H-Infinity Control of an Adaptive Hybrid Active Power Filter for Power Quality Compensation

This article highlights an optimal robust control technique called H-infinity, which thanks to a particular algorithm offers several solutions in the experi-mental implementation of harmonic compensators of systems with API-siemens modules. This control and command technique is directly tested on a TLC adaptive hybrid filter topology that provides benefits, such as reduced switching losses when injecting currents in the network, limitation of resonance problems and above all low power consumption at the DC bus level, thus allowing us to obtain results for 105 V to be compared with existing models in the literature which require 600 V for the same performance. This article therefore simultaneously offers two essential contributions to the optimization of harmonic pollution control. A first contribution is essentially based on the H-infinite algorithm and its particularity in its implementation on our TLC hybrid model. The second is on the advantages offered by the TLC-HAPF hybrid topology. The results obtained with this algorithm give us THDs conforming to the IEEE 519-1996 and which are very meaningful compared to the results obtained with other robust and stochastic control al-gorithms taken under the same conditions.

ern society is heavily dependent on the electric supply. The life cannot be imagined without the supply of electricity. At the same time, the quality of the electric power supplied is also very important for the efficient functioning of the end user equipment. The term power quality became most prominent in the power sector and both the electric power supply company and the end users are concerned about it [1]. The quality of power delivered to the consumers depends on the voltage and frequency ranges of the power. If there is any deviation in the voltage and frequency of the electric power delivered from that of the standard values, then the quality of power delivered is affected. Nowadays with the advancement in technology, there is a drastic improvement in the semi-conductor devices. With this development and advantages, the semi-conductor devices got a permanent place in the power sector helping to ease the control of overall system. Moreover, most of the loads are also semi-conductor based equipment. But the semi-conductor devices are non-linear in nature and draw non-linear current from the source. And also the semi-conductor devices are involved in power conversion, which is either AC to DC or from DC to AC. This power conversion contains a lot of switching operations which may introduce discontinuity in the current. Due to this discontinuity and non-linearity, harmonics are present which affect the quality of power delivered to the end user. In order to maintain the quality of power delivered, the harmonics should be filtered out. Thus, a device named Filter is used which serves this purpose. There are many filter topologies in the literature, like active, passive and hybrid. Installation of the current quality compensators is one of the solutions for the low power factor, harmonic current pollution and unbalanced problem. Different power quality compensators have been compared in historical order in the following: Shunt capacitor banks (CBs) were firstly applied in power systems in around 1900s for power-factor correction and feeder voltage control due to its advantages of low cost and flexibility of installation. However, CBs can easily get burnt if the current harmonics level is high. To compensate the current harmonics, the passive power filters (PPFs) were proposed in 1940s. Unfortunately, the PPFs have many disadvantages, such as low dynamic performance, resonance problem and lack of unbalanced compensation ability [2]. The thyristor controlled static var compensators (SVCs) were firstly proposed in 1960s [3]. And the SVCs are popularly for dynamic reactive power [4] [5] [6] [7] and unbalanced power compensations [8] [9] [10]. However, the SVCs suffer from drawbacks, such as resonance problem, harmonic current injection and poor harmonic compensation ability. To overcome the drawbacks and improve the performances of SVCs simultaneously, the inverter based controlled active power filters (APFs) were proposed in the year of 1976 [11] [12]. Unfortunately, APFs still cannot have large scale devel- compensators, the HAPFs are more cost-effective than the APFs [13]- [18].
However, the HAPFs have a quite narrow compensation range, which limits its compensation ability. When the loading reactive power is outside its designed range, it loses its low-inverter rating advantages [19]. In the year of 2014, the topology of thyristor controlled LC coupling hybrid active power filter (TCLC-HAPF) is proposed in [20], in which this state-of-the-art TCLC-HAPF has the characteristics of a wider compensation range than HAPFs and lower dc-link voltage than APFs for power quality compensation. Until now, the complete studies of characteristics, design techniques and applications of the TCLC-HAPF are still lacking to enhance the characteristics of passive filter and also the system; the active filter should be controlled properly. There are different control techniques for this purpose. The main aim of any control technique is to make active filter inject a voltage in to the system that compensates the harmonics. To achieve this output voltage, the active filter is controlled such that it is equal to a pre-calculated reference value. In this paper the proposed theory is validated by simulating it in MATLAB SIMULINK environment. The proposed control strategy is simulated for both balance and unbalanced load conditions. In this project the use of H-Infinity to control hybrid adaptive power filters for the improvement of electric power quality is studied and analyzed.

