Interconnection of Weak Power System by VSC-HVDC using Fuzzy based Compensation Control

Interconnection of Weak Power System by VSC-HVDC using Fuzzy based Compensation Control

G. Saraswathi Professor, EEE, JNTUK-UCEV Vizianagaram, Andhra Pradesh,

India.

Srilekha. V

1. G. Student, EEE, JNTUK-UCEV Visakhapatnam, Andhra Pradesh,

   India.

   Abstract : To enhance the power transmission capability and stability using VSCs became an attractive alternative technology. This paper proposes and addresses a Fuzzy based power compensation control to interconnection of weak power systems by VSC-HVDC. This proposed new control methodology improves the damping control performance of the active power and the AC voltage by adjusting the compensated values based on analysis and characteristics of PV curves. Small signal model of the integrated system is established to study the performance of the proposed control method. The impact of the compensation coefficients and steady state operating points on the system stability is studied. Electromagnetic transient simulation models are considered and verified in MATLAB / Simulink environment for the effectiveness of the method on the performance of operating limits, damping performance, disturbance withstanding capability, and fault ride-through capability.

   Index TermsVoltage source converter (VSC), HVDC, weak power system, power compensation control, stability analysis, small signal modeling, virtual impedance.

   NOMENCLATURE

   Reactive power compensation capacitor

   P,Q Active- and reactive power at PCC

   Angle difference between and

   Phase voltage amplitude of grid source

   Phase voltage amplitude at PCC

   Transformer impedance

   Reactive power output of VSC

   Reactive power generated by

   References of active power for inner Current reference calculation

   References of reactive power for inner current reference calculation

   0 Steady state reactive power

   0 Steady state active power, 0=P*

   0 , 0 Steady state d-and q-axis current of VSC

   0 , 0 Steady state d-and q-axis grid current

   Grid current virtual impedance

   VSC current virtual impedance

   () Closed-loop transfer function of PLL

   Angle outputted from PLL

   () Transfer function of PI regulator

   , , Grid equivalent Impedance, Resistance, Reactance

   1. INTRODUCTION

      For reduction of cost and improved reliability, most of the worlds electric power systems continue to be interconnected. Interconnections take advantage of diversity of loads, availability of sources and fuel price for supplying power to loads at minimum cost and pollution with a required reliability. In a deregulated electric service environment, an effective electric grid is essential to the competitive environment of reliable electrical service[1]. Increased demands on transmission, absence of long-term planning, and the need to provide open access to generating companies and customers have resulted in less security and reduced quality of supply[2]. Compensation in power systems is, therefore, essential to alleviate some of these problems. Series/shunt compensation has been in use for past many years to achieve this objective. Load compensation is the management of reactive power to improve the quality of supply especially the voltage and p.f levels. Here the reactive power is adjusted with respect to an individual load and the compensating device is connected to the load itself.

      Interconnection between AC and DC networks has become the attractive solution by using Voltage-source converters (VSCs) . It has been widely used in many applications, such as integration of renewable energy generation [4], high-voltage direct current (HVDC) transmission [5] and back-to-back systems, integration of energy storage systems [6], and railway traction systems [7]. The VSC-HVDC technology has been proposed as a promising solution for a back-to-back project in China to interconnect a weak AC power system to a strong AC system [8]-[11].

      LC or LCL output filters are usually adopted to mitigate the converter switching ripples. However, if the converters are not properly controlled, inherent resonances of the filters can introduce serious power quality and stability problems [7]. Power-angle control has been studied for HVDC, static-synchronous- compensator (STATCOM), and wind-turbine applications. But the disadvantage of power-angle control is that the control bandwidth is limited by a resonant peak at the grid frequency and the control system does not have the

      capability to limit the current flowing into the converter.[8] The pulse-width modulation (PWM) based voltage-source converter (VSC) is an emerging technology for HVDC transmission [16], [17]. In contrast to the conventional thyristor-based HVDC system, a VSC-HVDC system has the potential to be connected to very weak ac systems, as well as the capability to generate or consume reactive power depending on the operating conditions. Synchronous condensers have to be installed to increase the short-circuit capacity of the ac system. However, synchronous condensers can substantially increase the investment and maintenance costs of an HVDC project [11].

