1

Analytical modelling of a square-wave controlled cascaded multilevel STATCOM R. Sternberger, Student Member, IEEE, and D. Jovcic, Senior Member, IEEE

Abstract—The aim of this paper is to present an analytical, statespace model of an indirect, voltage-controlled cascaded-type multilevel Static Synchronous Compensator (STATCOM) with ‘square wave control’. The multilevel converter model is segmented into a dynamic and static part in order to accurately represent all internal feedback connections. Each voltage component is analyzed in detail and described mathematically by an averaged expression with an equivalent capacitance. The STATCOM model is linearised and linked with a DQ frame ac system model and the controller model, and implemented in MATLAB. The controller gains are selected by analyzing the root locus of the analytical model, to give optimum responses. The validity and accuracy of the proposed model is verified against non-linear digital simulation PSCAD/EMTDC in the time and frequency domain. The model is very accurate in the subsynchronous range, and it is adequate for most control design applications and practical stability issues below 100Hz. Furthermore, the developed model can be used for multilevel cascaded converters which are exchanging real power. Index Terms—Modelling, multilevel converter, STATCOM, static VAR compensators, state-space methods.

I. INTRODUCTION In recent years, the multilevel converters have become increasingly popular in high power transmission/distribution systems and in industry applications [1]-[5]. In contrast to a conventional two-level voltage source converter, that works with Pulse Width Modulation (PWM), such multilevel converters use a number of (low voltage) series-connected capacitors to generate a high ac voltage. This allows higher power handling capability with reduced switching power losses and harmonic distortion [3]. The main types of multilevel converters are: diode-clamped, flying-capacitor and cascaded inverter [3]-[5]. By comparing these different topologies, whilst considering harmonic level, losses and component costs, the cascaded multilevel converter with ‘square wave control’ is found to be the optimum solution for STATCOM applications [3],[5]-[8]. The modular structure of this converter, with a number of identical Hbridges, makes this converter very flexible in terms of power handling capability. The use of ‘square wave control’ results in a single switch-on and -off per cycle for each switch, which brings benefits of low switching losses. In addition, since the control angle at each capacitor can be used to cancel one harmonic, a low harmonic distortion can be achieved [6],[9]. The dynamics of such complex converters are mostly analyzed using EMTP type programs, like PSCAD/EMTDC The authors are with the Engineering department, University of Aberdeen, Aberdeen, AB24 3UE, UK: [email protected], [email protected]

[10]. However, these simulators provide only trial and error type studies in time domain. The demanding analysis and design tasks become very time-consuming since a laboured search has to be executed to find the best solution. Alternatively, a suitable and accurate analytical system model would provide faster design routes. Such a model would provide the ability to use dynamic systems analysis techniques (e.g. eigenvalue and frequency domain methods) and modern control design theories, resulting in a shorter design time and advanced controller configurations [8],[11]. Comparing to conventional voltage source converters, the analytical modelling of multilevel converters is more challenging. A conventional PWM-controlled voltage source converter has fixed structure and the switching frequency is high which implies that intervals between switchings are short and conventional averaging approaches can represent converter dynamics [12]. A multilevel converter with squarewave control, on the other hand, fundamentally changes its structure as the capacitors are switched in and out of the current path and therefore these changes may have significant influence on the system dynamics. Further, each cell is switched only twice per cycle and therefore variables may undergo notable excursions within a single conduction interval. One of the primary challenges in the dynamic modelling of a square-wave converter is therefore the adequate dynamic representation of cell variables between the switching instants. The multilevel converter model in [8] is in convenient statespace form but it adopts overly simplified equivalent capacitance which is only valid for a particular number of levels and does not consider firing angles on individual cells. Similarly, the state space model in [9] is of little practical use, since the equivalent capacitance has to be tuned for every change in converter parameters, using identification methods. The researchers in [13] developed a frequency domain model of a square-wave controlled converter. Since it only considers a single cell with uncontrollable, full 180deg conduction periods, this model does not address the aforementioned modelling challenge. The research presented in [14] represents the structure of a multilevel converter at a wide range of frequencies in order to provide a model for phasor studies and harmonic analysis. However, this is essentially a static model, and it ignores completely the control influence. Moreover, since the model neglects the transfer of active power, it cannot be used for dynamic modelling. This paper aims to develop a dynamic analytical model of a multilevel cascaded STATCOM converter that is convenient for system stability studies and for analysis of interactions with ac systems. We attempt to establish generic modelling principles that are applicable for a range of multilevel