Solutions to Power Quality Problems
The most effective solution to improve the power quality is the use of filters to reduce harmonics. The basic idea of using a filter is explained in Figure 1, where the filter injects a compensating current that compensates the harmonics in load current. There are different filter topologies in the literature such as-active, passive,  The series APF works as a harmonic isolator and used to reduce the negative sequence voltage [2]. The combination of passive filter and APF known as Hybrid Filter has also been study but has a narrow compensation range. There is another filter topology which is the TCLC-HAPF which is study in this project.

Objectives
The main objective of this project is to control the thyristor controlled LC coupling hybrid active power filter (TCLC-HAPF) such that the harmonics in the current waveform are reduced. The control algorithm has the following objectives:  To control the voltage injected by TCLC-HAPF such that it compensates the reactive power and load current harmonics  To improve the passive filter performance  To make the whole compensating equipment to act as linear, balanced, resistive load on the system

Filter Classification
The different filters present in the literature are classified into three basic types.
They are Active Filters and Passive Filters and Hybrid filter. Each type has its own sub classification. Figure 2 shows the detailed classification of the filters.

Comparisons among Thyristor Controlled LC-Coupling Hybrid Active Power Filter (TCLC-HAPF) and Other Different Power Quality Filters
Installation of the current quality compensators is one of the solutions for different power quality problems such as the low power factor, harmonic current pollution and unbalanced problem. The historical review of different power   [9]. Since the rating of the active inverter part is proportional to the cost of compensators, the HAPFs are more cost effective than the APFs.
However, the HAPFs have a quite narrow compensation range, which limits its compensation ability. When the load reactive power is outside its designed range, it loses its low-inverter rating advantages. From 2014 onwards, the thyristor controlled LC coupling hybrid active power filters (TCLC-HAPFs) have been widely studied in [10]- [16], which have the desirable characteristics of a wider compensation range than HAPFs and lower dc-link voltage than APFs for power quality compensation. Based on above discussions and Table 1, the traditional power quality compensators, PPFs and SVCs have inherent problems like resonance problem, slow response, poor harmonic compensation ability, etc.
The above mentioned inherent problems can be solved if the active inverter part has been added to the topologies such as APFs, HAPFs and TCLC-HAPFs.
However, the cost of PPF and SVC are lower than that of the active inverter part, thus the reduction of the active inverter part rating can lead to a decrease in the total cost of APFs, HAPFs and TCLC-HAPFs. After serious compariation the conclution was drawn, the comparisons among the APF, HAPF and TCLC-HAPF have been provided in terms of V-I characteristic (compensation range and Compared TCLC-HAPF with APF and HAPF, it has higher reliability and lower power loss than both APF and HAPF. Besides, the TCLC-HAPF can be more cost effective than the APF for medium/high voltage level applications (≥10 kV).
In addition, the TCLC-HAPF has wider reactive current compensation range than the HAPF and lower DC-link voltage than the HAPF. Therefore, the TCLC-HAPF has a large potential to be further developed for medium/high voltage level applications.

The Circuit Configuration and Modeling of the Three-Phase TCLC-HAPF
The circuit topology of a three-phase three-wire TCLC-HAPF is provided in  [22]. Therefore, the three-phase modeling is proposed in Figure 3 for TCLC-HAPF compensation analysis.
The equivalent compensating circuit is shown in Figure 3. The active inverter part can be considered as the adjustable active impedance to improve the TCLC part fundamental and harmonic current compensation ability [20] and [21]. Therefore, the three-phase modeling for TCLC-HAPF unbalanced compensation analysis is proposed in Figure 4 & Figure 5. At the fundamental frequency (Figure 4), the active impedance X ACTxf is used to help the TCLC part impedance X af to compensate fundamental reactive power and balance active power. At the harmonic frequency ( Figure 5), the active impedance X ACTxn changes the equivalent TCLC-HAPF impedance to be zero, so that the load harmonic current will not pollute the source side. The fundamental and harmonic active impedance X ACTxf and X ACTxn are proportional to the inverter voltage. To keep active inverter working at low rating, the X ACTxf and X ACTxn need to be designed as small as possible.

Proposed TCLC-HAPF Parameter Design
In this section, a parameter design method is discussed and explained into three parts. In Part A, the relationship between the required TCLC-HAPF fundamental impedances (X af + X ACTxf ) and the load powers is deduced based on power flow analysis. The parameter design of the required fundamental dc-link voltage V DCf , C PF and L PF is proposed under the fundamental frequency consideration. In Part B, the parameter design of the required harmonic dc-link voltage V DCh is proposed under harmonic frequency consideration. In Part C, the design of L c is given.