      To enhance the power transmission capability of a VSC-HVDC connected to a very weak power system. Some control strategies were introduced such as power synchronization control (PSC) [8]-[11], [16], virtual

      synchro-converter control [6], [15], [17]-[19], and advanced vector control [11]. The operating performance of the VSC-HVDC that is interconnected with a weak power system was investigated in [11], which indicates that the stability of the VSC was mainly affected by the phase angle difference of the AC voltage when AC voltage control mode was adopted. Hence, to improve the power transmission capability PSC method was adopted. A backup PLL was needed for synchronous loop under AC system faults. Since the power controller has the capability of regulating the AC voltage and frequency, the proposed control method provided an effective solution for large- scale renewable integration. However, the control method did not provide a solution for the interconnection of a weak power system. A series of control strategies such as nonlinear power damping controller [15], comprehensive control framework [18], and multi-variable droop control

      [19] were proposed to imitate VSCs as synchronous machines for integrating renewable power generation into weak power systems.

      The virtual synchro-converter control and PSC can provide great control performance for a VSC-HVDC system. In order to enhance the transmission capability, dominant control strategy has been used for the practical applications is the vector control comprising an outer loop and an inner loop [11]. Conclusions drawn in [11] indicates that it was the outer loop limits the power transmission capability when compare with the inner current loop. An advanced current vector control method was presented using a novel outer loop. During the dynamic state the parameters of the outer loop controllers designed using the H-infinity method needs to be updated.

      A ± 420kV/1250MW back-to-back VSC-HVDC project is planned to interconnect the AC systems between Chongqing City and Hubei Province in China. The project is proposed to employ symmetrical monopole

      Corresponding small signal modeling is established and the performance of the proposed method is studied.

   2. SYSTEM MODEL

      A VSC-HVDC system is presented in Fig. 1, where the basic controller for the rectifier and inverter stations are also presented. Normally, power is sent from he rectifier station to the inverter station. Power control is implemented at the rectifier station, while the inverter station controls dc-link voltage to ensure power balance. An advantage of VSC-HVDC over line commutated Converters HVDC (LCC-HVDC) is its reactive power compensation capability, which is provided by the reactive power controller or ac voltage controller [21].

      Fig.1: Two terminal VSC-HVDC system

      For a balanced three-phase system, Fig. 2 shows the equivalent circuit of a VSC connecting with an ac grid via a coupling inductor, where the resistor is neglected. The ac grid is modeled as an ac voltage source with an impedance.

      Fig.2: Equivalent circuit representation of a VSC

      configuration based on modular multilevel converter

      =

      sin (1)

      (MMC) topology. The main purpose of this paper is to

      2 cos

      improve the stability of the project. To restrain the AC system resonance, active damping impedance has been

      =

      (2)

      combined in the inner loop, such that this control strategy which automatically adjusts the compensated values.

      Where, P and Q are the active and reactive powers between two electrical nodes in ac systems with voltage magnitudes

      and . The quantities and are the phase-angle difference and line impedance between the two nodes. From (1 &2) it follows that the active power is mainly related to the phase angle , while the reactive power is more related to the voltage-magnitude difference. i.e., the active power is controlled by the phase angle of the VSC voltage, while the reactive power is controlled by the magnitude of the VSC voltage

      1. Characteristics of Weak AC Systems

         A weak ac system characterized typically by its high impedance [8]. As the ac-system impedance increases, the voltage magnitude of the ac system will become more sensitive to power variations of the HVDC system. The difficulty is measured by the short-circuit ratio (SCR), which is defined as a ratio of the ac-system short-circuit capacity to the rated power of the HVDC system

         This phenomenon indicates that the PCC needs to absorb reactive power to keep the AC voltage under steady states. Another important phenomenon is that the reactive power is not significantly required when |P| 0.5 (i.e. ||

         30°) [9].However, its demand increases dramatically when |P|>0.5 (i.e.||>30°) especially in the high power level range (|i.e. P| 0.7, ||45°) due to the strong

         nonlinearity of weak power system (SCR=1). Therefore, the reactive power providing capability of VSCs which interconnect with a very weak power system determine the dynamic performance of the weak power system. With dramatically increased demand for reactive power, a capacitor Cf is connected to the PCC to prevent the VSC from over current under steady states in practical applications.