2

II. MULTILEVEL TEST STATCOM CONFIGURATION The selected test STATCOM, which contains a cascaded (chain-circuit) multilevel converter and its connection to the ac system, is shown in Figure 1. The multilevel STATCOM system includes a 3-phase transformer and three strings of single-cells that are connected in series. A single-cell is here a full-bridge, single-phase inverter with a capacitor as dc source [15]. Note that only chain circuit A with its IGBTs, diodes and capacitors is shown, but similar groups of single-cells are also connected in branches B and C. The STATCOM parameters that determine the main circuit are: number of cells per phase X, dc capacitances CDC1 to CDCX, buffer inductance LDC, transformer leakage inductance LTR and transformer ratio nTR. The relationship between the number of single-cells per string X, and the number of levels N for a cascaded multilevel converter is given by X=(N-1)/2. In further studies it is assumed that all capacitances are equal CDC=CDC1=…=CDCX and that the steady-state voltages across these capacitors are also identical VDC0=VDC01=…=VDC0X (with the help of a suitable controller). To ensure low converter losses and dc voltage ripple, the STATCOM system parameters are optimized by following the algorithm proposed in [16] and their values can be found for the selected test systems in Table I in the Appendix. At the Point of Common Coupling (PCC), the STATCOM is connected to ac circuit represented by resistance RAC and inductance LAC. The ratio between XAC (ω*LAC) and RAC is assumed to be constant (XAC/RAC=10). The converter switching losses are reduced by the use of ‘square wave control’ where each device has only one turn-on and turn-off per cycle. The harmonic distortion is reduced by calculating the firing angle at each cell to cancel one harmonic. In view of the fact that triplen harmonics are cancelled in the converter delta connection, the firing angles

Y/∆

RAC VAC 0

230kV 50Hz

abc

Phase Locked Loop

magnitude

VAC +

Vs ϕm

o

CDC1

VGrid

CDCX

LAC .

VDC0X

Transformer STR, LTR, nTR

PI Controller

+ -

VDC01

PCC

LDC

3-phase ∆ -connected cascaded converter with X cells

IS

ϕm

Firing pulses Gate Pattern Logic

αa=constant VRefAC Controller Fig. 1: Multilevel STATCOM test system with indirect control.

Mm=1

for the individual cells are calculated for cancellation of the lowest odd, non-triple harmonics [9], and their values are shown for selected test systems in Table II in Appendix. Note that these calculated firing angles can cancel only steady-state harmonics but as loading changes, harmonics increase [14]. The ideal stepped output voltage of a 13-level converter is shown in Figure 2. The STATCOM uses indirect control [15], as shown in Figure 1. This control has as benefit a constant modulation ratio Mm=1, resulting in constant control angles αι and therefore in low harmonics even at low converter voltages. The dc voltage is affected by controlling the angle ϕm (i.e. the active power transfer) [15]. VS , VAC [pu]

cascaded converters, even those which exchange real power. A modular modelling approach is adopted to represent the complexity of the system with the benefit that each individual subsystem can be analysed independently. This analytical model employs all parameters and variables with physical meaning and therefore, it can study variations in ac and dc system structures.

6

ϕm

4 α 2 2 α1

α

0 −π/2

α4

α

α

vAC(t) 6

vS(t)

5

3

−π/4

0 π/4 π/2 ω*t [rad] Fig. 2: Ideal stepped output voltage of 13-level converter (ω=2*π*f)

III. CASCADE MULTILEVEL CONVERTER MODEL A. General Voltage and Current Representations The phasor diagram of the STATCOM system is shown in Figure 3. It is assumed that the system is balanced and all variables are represented with fundamental frequency phasor. Note that all ac voltages are shown as line-neutral magnitudes. With reference to Figure 3, the instantaneous ac voltage vAC(t) and the ac current iS(t) can be presented as: v AC (t ) = VAC ⋅ cos(ωt )

(1)

iS (t ) = I S ⋅ cos(ωt + ϕ I ) .