Design of Vdcf, CPF and LPF Based on Power Flow Analysis under Fundamental Frequency Consideration
Referring to Figure 4, the required TCLC-HAPF impedance (X xf + X ACTxf ) can be calculated by applying the Ohm's Law as: where V xf and I cxf are the fundamental load voltage and compensating current phasors respectively, where x stands for phase a, b and c. V n is the fundamental common point voltage. By using the Kirchhoff's circuit laws (KCL), the compensating current relationship can be expressed as: Simplifying (2.2), the expression of V nf can be obtained as: The I cxf can be expressed in terms of V fx (V x = V fx for V x is assumed to be pure sinusoidal without harmonic components [11]- [18] and the compensating active and reactive power P cx and Q cx as: where the note "*" denotes the conjugate. For unbalanced compensation, the Referring to Figure 2(a), the fundamental inverter voltage V invxf can be obtained as: where X ACTxf and I cxf are the fundamental active impedance and compensating current, and I cxf is design to compensate load fundamental reactive current ( ) Lxfq cxf Lxfq . The relationship between the V invxf and V DCxf can be expressed as: In (9), the scale of ( ) 2) To guarantee the sufficient V DCxf , the peak value of fundamental inverter voltage needs to be considered Moreover, the final required V DCf is designed to be the maximum value among each phase. Therefore, the final required V DCf can be expressed as: The X ACTxf is directly proportional to the required V DCf . The low dc-link voltage is one of the major advantages of TCLC-HAPF. This can be achieved when the value of X ACTxf is designed to be zero (X ACTxf ≈ 0). In other words, the value of V DCf is minimized (V DCf ≈0). With such minimum V DCf design, the TCLC part is mainly used to compensate reactive power and balance the active power, while the active inverter part is mainly used to improve the harmonic compensation ability of TCLC part. The TCLC part is an L c (X Lc ) in series with a paralleled combination of a L PF (X LPF ) and a C PF (X CPF ), in which the X xf can be deduced as: In (12), X Lc , X CPF , X LPF are the fundamental impedances of L c , C PF and L PF . α x is the firing angle of the thyristor. The TCLC part has two back-to-back connected thyristors T1x, T2x, and they are triggered alternately in every half cycle. When α x = 180˚ (thyristors are opened for the whole cycle), the TCLC part has the Energy and Power Engineering maximum capacitive impedance XCap(Max) (<0). On the other hand, when the firing angle α x = 90˚ (one of thyristors is closed for whole cycle), the TCLC part has the minimum inductive impedance XInd(Min) (>0). Therefore, XCap(Max) and XInd(Min) can be expressed as: where ω (=2πf) is the angular frequency. To guarantee the TCLC part has inductive compensation range and capacitive compensation range, the basic conditions of XCap(Max) < 0 and XInd(Min) > 0 need to be satisfied. Thus, from (13) and (14), the following relationships can be obtained:

Control Strategy
The TCLC-HAPF should be controlled such that the voltage injected by it should compensate the harmonics present in the system and should help in improving the quality of power. To achieve the above purpose, the output voltage of the APF should be controlled. For this to happen, at first a reference voltage is generated which when injected by APF will serve the desired purpose. Then the actual output voltage of the series connected APF is controlled using a PI controller such that the actual output voltage generated is equal to the reference value. The compensation strategy to compensate the harmonics is designed where R e is the equivalent resistance Thus, the average power supplied by the source is given as-

Reference Vector Generation
To control the series connected APF the reference vector should be generated and compared with the actual voltage vector [1]. The reference voltage vector given by Equation (21)

Presentation of H-∞ Command
The H-∞ command (still called advanced frequent control or multivariable robust command) is a new approach to the Frequency Automatic; it was initiated by Zames in the early 1980s and developed, in particular by Doyle, Glover, Khargonekar and Francis; it has in recent years become one of the flagship methods of "robust control" [23]; it is used for the rapid development of robust  -Reference trajectory tracking (guidelines): this is to study the influence of the reference signal r(t) on the error signal E(t) -Rejection/mitigation of disturbance signals: this is the to study the influence of the b(t) disturbance signal on the Error Signal E(t) -Measurement Noise Mitigation: it was intended to study the influence of noise signals w(t) on the signal U(t) and on the output signal y(t) -Moderate control: it is a question of studying the influence of the reference signals r(t) and the disturbance signal b(t) on the command signal u(t)

Standard Problem of H-∞
The H-∞ synthesis uses the concept of standard problem, which is shown in Figure 7: the P(s) transfer matrix models the dynamic interactions between 2 sets of inputs and 2 sets of outputs: vector IV represents external inputs, such as reference signals, disturbances, noises; The vector II represents the commands; the e signals are chosen to characterize the right functioning of "system"; Finally, y represents the measures available to develop the order.