   3. PV curve analysis

      SCR=

      (3)

      The impact of active power on the PCC voltage can

      where, is the short-circuit capacity of the ac system at the filter bus, is the rated dc power of the HVDC link.

      1. Operating limits

         When operating a VSC-HVDC link, it is essential to take into account the limitations of the converter in terms of active and reactive power transfer capability. One of the limit is the converter-current limitation, which is imposed by the current rating of the converter valve. Since both the active power and reactive power contribute to the current

         be unambiguously demonstrated by the natural PV analysis. To clarify the dynamic process, we suppose that only one regulator in the two channels is enabled during the dynamic states. The first progress enables the power control loop with reactive power unchanged. The second progress enables the AC voltage control loop with active power unchanged.

         In the first progress, that the value of the phase angle difference increases due to

         flowing through the valve, this limitation is manifested as a

         = sin1

         (4)

         circle.

         Fig.3: Operation limitation of a typical VSC- HVDC converter

         Accordingly, if the converter is intended to support the ac system with reactive power supply/consumption, the maximum active power has to be limited to make sure that the converter current is within the limit. Another limitation which determines the reactive- power capability of the VSC is the over or under-voltage magnitude of the VSC. The over-voltage limitation is imposed by the direct voltage level of the VSC. The under- voltage limit, however, is limited by the active-power transfer capability, which requires a minimum voltage magnitude to transmit the active power. Fig. 3 represents the PQ diagram for a typical VSC-HVDC converter [9].

         In the second state, the active power holds unchanged and the AC voltage controller regulates the q-axis current to produce more reactive power to the PCC.

         The response speed of the reactive power injected into the PCC determines the fluctuation of the PCC voltage in the analyzed dynamic process. However, the response speed of the reactive power is directly related to the AC voltage loop. Hence, whether the response speed of the AC voltage loop can match that of the active power loop will determine the system performance when the active power changes from a low level to a higher level. Consequently, reducing the response speed of the active power and improving the response speed of the AC voltage loop are a reliable way to restrain the fluctuation of the PCC voltage and enhance the system stability when the VSC is interconnected with a very weak power system.

         Pcom =

         UUs

         s

         s

         U

         U

         s

         s

         s

         |Q|. sign(p). d2 (5)

         Qcom =

         UUs

         U

         U

         s

         |P|. d1 (6)

         The reference values of active and reactive power

         generated from outer loop can be expressed as

         = + (7)

         = (8)

         = 1.5 ()( ) (9)

         = + ( )3 (10)

         Fig.4: Flowchart for perturbation and observation technique.

   4. COMPENSATION STRATEGY

      Where, () is the transfer function of PI regulator. d1 and d2 are the damping compensation coefficients. d3 is the tuning coefficient for the AC voltage loop

      1. Current control: The VSC-HVDC system consists of two terminal stations: one is the rectifier station and the other is the inverter station. Fig. 6 presents the controller of the rectifier station, in which active power and reactive power are to be regulated. Depending on the requirements of the application, the reactive power control could be substituted with ac grid voltage control. and

         are the converter output voltages and could be derived

         as (11) &(12), which is essentially the inner current control loop.[24]

         = ( ) + 1 + +

         (11)

         =

         (

         ) +

         +

         +

         1

         Fig.5: outer loop of compensation strategy

         A new control strategy which can achieve two objectives. One is that it would be better for the new control strategy to have the ability to restrain the rate of response of the active power. By doing so, more time is left to the AC voltage controller to generate corresponding reactive power. The other one is that control method should have the ability to enhance the response of the AC voltage loop to suppress the PCC voltage fluctuation[3].