(2)

The equation (2) can also be represented using dqcomponents in the rotating dq frame, where d and q indices appropriately label axes: iS (t ) = I Sd ⋅ cos(ωt ) − I Sq ⋅ sin(ωt )

(3)

where: I Sd = I S ⋅ cos(ϕ I )

(4)

I Sq = I S ⋅ sin(ϕ I )

(5)

q IS

ISq

ϕm

ϕI ISd

VAC VS

d

VAC - AC voltage at PCC ϕI - Current phase angle VS - STATCOM voltage ϕm - Control output (STATCOM voltage phase angle) IS - STATCOM current

Fig. 3: STATCOM phasor diagram where the general reference frame is linked with VAC.

B. Instantaneous Non-linear Dynamic Model of a Single-cell In the first part, we investigate a single-cell of a cascaded multilevel converter, as shown in Figure 4. The single-cell

S v DC (t ) S − v DC (t )

;− π2 + α − ϕ m ≤ ωt ≤ ; π2 + α − ϕ m ≤ ωt ≤

π 2

3π 2

− α − ϕm − α − ϕm

(6)

; elsewhere

0

It is also seen that the converter is supplied by a rectified ac current (positive: S1,S4 ON; negative: S2,S3 ON) and that the current iSDC(t), which flows through the capacitor, has the following form:   S i DC (t ) =   

i S (t )

;− π2 + α − ϕ m ≤ ωt ≤

− i S (t )

; 2 + α − ϕ m ≤ ωt ≤ π

0

π 2

3π 2

− α − ϕm − α − ϕm

(7)

; elsewhere

The single-cell variables in Figure 5 are shown under steadystate conditions for ideal converter. Under dynamic transients and assuming some internal losses, the instantaneous variables will have different waveforms. Figure 6 shows the single-cell variables assuming non-zero control angle ϕm, which results in an active power transfer. The transient curves in Figure 6 have different traces comparing with Figure 5 since: • The current will have some non-zero average value, ISDCa. • The dc voltage vSDC(t) is not constant, but it is increasing along a sine curve in the conducting interval (from initial value VDC0). The capacitor dc voltage vSDC(t) can be represented as: V DC 0 ;0 ≤ ωt ≤ − π2 + α − ϕ m  (8)  S S v DC (t ) = V DC 0 + v DCdyn (t ) ;− π2 + α − ϕ m ≤ ωt ≤ π2 − α − ϕ m  S ; π2 − α − ϕ m ≤ ωt ≤ π2 V DC 0 + V DCdyn (π2 − α − ϕ m )

where vSDCdyn(t) stands for the dynamic voltage component which depends on the current and the control angle, and has therefore a direct influence on the converter dc and ac voltage: S1 iS(t)

vSS(t)

S3

D1 CDC

vSDC(t) D3

S2

iSDC(t) S4

D2 Fig. 4: Single-cell of a cascaded converter

D4

VS [pu] IS [pu] DC DC VS [pu] S

1

v

AC

vS (t) S

(t)

0

iS(t)

−1 S1,S4

S2,S4

1 0 −1 1

S2,S3

S1,S3

S2,S4

iS (t) DC V

0 1

V

0

vS (t)

DC0

DC

(t) vS S DC0

ϕm −1 α −π/2 −π/4

α ϕ m 0

π/4

π/2 3∗π/4

π

ω *t [rad] Fig. 5: Currents and voltages of a STATCOM system in steady-state

5∗π/4

 1 S (t )dt ;− π2 + α − ϕ m ≤ ωt ≤ π2 − α − ϕ m (9) ⋅ i DC  S v DCdyn (t ) =  C DC ∫  0 ; elsewhere 

VDC0 is the average steady-state value of dc voltage of each single-cell. It is shown in [16] that VDC0 is constant and can be found using static (power flow) equations:   1 V DC 0 = VS − ⋅ IS    ω C ⋅ eq1   with Ceq1 =

4 X   ⋅ ∑ [cos(α i )] ⋅ M m  π i =1  

π X

X

i =1

i =1

X ⋅ π − 2∑ [α i ] − ∑ [sin( 2α i )]

⋅ C DC

(10) (11)

where Mm defines the converter modulation ratio (Mm=1 if STATCOM is using indirect control). We observe that VDC0 expression does not involve any dynamics, but it will have a direct static influence on the magnitude of the converter ac voltage. This component depends on the current magnitude IS, the firing angles α and capacitance CDC, but not on the control angle ϕm. S1,S4