Riccati Equation Solution
From the following model below; So there is a K(p) corrector solution to the problem Hꚙ standard 1) The Hamiltonian Matrix has no values of its own on the imaginary axis and there is a sysmmetrical matrix 0 X ∞ ≥ such as: ( ) 2) The Hamiltonian Matrix has no values of its own on the axis imaginary and there is a symmetrical matrix 0 Y ∞ ≥ such as ( ) or ρ corresponds to the module of the highest value (spectral radius). In addition, all K(P) correctors responding to the problem are given by A particular corrector can be obtained as the central corrector, obtained by taking 0, which gives: 4) Equivalent model of the HAPF (applying H-∞ to filter) (Figure 8) The equivalent reactance gives us: with the matrice obtain above will permit us to have a good functioning of the H∞ corrector.

Simulation Results and Discussions
The proposed control strategy is simulated in MATLAB SIMULINK environment to check the performance of the control strategy in improving the system behavior. The simulation is carried under two different conditions:

Unbalance Non-linear Load
Balance Non-linear Load The performance of the system with the proposed control strategy under different conditions is discussed in detail in the following section.

Simulation Results with Non-Linear Load (Table 2)
To study the performance of the system when the source voltage is in equilibrium and connected to a non-linear load with R l and L c . The load turns to inject harmonic back to the source current coursing other linear load to suffer malfunction. The simulink model of the system is shown in Figure 9.
As observed from the above current curve of a single phased compared with that of Figure 10, it's clearly seen that due to the injected harmonics to the system has changed the sharp of the curve, therefore by applying our filter topology to smoothing the sharp of the curve (Figure 11).

Unbalanced Non-Linear Load (Table 3)
The simulink model is same as that of the balance load while maintaining similar   parameters, with an additional unbalanced load. See simulink model in Figure   12.
To study the performance of the system when the load is unbalance with the    above parameters, the simulation is carried out. And the simulation result of phase1 is presented in Figure 12, showing the difference in amplitude.
From the above analysis it is obvious that each line with respect of the unbalanced load injected to the system has caused different distortion in the lines (phase) making each line to observed different THD 23.98%, 22.34%, 30.48% as shown in Figure 13.

Comparative Study of System before Filtering
A comparative study is made to analyze the performance of the system at various operating conditions when operating with balanced load, unbalanced load and balanced voltage. The comparison is given in Table 4. From the results it is clear that the system behavior can be improved after the filter is connected and the source current THD will be very less and is within the IEEE permissible standards.

Simulation Results of Non-Linear Load after Filtering
The power system may experience unbalanced load and voltage source conditions at many times. Thus, the behavior of the proposed control strategy is analyzed  by simulating it under non-linear load, unbalanced non-linear load and voltage condition. Here the non-linear load is created by connecting three single-phase uncontrolled rectifiers and resistor. The load voltage values are given in Table 5.
A passive filter is connected at PCC to eliminate fifth order harmonics. Also an active filter is also connected at the output of the VSI. The values of these filters along with load values are given in Table 5. The filter impedance should be less  From observation and assumption the phase 3 of the unbalanced non-linear load will produced a level of THD with after filtering will respect the norms of THD. With the system parameters in Table 5, the proposed control strategy is simulated and the circuit diagram is shown in Figure 14. The MATLAB SIMULINK results are presented in Figures 15-17 respectively. Figure 18 shows the source current after compensation. The THD of this current is shown in Figure 16 which is (3.16%) Now the THD of the current is less and the harmonic analysis is shown in Figure 17. The use of adaptive hybrid active power filter increases the performance of the system and the overall power factor is also improved. In addition, 5th order harmonics are greatly reduced.

Case of an Unbalance Non-Linear Load after Filtering
In the case of an unbalanced non-linear load the simulation is done and observation is taken per phase since an additional non-linear load is added to each phase with different values causing each phase to obtain different THD. The table of values for the additional load is given in Table 6 below.
The h-infinity command here at this level allows the filter to produce harmonic currents adapted to the disturbance of the network depending on the nature of the disturbance of each phase with THD concerned of 1.8%, 3.29%, 2.78%.

Comparative Study under Various Conditions
A comparative study of the three phase source current THD during unbalanced and balanced load at various operating conditions is presented in Table 7. From these results it is clear that the proposed control strategy works better at almost all operating conditions and thus helps in improving the quality of electric power delivered to the end user.

Conclusion
In this article, it was a question of simultaneously using two aspects for the op-