         In Fig.5, P* is the commanded active power ordered from the dispatching center, Pref is the active power from

         (12)

         Fig.6: Controller of the rectifier station

         The currents in the inner loop can be written as

         = 2 /3( ) (13)

         = 2 /3( ) (14)

         p>outer loop (Pref =P* in steady state). |·| is the operation for

         absolute value, sign(·) is sign function, Pcom and Qcom

         The active damping impedances are described as follows

         are the compensated values of the active and reactive power, respectively. From the structure of the proposed

         =

         =

         .

         +

         +

         + .

         (15)

         (16)

         method, it uses the product of the PCC voltage fluctuations

         +

         and instantaneous power to generate the compensated power automatically. Hence, it does not need to dynamically adjust the compensation coefficients based on

         Where, and are the active damping resistors

         and are the gain coefficients of the HPFs

         and are the bandwidths

         the transmitted active power level when compared to [9]. The expression for compensated power can be written as,

         =

         =

         +

         and are the PI regulators of current loop in d and q axis.

      2. Power control: The rectifier station controls the active power transferred from the left-hand side grid to the converter, which is the outer control loop in dual dq control loops. Equation (17) computes the d -axis current reference that controls the active power flow

   5. PROPOSED METHOD

      Fuzzy Logic Control: L. A. Zadeh presented the first paper on fuzzy set theory in 1965. Since then, a new language was developed to describe the fuzzy properties of reality, which are very difficult and sometime even impossible to be described using conventional methods.

      = 2

      3

      (17)

      Fuzzy set theory has been widely used in the control area

      with some application to power system [29]. A simple

      The small-signal representation of the -axis current reference is then expressed as follows:

      = 2 [ ] (18)

      fuzzy logic control is built up by a group of rules based on the human knowledge of system behavior. Matlab/Simulink simulation model is built to study the

      3 2

      Where,

      is

      the

      rectifier PCC steady-state voltage.

      dynamic behavior of converter. Furthermore, design of

      fuzzy logic controller can provide desirable both small

      Since the direct reactive power compensation control is used throughout this paper, the q-axis current reference can be derived similarly as above

      signal and large signal dynamic performance at same time, which is not possible with linear control technique. Thus, fuzzy logic controller has been potential ability to improve

      = 2

      3

      = 2 [

      (19)

      ] (20)

      the robustness of compensator.

      3

      3

      2

      The basic scheme of a fuzzy logic controller is shown in

      1. DC voltage control: The inverter station regulates the dc-link voltage and maintains the active power balance. The inner current control loop is identical to the rectifier station shown in Fig. 3, whereas the outer loop is replaced by a dc voltage controller. The dc voltage controller is a typical PI controller.

      Fig.7 Small signal model of the integrated system with the proposed method

      Considering the dynamics on dc capacitor , the energy stored in the capacitor is

      Fig 8 and consists of four principal components such as: a fuzzification interface, which converts input data into suitable linguistic values; a knowledge base, which consists of a data base with the necessary linguistic definitions and the control rule set; a decision-making logic which simulating a human decision process, infer the fuzzy control action from the knowledge of the control rules and linguistic definitions; a de-fuzzification interface which yields non fuzzy control action from an inferred fuzzy control action [29].