S1,S3 IS [pu] DC

  v SS (t ) =   

VDC [pu]

consists of one capacitor and four switching devices (each contains one IGBT and an antiparallel diode). The instantaneous ac voltage of such a single-cell vSS(t) is shown in Figure 5 together with the ac voltage at the PCC vAC(t), the capacitor voltage VDC(t), and the converter ac current iS(t). The variables iS(t) and vSS(t) (all variables in top graph of Figure 5) are at system-level, for a multilevel converter. These variables are assumed to be of ideal sine shape because of averaging action of multiple levels and filtering action of passive components. Since a single-cell is switched into the electrical circuit within the interval -π/2+α-ϕm≤ωt≤π/2-α-ϕm, (S1,S4 ON or S2,S3 ON), the converter ac voltage vSS(t) in time-domain can be represented as:

VAC, I S [pu]

3

1 0 α -ϕm −1

S2,S4

iS (t) DCa

α +ϕm

iS (t) DC

vS (t) DCa vS (t) DC

1

vS (t) DCdyna

VDC0

0 −π/2

−π/4

0

π/4

π/2

ω *t [rad] Fig. 6: Dynamic dc current and dc voltage of a single-cell

C. Dynamic DC Voltage Component VDCdyn The dynamic component of the STATCOM ac voltage in (9) plays an important role in the dynamics of the overall system. Since the non-linear current waveform in (7), which is driving the dynamic component VSDCdyn, cannot be represented directly

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in a continuous model, an average dc current iSDCa(t) will be derived firstly. This average current, also illustrated in Figure 6, will result in average value of dynamic dc voltage component VSDCdyna that shows the same cumulative dc voltage change over one half-cycle as the instantaneous dynamic component VSDCdyn, i.e. vSDCdyna(π)=vSDCdyn(π). Since system dynamics are assumed to be slower than structural changes within one-half cycle, there will be no loss in accuracy if all variables are represented as constant/linear curves, which give accurate values at the end of each half-cycle. The average dc current ISDCa of the non-linear current waveform iSDC(t) over one half-cycle can be found averaging (7):

VSdS = VSqS =

S I DCa =

1

π

⋅ −

∫i π

S DC

(t ) ⋅ d (ωt ) =

2

1

π

⋅

π 2 −α −ϕ m

∫i

Placing (3) in (12), we can derive expression for the averaged current. At this stage, we assume that ISd and ISq in (3) are constant over the integration period in (12). This can be justified since the current dynamics are slower than the voltage dynamics and can be neglected over the short segment of half fundamental cycle. The dc current equation becomes:

[

]

(13)

The voltage expression in (9) can not be directly averaged since the essential dynamic properties of the converter would be lost. Firstly, the equation (9) is differentiated, which gives a non-linear but algebraic expression on the right side. The resultant expression on the right side of (9) can be averaged as in (12). Following these steps we obtain dynamic equation for the average dynamic component of STATCOM voltage in sdomain (LaPlace transformation): S S s ⋅ V DCdyna = 1 C DC ⋅ I DCa

(14)

where s denotes coefficient of Laplace transformation. The average dynamic voltage VDCdyna in (14) uses linear approximation to represent a sine-shaped dynamic voltage component, as seen in Figure 6. Replacing (13) in (14), the formula for dynamic dc voltage component is obtained as: S s ⋅ V DCdyna =

2

π ⋅ C DC

[

⋅ cos(α ) ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m )

S DC −π 2 + − m

π 2 −α −ϕ m

2

∫v

S DC −π 2 +α −ϕ m

π

(t ) ⋅ cos(ωt ) d (ωt )

(16)

(t ) ⋅ cos(ωt + π 2)d (ωt )

(17)

S VSdS = VSdynad + VSS0 d

=V

S Sq

S Sdynaq

+V

(18) (19)

S S 0q

(12)

(t ) ⋅ d (ωt )

S −π 2 +α −ϕ m

S I DCa = 2 π ⋅ cos(α ) ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m )

∫αv ϕ

π

The Fourier coefficients in (16)-(17) are chosen considering the coordinate frame alignment in Figure 3. Replacing (8) in (16)-(17), the ac voltage will consist of two components:

V π 2

π 2 −α −ϕ m

2

]

(15)

Assuming a typical STATCOM application, we will have ϕm≈0 and it can be seen in (15) that the dynamic ac voltage will depend mainly on the active current ISd and the control angle α. Therefore, in a typical STATCOM application, the component ISq in (15) can be neglected with a limited loss in accuracy. However, the converters with active power flow should include the complete model. D. Converter dynamic Model at Fundamental Frequency The fundamental ac voltage of the switched dc voltage components can be found if the Fourier Transformation at frequency ω, (Fω) is used. We calculate first coefficients of the Fourier series to obtain d and q axis components of ac voltage:

