are given in ( 3)-( 7): with: The model of the DC side capacitors voltages is given by ( 8): Therefore, by using the d-q frame transform, the nonlinear state-space model of the ASVC based on the five-level inverter combined the DC circuit equation can be expressed by (9): where m is the ratio relating the AC to the DC voltage.
The modulation index relates the maximum phase voltage V o,Peak to the total DC side voltage V dc is given by (10): (10) The obtained state equation is non-linear, with respect to the control variable  which is related to the phase difference between the source voltage and inverter output voltage.In the range of small values of  (||< 5°), the small-signal equivalent state equations and the reactive power delivered by the ASVC-based five-level inverter system in the d-q frame is expressed as (11) and ( 12) [18]: from ( 11) and ( 12), the transfer function of the ASVC-based five-level inverter system is given by:  The PI parameters are calculated using root locus design for a damping factor 0.7.

The State feedback controller
The overall closed loop control of the ASVC-based five-level inverter with SFC control technique is shown in Figure 3(a) and the cascade control system of the state-feedback configuration is given in Figure 5 [31], [32].The controlled variable Q c is compared with the set-point value Q ref and the control error is fed back to an integrator.The former feed forward gain k 1 is now the gain of the integrator.This configuration shows that the gain K in the internal closed-loop is a feedback parameter.
The basic principle of the designed system is to insert an integrator in the feed forward path between the error comparator and the process [31].We assume that the state equation given by ( 17) is completely state controllable.The augmented dynamic system can be described by an equation that is combination of ( 17) and (19). where: We shall design an asymptotically stable system such that x(∞), xe(∞), and u(∞) approach constant values, respectively.Then, at steady state 0 The aim of the proposed SFC controller is to regulate the total DC side voltage V dc which is the sum of the four capacitors voltages on the DC side of the inverter, the AC-current components I dq and the reactive power response Q c .
The state space representation of the ASVC based upon five-level inverter controlled by SFCcontroller can be described by an equation that is combination of ( 11) and (12).
To achieve a good dynamic response of the closed loop system, the poles are determined by the Ackermann algorithm where:

SIMULATION RESULTS AND DISCUSSION
A digital simulation is done based on the detailed ASVC system shown by Figure 3.The PI and the SFC controllers' parameters are listed in Appendix.
The PI and SFC controllers were evaluated under more realistic simulation condition when the ASVC was controlled by PWM control circuit.
Figure 6 displays the simulation run for a step change in the Q ref reference from 10 Kvar (inductive) to -10 Kvar (capacitive) in order to swing the system from leading to lagging mode at time 0.1 sec and standby mode at 0.2 sec.
From Figure 6, a performance comparison between the two controllers is summarized in Table 2.The SFC controller has a rise time and settling time of 12.4 msec and 30 msec respectively, where the PI controller has a rise time and settling time of 14.2 msec and 45.65 msec respectively.It can be seen that the rise time and the settling time of the SFC controller are 12.7 % and 35.5 % respectively less than those of the PI controller.Moreover, the SFC controller does not have any overshoot when the PI controller in transient state exhibits 5% of overshoot.7 shows a phase current waveforms dynamic behavior.It is observed that the current injected into the transmission line controlled by the SFC controller is faster that the one controlled by PI. Figure 8 (a) and (b) show the source phase voltage and the inverter current waveforms respectively with PI and SFC controllers.At first, the ASVC is generating 10 kVar inductive reactive power and at 0.1 sec is fastly absorbing also 10 kVar but as a capacitive reactive power.From 0.2 sec the ASVC is in standby mode, it provides no reactive power (Qc=0 kVar).Figure 9

CONCLUSION
In this paper, a simulation study of the dynamic performance and robustness of state feedback control have been proposed, evaluated and compared to a conventional PI controller applied to an ASVC compensator using five-level NPC-VSC inverter.The five-level NPC inverter has been used for its high quality waveshaping outputs.
Simulation results show the effectiveness of the proposed control strategy.The SFC control algorithm shows that reactive power compensation is achieved by controlling the exchange of active power between DC side of inverter circuit and AC system.It can be concluded that the SFC controller leads to improve transient response and hence provide fast reactive power compensation with balancing the total DC capacitors voltage.Also the SFC controller is better than the PI controller and can be easily tuned.

Figure 1 .
Figure 1.Power circuit of the ASVC-based five-level NPC-VSI inverter

Figure 4 .
Figure 4. Block diagram of the ASVC system with PI control

Figure 5 .
Figure 5. Block diagram of the ASVC system with SFC control

𝑝 1 =
−500,  2 = −450,  3 = −300 + 305.65,  4 = −300 − 305.65The necessary state feedback gain matrix K ˆare determined using the pole placement technique, where  ̂= [ ⋮  1 ] = [−0.0320− 0.0174 − 0.0023 − 0.0382] Figure7shows a phase current waveforms dynamic behavior.It is observed that the current injected into the transmission line controlled by the SFC controller is faster that the one controlled by PI.Figure8(a) and (b) show the source phase voltage and the inverter current waveforms respectively with PI and SFC controllers.At first, the ASVC is generating 10 kVar inductive reactive power and at 0.1 sec is fastly absorbing also 10 kVar but as a capacitive reactive power.From 0.2 sec the ASVC is in standby mode, it provides no reactive power (Qc=0 kVar).Figure9and Figure 10 represent transient DC side voltage across the four capacitors after a sudden change in reference under PI and SFC controllers respectively, it can be noticed that voltage fluctuation of V dc1,2,3,4 with SFC controller were reduced by 50 % compared to those obtained with PI controller.Figure 11(a)(b) and Figure 12(a)(b) show the transient response of the total DC side voltage V dc , and I d , I q AC-current components respectively with PI and SFC controllers.It can be seen that total side voltage controlled by the SFC is better that the one controlled by PI.The source voltage and inverter output voltage waveforms of the five-level NPC inverter for reactive power compensation with PI and SFC controllers are depicted by Figure 13 (a) and (b) respectively.

Figure 11 (
Figure7shows a phase current waveforms dynamic behavior.It is observed that the current injected into the transmission line controlled by the SFC controller is faster that the one controlled by PI.Figure8(a) and (b) show the source phase voltage and the inverter current waveforms respectively with PI and SFC controllers.At first, the ASVC is generating 10 kVar inductive reactive power and at 0.1 sec is fastly absorbing also 10 kVar but as a capacitive reactive power.From 0.2 sec the ASVC is in standby mode, it provides no reactive power (Qc=0 kVar).Figure9and Figure 10 represent transient DC side voltage across the four capacitors after a sudden change in reference under PI and SFC controllers respectively, it can be noticed that voltage fluctuation of V dc1,2,3,4 with SFC controller were reduced by 50 % compared to those obtained with PI controller.Figure 11(a)(b) and Figure 12(a)(b) show the transient response of the total DC side voltage V dc , and I d , I q AC-current components respectively with PI and SFC controllers.It can be seen that total side voltage controlled by the SFC is better that the one controlled by PI.The source voltage and inverter output voltage waveforms of the five-level NPC inverter for reactive power compensation with PI and SFC controllers are depicted by Figure 13 (a) and (b) respectively.