      Fig.8. Block diagram of the Fuzzy Logic Controller (FLC) for Proposed method

      = 1

      2 (21)

      2

      1 2

      2

      2

      = P – = (22)

      =P-

      (23)

      Fig.9. Membership function of input, change in input

      Fig.10. Membership function of output

      Rule Base: The elements of this rule base table are determined based on the theory that in the transient state, large errors need coarse control, which requires coarse in- put/output variables; in the steady state, small errors need fine control, which requires fine input/output variables. Based on this the elements of the rule table are obtained as shown in Table, with Vdc and Vdc-ref as inputs

      Table1.1 Rule Base Table of 5*5 Matrix

   6. SIMULATION RESULTS:

      1.Dynamic Performance Analysis:

      The system with compensation method of AC voltage loop has a faster response than of without compensation method. It should be noted that enlarging the parameters of AC voltage PI regulator is able to accelerate the response speed of the without compensation. However, this is limited to a narrow range since it is easy to bring oscillatory instability if the system is underdamped

      Table 1.2 Dynamic Response comparison

      1. Stability Performance Analysis: The stability performance is analyzed by taking the configurations for the electromagnetic simulation model. The active power reference P* ordered from the dispatching center steps -0.1

         p.u. per 0.5s for both methods. The parameters of PI regulator are taken as = 0.005 , = 0.05 and the parameters of AC voltage regulator are taken as =

         0.035 , = 0.35 and the corresponding waveforms are plotted as shown in fig . However with this method of compensation strategy it presents satisfactory performance with PCC voltage almost unchanged during dynamic process.

      2. Stability limit simulation: The operating limits of electromangentic transient model are tested in both rectifier and inverter mode. The first graph plots the active power generated from the VSC. The PCC voltage in the middle graph. In the first dynamic process, voltage continuously sags before t=2.15s because | + | < ||and it gradually restores to steady point around t=2.54s.. (rectifier mode) In second dynamic process, voltage increased to maximum value 1.21p.u at t=4.06s due to | + | >

         ||.( inverter mode).

      3. Damping Performance Analysis: The damping performance of the method is simulated in order to validate the system as it cannot operate in steady state if a single virtual impedance is adopted. Two cases are studied i.e accelerating stability( P= -1.04p.u) and oscillatory stability. (P= 0.94p.u) . The system does not become unstable as long as does not exceed its limited values. The practical situations of the weak power system are simulated i.e. NPS and PPS are superimposed to fundamental voltage. Based on such operating conditions, four desired active power, P*=0.67p.u. and P*=0.4p.u. in rectifier mode, P*=- 0.74p.u. and P*=-0.4p.u. in inverter mode are simulated.

      4. Disturbance Rejection Capability Analysis: The practical situations of the weak power system are simulated

         i.e. NPS and PPS are superimposed to fundamental voltage. Based on such operating conditions, the rectifier and inverter mode waveforms are plotted as follows

      5. Fault Ride through Capability: When two percent 5 Negative Phase Sequence(NPS) and one percent 7 Positive Phase Sequence(PPS) harmonics are superimposed to the fundamental voltage to simulate the practical power grid.

      MAIN PARAMETERS OF CIRCUIT AND CONTROL SYSTEM:

      Parameters	Values
      DC voltage	±420
      Rated AC voltage (RMS)	525KV
      Rated active power	1250MW
      Grid mpedance(SCR=1)	220.5
      Compensation capacitor	4.1
      Transformer leakage	0.098H
      Inductance per arm 1 , 2 , 3	140mH
      	2.5 ,1.5,6
      	0.005 0.05
      	0.45 2.95
      ,	0,0.6
      ,	0.45,0.6

      Case1: PI Controller

      Fig 11 PCC Voltage Steps to 0.1p.u with and without compensation

      Fig 12 Active Power Steps to 0.1p.u with and without compensation

      Fig.13 Stability Performance of system upper: active power at PCC; middle: PCC voltage; lower: phase angle .when = 0.005 , = 0.05

      Fig.14 Stability Performance of system upper: active power at PCC; middle: PCC voltage; lower: phase angle . when = 0.035 , = 0.35

      Fig.15 Limits testing of the method. Upper: active and reactive power injected by VSC; middle: PCC voltage; lower: phase angle .

      Fig.16 Damping performance of the system for the step value of P= 1.04p.u upper: active power at PCC; middle: PCC voltage; lower: phase angle

      Fig.17 Damping performance of the system for the step value of P = 0.94p.u upper: active power at PCC; middle: PCC voltage; lower: phase angle .