1) Dynamic Voltage Component VSSdyna: In order to preserve the dynamic nature of the converter voltage in (15), we need to apply Fourier transformation to differential of converter voltage. For a given function y, the following property of Fourier transformation is employed:

sFω(y)= Fω(sy)

(20)

Placing (15) in (16)-(17), using (20) and assuming a constant current magnitude IS across the integration interval, the dynamic equations for the ac voltage components are:

[ ⋅ [I

] )]sin(ϕ

S s ⋅ VSdynad = 1 CeqS 2 ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m ) cos(ϕ m )

s ⋅V

S Sdynaq

=1 C

with C eqS 2 =

S eq 2

Sd

⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m

π2 8 ⋅ cos 2 (α )

⋅ C DC

m

)

(21) (22) (23)

2) Static Voltage Component VSS0: Placing (10) in (16)-(17), the corresponding static ac voltage components become: VSS0 d = k S ⋅ VDC 0 ⋅ cos(ϕ m )

(24)

= k ⋅ VDC 0 ⋅ sin(ϕ m )

(25)

V

S S 0q

S

with k S = 4 ⋅ cos(α ) π

(26)

The expressions for equivalent capacitance in (11) and (23) represent some imaginary capacitance which enables us to represent multilevel converter dynamics using simple equations for a single-capacitor system. It is observed that these capacitances depend only on the capacitance CDC and the control angle α, which are constant for a given STATCOM topology. Therefore, the equivalent capacitance will not change as the operating conditions are changing. It is noted that the q components in (22) and (25) will be small in a typical STATCOM. E. Non-linear Dynamic Model for a Multilevel Converter The above modelling approach can be adopted for each cell of a multilevel converter. The fundamental component of ac voltage for a cascaded STATCOM with X cells is obtained as: x

V Sd = ∑ V SdS _ i i =1

(27)

5 x

V Sq = ∑ V SqS _ i

(28)

i =1

VGrid

VSd, VSq

DC Electrical System

AC System ISd, ISq

The capacitor voltage balancing is normally used with multilevel converters and such control function is very fast compared with main control responses [17]. Consequently, the DC voltages on individual cells can be assumed identical. This enables us to use same equations for each cell (21)-(26), and only the firing angles will differ. Replacing (21)-(26) in (27) and (28), it can be shown that the fundamental ac voltage components of a multilevel converter are similar to (21)-(22) and (24)-(25). The equivalent capacitance Ceq2 and the parameter k for a multilevel converter are different, however, and result in:

[

]

 x  C eq 2 = π 2  8 ⋅ ∑ cos 2 (α i )  ⋅ C DC  i=1  x   k =  4 ⋅ ∑ [cos(α i )] π  i =1 

(29) (30)

where αi are the control angles on individual cells. The final d-component VSd of the multilevel converter model is therefore obtained using (21) and (24):

[

]

s ⋅ VSdynad = 1 Ceq 2 ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m ) cos(ϕ m )

(31)

VS 0 d = k ⋅ VDC 0 ⋅ cos(ϕ m )

(32)

and using (22) and (25), the q-component VSq is:

[

]

s ⋅ VSdynaq = 1 Ceq 2 ⋅ I Sd ⋅ cos(ϕ m ) + I Sq ⋅ sin(ϕ m ) sin(ϕ m )

(33)

VS 0 q = k ⋅ VDC 0 ⋅ sin(ϕ m )

(34)

IV. ANALYTICAL STATCOM SYSTEM MODEL A. Model Structure The STATCOM system consists of the multilevel converter, a transformer, a control system with Phase Locked Loop (PLL) and it is connected with an ac system. The STATCOM model schematic diagram is shown in Figure 7 with all linking variables between the sub-systems. It can be seen that the STATCOM model is segmented in two main sub-systems: an ac system and a dc system, which is a conventional approach with converter modelling [11]. Each of these subsystems is represented as a stand-alone state-space model, and the dc system includes the converter model, the controller model and the PLL model. The final state-space model can be written in the following matrix form: [9], [11] s ⋅ x S = AS ⋅ x S + BS ⋅ u Ref

(35)

y S = C S ⋅ x S + DS ⋅ u Ref

(36)

where:

BDCAC C AC   ADC AS =  AAC   B ACDC C DC T u Ref = [VRefAC VGrid ] y S = [V AC ]

(37)

ϕAC

ϕPLL

PLL System

ϕm VAC

PI Controller

Feedback Filter ω²f s²+2ζωfs+ω²f

KP

-

+

+

KI 1/s

- +

ϕRefAC

Delay Filter 1 Tf*s+1

DC Model

VRefAC

ϕRefAC - Calculated firing angle ϕPLL - PLL reference angle ϕm - Actual firing angle

VRefAC

- Reference ac Magnitude

VAC, ϕAC - Magn., Phase angle of νAC νAC=VAC cos(ωt+ϕAC)

Fig. 7: Structural diagram of STATCOM system model

In (35) and (36), xS is the state-variable vector, uRef represents the input variable vector (including reference voltage VRefAC and disturbance voltage VGrid) and yS stands for the output variable vector. The sub-matrices in the system matrix AS are associated with the individual subsystems: dc for dc subsystem and ac for ac subsystem. The indices with input matrix B and output matrices C indicate the actual model (first index) and the linking subsystem (second index). B. AC Model Assuming a symmetrical and balanced ac system, the ac grid can be modelled as a single-phase dynamic system as shown in Figure 8. The instantaneous circuit variables are used as the states. Since the ac grid voltage VAC cannot be measured directly in a state-space model, an artificial capacitance CART, which has a very small value (no dynamic effects to the system), is used to accommodate state-space modelling. It is shown in [16] that conduction losses are dominant in a converter with square-wave control and therefore, the converter losses are represented in the model as a seriesconnected resistance Rloss. The final model is subsequently converted to the dq rotating frame using Park’s transformation as described in [11]. IAC

IS RAC

VGrid

LTR+LDC*nTR² Rloss*nTR²

LAC CART

VAC

VS

Fig. 8: Equivalent ac circuit

C. Converter Model The final STATCOM model is presented in the dc submodel and the ac voltage is the sum of dynamic and static voltage components (31)-(34). Since these expressions are strongly non-linear, they are linearised in a usual manner. The linearised STATCOM model is linked with the PLL and the controller model, as shown below in the next section, to create the final dc model. D. Controller and PLL The STATCOM system uses indirect control, which consists of a proportional plus integral (PI) controller, to regulate the ac voltage as seen in Figure 7. The ‘PI’-controller produces

6

Imag

Gain increase

0 Selected Eigenvalue -50

-100

-60

-30 -20 -10 Real Fig. 9: Dominant branch of the root locus for test system 1.

-50

-40

0

B. Reference Step Input 1) 13-level converter A reference step response for test system 1 (13-level converter) is shown in Figure 10. It can be seen that the developed model shows good accuracy for all considered variables, i.e. the voltage magnitude, the STATCOM current and the control angle ϕm. Figure 11 shows the model verification for test system 2 (13level converter). It is concluded that the analytical model shows very good accuracy even for different system topologies. 2) 9-level converter Figure 12 shows the system response when test system 3 is used, i.e. with a 9-level converter. It is seen that excellent model matching occurs for all compared variables. C. Reference Disturbance Input 1) 13-level converter The perturbation on the remote ac source tests the model ability to represent perturbations on remote ac source and the coupling between the ac and dc sub-models. The model verification for disturbance inputs is shown in Figure 13 and 14 for test system 1 and test system 2, respectively. It can be seen that a good response matching exists for both systems, and similar matching was confirmed for all the other variables that are not shown. 230.67 230.66

Reference Average model PSCAD

230.65 0

0.05

ωt [sec]

0.1

0.15

a) System voltage 0.176 0.174 0.172 0.17

Average Model PSCAD 0

0.05

ωt [sec]

0.1

0.15

b) AC Current IS Control angle ϕm [deg]

A. Selection of Controller Gains The controller gains are selected in this study by analysing the dominant eigenvalues of the system and they are tuned to meet the following requirements: - Rise time less than 3 cycles of fundamental frequency, - Overshoot below 10%, - Settling time less than 10 cycles of fundamental frequency. Figure 9 shows the root locus of test system 1 as the open loop gain changes with the nominal system configuration. The PI controller zero is selected for this particular test system at z1=–KI /KP=-2.8, which is close to the slowest system pole, in order to improve speed of response and robustness. It can be seen that for small open-loop gains the dominant eigenvalues are only real poles resulting in slow rise times and large damping. It can also be seen that large controller gains will result in significant overshoots, faster rise times and a reduced damping. Best controller gains are then selected for each individual test system from its root locus to give optimum responses according with the above requirements, and final controller gains are given in Table I in the Appendix.