      Fig.18 Disturbance resisting performance in the rectifier mode. Upper: active and reactive power injected by the VSC; middle: PCC voltage; lower: phase angle difference

      Fig.19 Disturbance resisting performance in the inverter mode. Upper: active and reactive power injected by the VSC; middle: PCC voltage; lower: phase angle difference .

      Fig.20 Fault ride-through performance of the proposed method. First: active and reactive power, second: voltages of PPS and NPS; third: phase angle ; fourth: DC voltage and current.

      CASE2: FUZZY Controller

      Fig.21 Dynamic Performance of the proposed method. Upper: Active Power Lower: PCC Voltage when Step 0.1p.u

      Fig.22 Stability Performance of the proposed method upper: active power at PCC; middle: PCC voltage; lower: phase angle . when = 0.005 ,

      = 0.05

      Fig.23 Limits testing of the proposed method. Upper: active and reactive power injected by VSC; middle: PCC voltage; lower: phase angle

      Fig.24 Damping performance of the proposed method for the step value of P= 1.04p.u upper: active power at PCC; middle: PCC voltage; lower: phase angle

      Fig.25 Damping performance of the proposed method for the step value of P= 0.94p.u upper: active power at PCC; middle: PCC voltage; lower: phase angle

      Fig.26 Disturbance resisting performance in the rectifier mode. Upper: active and reactive power injected by the VSC; middle: PCC voltage; lower: phase angle difference

      Fig.27 Disturbance resisting performance in the inverter mode of the proposed method. Upper: active and reactive power injected by the VSC; lower: phase angle difference

      Fig.28 Disturbance resisting performance in the inverter mode of the proposed method.PCC Voltage

      Fig.29 Disturbance resisting performance in the inverter mode of the proposed method.PCC Voltage

      Fig.30 Fault ride-through performance of the proposed method. First: active and reactive power, second: voltages of P.S and NPS; third: phase angle ; fourth: DC voltage and current.

      1. Comparative Analysis

         This section presents a comparative analysis between the traditional method and proposed method with respect to the time. From the above simulation results, we conclude that FUZZY controller is effectively controls and mitigate the Active power and PCC voltage when compared to PI controller. The Performance comparison of different control methods is shown

         Table5.1 Dynamic performance comparison

         Controller	Active Power(p.u)	PCC voltage(p.u)
         PI	1	1.25
         Fuzzy	0.98	1.15

         Table 5.2 Comparison of the testing of the system

         Controller	Active Power(p.u)	PCC voltage(p.u)
         PI	1	1.18
         Fuzzy	0.98	1.06

         Table 5. 3 Comparison of disturbance rejection of a system

         Controller	Rectifier Mode		Inverter Mode
         	Active Power	PCC voltage	Active power	PCC voltage
         PI	0.6	0.97	0.6	0.96
         Fuzzy	0.65	0.994	0.65	0.985

         Table 5. 4 Comparison of fault analysis

         Controller	Active Power(p.u)	PCC voltage(p.u)
         PI	0.5	0.5
         Fuzzy	0.2	0.2

   7. CONCLUSIONS

To enhance the operating limits and stability of VSCs which interconnect with weak power system, a power compensation control strategy combined with active damping control in inner loop has been proposed based on the analysis of PV curves. With the use of Fuzzy controller compensation strategy, the impact of compensation coefficients and steady state operating points on system performance has been studied based on the small signal model of the integrated system. The feasibility and effectiveness of the proposed control method has been validated. Electromagnetic transient simulation models are

considered and verified in MATLAB/Simulink environment for the effectiveness of the method on the performance of operating limits, damping performance, disturbance withstanding capability, and fault ride-through capability.

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25. L. Harnefors, A. G. Yepes, A. Vidal, and J. Doval-Gandoy, "Passivity based controller design of grid connected VSCs for prevention of electrical resonance instability ," IEEE Trans. Ind. Electron., vol. 62, no. 2, pp. 702-710, Feb. 2015.