50

AC voltage VAC [kV]

V. MODEL VERIFICATION The derived analytical model is implemented in Matlab and tested in the time domain against the detailed, non-linear simulation software PSCAD/EMTDC. Three, largely different, test systems are simulated in order to verify the accuracy and flexibility of the analytical model, and all parameters are given in Table I in the Appendix. The test system 1 is a 30kV, 70Mvar 13-level (X=6) STATCOM system, whereas test system 2 is operating at 15kV and has a rated reactive output power of 25Mvar. To confirm the model accuracy with different ac system parameters, the grid strength in test system 2 is reduced by the factor of 3, compared with test system 1. A third test system is introduced to study the model robustness for a STATCOM system with different number of levels. This system is similar to test system 1, but it involves a 9-level converter (X=4). The operating point of all systems is set close to their rated reactive power QST. The MATLAB model enables insightful study of changes in parameters or structure since all parameters are directly reflected in the symbolic form in the final model. The perturbation signal used in step responses is the reference voltage VRefAC, as seen in Figures 1 and 6. In disturbance tests, a step on the remote source voltage magnitude, given by VGrid in Figures 1 and 7, is introduced.

100

AC current IAC [kA]

the phase angle ϕm, which is common for all cells. The reference signal VRefAC is provided by an external or a system control [15]. The delay filter in Figure 7 is introduced to represent firing control circuit delay and does not represent actual system dynamics. This filter accounts for the delay in the signal transfer in a discrete system [18], and the delay time constant is determined as Tf=1/(50*3*2*X), where 50 represents fundamental frequency, ‘3’ is the number of branches and ‘2’ considers two switchings per cycle per each branch. The main role of PLL is to synchronise the converter firing signals with the ac system, and the latest D-Q-Z type [18], which has the advantage of accurate phase information even with a distorted ac system voltage waveform, is used here.

Average model PSCAD

0.23 0.21 0.19 0.17

0

0.05

ωt [sec]

0.1

0.15

c) Control angle ϕm Fig. 10: Test system 1 response following a 0.02kV voltage reference step

Reference Average Model PSCAD

230.75 230.7

0

0.05

ωt [sec]

0.1

0.15

0.07

230.65

0.066 Average Model PSCAD

0.062 0.05

ωt [sec]

0.1

0.15

Average Model PSCAD

0.5 0.4 0.3 0.2

0

0.05

ωt [sec]

0.1

0.15

AC voltage VAC [kV]

b) Control angle ϕm Fig. 11: Test system 2 response following a 0.1kV voltage reference step

230.75 Reference Average model PSCAD

230.7

230.65 0

0.05 ωt [sec]

0.1

0.15

AC current IAC [kA]

a) System voltage 0.2 0.19 0.18

Average Model PSCAD

0.17 0

0.05

ωt [sec]

0.1

0.15

Control angle ϕm [deg]

b) AC Current IS Average model PSCAD

0.8

0.4 0.2 0.05

ωt [sec]

0.1

AC voltage VAC [kV]

230.65

230.63

Reference Average model PSCAD 0

0.05

ωt [sec] 0.1

0.15

Fig. 13: Test system 1 response following a 0.02kV disturbance step change

0.15

230.71

0 -10 -20 -30 -40

Average model PSCAD

-200 -300 -400 10

Average model PSCAD 30

100 Frequency [rad]

300

600

Fig. 16: Test system 1 frequency response of the linear model compared with PSCAD/EMTDC frequency response 0 -10 -20 -30 -40

Average model PSCAD

-100

V

AC voltage VAC [kV]

ωt [sec] 0.1

-100

0.15

c) Control angle ϕm Fig. 12: Test system 3 response following a 0.1kV voltage reference step

230.64

0.05

D. Frequency Response of AC Grid with STATCOM Since a time domain study cannot reveal the frequency bandwidth for the model accuracy, the test systems 1 and 3 are also tested against PSCAD in the frequency domain. As PSCAD does not possess the frequency response ability, the individual frequency components are injected (1Hz step) with measurement of the gain and phase of the system voltage VAC at the PCC. The experimental frequency response is performed in the frequency range 1.6Hz