26. J. He, Y. Li, D. Bosnjak, and B. Harris, "Investigation and active damping of multiple resonances in a parallel inverter based microgrid," IEE Trans.Power Electron., vol. 28, no. 1, pp. 234- 245, Jan. 2013.

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28. X. Guo, W. Wu, and Z. Chen, "Multiple complex coefficient filter based phase locked loop and synchronization technique for three phase grid interfaced converters in distributed utility networks ,"

    Fig. 31. Scheme of the linearized system

    APPENDIX B LINEARIZED DYNAMIC EQUATIONS

    The dynamic system equations are linearized as independent systems and they are connected according to Fig. 31 for the inner loop and outer loop control. The subscript indicates the value at the linearization point, indicates an average of the variable quantity and the superscript means that the variables have been transformed by means of the linearized park transformation

    1. Linearized Electrical System Equations: The electrical system is composed by the coupling filter and the electrical gridn(see Fig. 1). The state space representation of the linearized system is defined by

       = + (28)

       Y = (29)

       Where the state variables , inputs and outputs are

       = [ ] (30)

       = [ ] (31) Where the matrix is defined as

       1 0 0 0 0 0

       IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1194-1204, Apr.2011.

29. Suranjana Bharadwaj , Application of Fuzzy Logic in Load Frequency Control of Two Area Power System International

0 1 0 0 0 0

0 0 1 0 0 0

0 0

0 0

= 0 0 0 1 0 0

(32)

Journal of Scientific & Engineering Research, Volume 5, Issue 11, November-2014

0

0

0 0 0

0

APPENDIX A

[0 0

30

2

30

2

30

2

30

2 ]

UTILISED PARK TRANSFORMATION

The used park transformation is defined as

|| = |()||| (24)

1. Linearized PLL Equations: The PLL is used in order to orient a control with the electrical grid angle. In the linearized model the PLL introduces the angle deviation when the linearized system is moved from the linearization

|

| = ()1|

| (25)

point. The PLL has been linearized following [11]. The

PLL linearised transfer function representation is

where|xabc| is a vector with the three-phase quantities

in the 0frame and is a vector with the transformed

+

=

2++

(33)

quantities in the frame.

The transformation [Tdq0] matrix can be written as cos cos( 2) cos( + 2)

3). Linearized Park Transformation and Inverse- Transformation Equations: The linearized Park transformation (see Appendix B) expressed is given by

3 3 [

] = [

][

] (34)

[()] = 2

sin sin( 2) sin( + 2) (26)

3 3 3

1 1 1

Where [ ] is

[ ] =

[ 2 2 2 ]

and its inverse

1

1 2 ( 2) ( 2) 1

cos 0 sin 0 sin 0 0 cos 0 0 [ sin cos cos sin ] (35)

0 0 0 0 0 0

0 0 0 0 0 0

and the linearized inverse transformation is

[()] = [ 3 3

3 ] (27)

1

( + 2) ( + 2) 1

[ ] = [ ] [ ]

(36)

3 3 Where [ ]1 is

[ ]1=[ cos 0 sin 0 cos 0 0 sin 0 0 ](3

5) Classic Outer Loop Equations: For the study of the

7)

sin 0 cos 0 cos 0 0 sin 0 0

whole classic system, the outer loop equations are

= (46)

4) Inner Loop Equations: The vector current control equations are

= (38)

= + (39)

Where the state variables, inputs and outputs are

= [ ] (40)

=

[ ] (41)

= + (47)

where the state variables, inputs, and outputs are

= [ ] (48)

= [ ] (49)

= [ ] (50)

are the active power and voltage error.

[

[

Where the matrix gains are defined as

= [

] (42)

= 1 0

1 0 ] (51)

is the current error, defined as the difference

0 1 0 1

0

between . The matrix gains are

= [

0

] (52)

= [

1 0 1

0 1 0

0 0 0

1 0 0 ] (43)

= [

0

0 ] (53)

=

=

[

0 ] (44)

0

0

0

0

= [ 0

1 0 ](45)

0 1