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Implement phasor model of three-phase static synchronous series compensator

FACTS/Power-Electronics Based FACTS

The Static Synchronous Series Compensator (SSSC) is a series device of the Flexible AC Transmission Systems (FACTS) family using power electronics to control power flow and improve power oscillation damping on power grids [1]. The SSSC injects a voltage Vs in series with the transmission line where it is connected.

**Single-line Diagram of a SSSC and Its Control
System Block Diagram**

As the SSSC does not use any active power source, the injected voltage must stay in quadrature with line current. By varying the magnitude Vq of the injected voltage in quadrature with current, the SSSC performs the function of a variable reactance compensator, either capacitive or inductive.

The variation of injected voltage is performed by means of a Voltage-Sourced Converter (VSC) connected on the secondary side of a coupling transformer. The VSC uses forced-commutated power electronic devices (GTOs, IGBTs or IGCTs) to synthesize a voltage V_conv from a DC voltage source.

A capacitor connected on the DC side of the VSC acts as a DC voltage source. A small active power is drawn from the line to keep the capacitor charged and to provide transformer and VSC losses, so that the injected voltage Vs is practically 90 degrees out of phase with current I. In the control system block diagram Vd_conv and Vq_conv designate the components of converter voltage V_conv which are respectively in phase and in quadrature with current. Two VSC technologies can be used for the VSC:

VSC using GTO-based square-wave inverters and special interconnection transformers. Typically four three-level inverters are used to build a 48-step voltage waveform. Special interconnection transformers are used to neutralize harmonics contained in the square waves generated by individual inverters. In this type of VSC, the fundamental component of voltage V_conv is proportional to the voltage Vdc. Therefore Vdc has to vary for controlling the injected voltage.

VSC using IGBT-based PWM inverters. This type of inverter uses Pulse-Width Modulation (PWM) technique to synthesize a sinusoidal waveform from a DC voltage with a typical chopping frequency of a few kilohertz. Harmonics are cancelled by connecting filters at the AC side of the VSC. This type of VSC uses a fixed DC voltage Vdc. Voltage V_conv is varied by changing the modulation index of the PWM modulator.

The SSSC (Phasor Type) block models an IGBT-based SSSC (fixed DC voltage). However, as details of the inverter and harmonics are not represented, it can be also used to model a GTO-based SSSC in transient stability studies.

The control system consists of:

A phase-locked loop (PLL) which synchronizes on the positive-sequence component of the current I. The output of the PLL (angle Θ=ωt) is used to compute the direct-axis and quadrature-axis components of the AC three-phase voltages and currents (labeled as Vd, Vq or Id, Iq on the diagram).

Measurement systems measuring the q components of AC positive-sequence of voltages V1 and V2 (V1q and V2q) as well as the DC voltage Vdc.

AC and DC voltage regulators which compute the two components of the converter voltage (Vd_conv and Vq_conv) required to obtain the desired DC voltage (Vdcref) and the injected voltage (Vqref). The Vq voltage regulator is assisted by a feed forward type regulator which predicts the V_conv voltage from the Id current measurement.

The SSSC block is a phasor model which does not include detailed representations of the power electronics. You must use it with the phasor simulation method, activated with the Powergui block. It can be used in three-phase power systems together with synchronous generators, motors, dynamic loads, and other FACTS and Renewable Energy systems to perform transient stability studies and observe impact of the SSSC on electromechanical oscillations and transmission capacity at fundamental frequency.

The SSSC parameters are grouped in two categories: ```
Power
data
```

and `Control parameters`

. Use the **Display** listbox to select which group of parameters
you want to visualize.

**Nominal voltage and frequency**The nominal line-to-line voltage in Vrms and the nominal system frequency in hertz. Default is

`[ 500e3, 60 ]`

.**Series Converter rating**The nominal rating of the series converter in VA and the maximum value of the injected voltage V_conv on the VSC side of the transformer (see single line diagram), in pu of nominal phase-to-ground voltage. Default is

`[ 100e6, 0.1]`

.**Series Converter impedance**The positive-sequence resistance and inductance of the converter, in pu based on the nominal converter rating and nominal voltage. R and L represent the resistance and leakage inductance of the coupling transformer plus the resistance and inductance of the series filtering inductors connected at the VSC output. Default is

`[ 0.16/30, 0.16 ]`

.**Series Converter initial current**The initial value of the positive-sequence current phasor (Magnitude in pu and Phase in degrees). If you know the initial value of the current corresponding to the SSSC operating point you may specify it in order to start simulation in steady state. If you don't know this value, you can leave [0 0]. The system will reach steady-state after a short transient. Default is

`[0, 0]`

.**DC link nominal voltage**The nominal voltage of the DC link in volts. Default is

`40000`

.**DC link total equivalent capacitance**The total capacitance of the DC link in farads. This capacitance value is related to the SSSC converter rating and to the DC link nominal voltage. The energy stored in the capacitance (in joules) divided by the converter rating (in VA) is a time duration which is usually a fraction of a cycle at nominal frequency. For example, for the default parameters, (C=375 µF, Vdc=40 000 V, Snom=100 MVA) this ratio is 3.0 ms, which represents 0.18 cycle for a 60 Hz frequency. If you change the default values of the nominal power rating and DC voltage, you should change the capacitance value accordingly. Default is

`375e-6`

.

**Bypass Breaker**Specifies the status of the bypass breaker connected inside the block across terminals A1, B1, C1 and A2, B2, C2. Select either

`External Control`

(default),`Open`

or`Closed`

. If the bypass breaker is in external control, a Simulink^{®}input named Bypass appears on the block, allowing to control the status of the bypass breaker from an external signal (0 or 1).**Injected voltage reference Vqref**Specify the quadrature-axis component of the voltage injected on the VSC side of the series transformer, in pu. Default is

`0.05`

.**External**When

**External**is selected, a Simulink input named Vqef appears on the block, allowing you to control the reference voltage from an external signal (in pu). The**Injected voltage reference Vqref**parameter is therefore unavailable. Default is cleared.**Maximum rate of change for Vqref**Maximum rate of change of the Vqref voltage, in pu/s. Default is

`3`

.**Injected voltage regulator gains: [Kp Ki]**Gains of the PI regulator which controls the injected voltage. Specify proportional gain Kp in (pu of Vq_conv)/(pu of V), and integral gain Ki, in (pu of Vq_conv)/(pu of V)/s, where V is the Vq voltage error and Vq_conv is the quadrature-axis component of the voltage generated by the VSC. Default is

`[0.03, 1.5]/8`

.The feed forward gain is computed from the

**Series Converter impedance**parameters.**Vdc regulator gains: [Kp Ki]**Gains of the DC voltage PI regulator which controls the voltage across the DC bus capacitor. Specify proportional gain Kp in (pu of Vd_conv)/Vdc, and integral gain Ki, in (pu of Vd_conv)/Vdc/s, where Vdc is the DC voltage error and Vd_conv is the direct-axis component of the voltage generated by the converter. Default is

`[0.1e-3, 20e-3]`

.

`A1 B1 C1`

The three input terminals of the SSSC.

`A2 B2 C2`

The three output terminals of the SSSC.

`Bypass`

This input is visible only when the

**Bypass Breaker**parameter is set to`External Control`

.Apply a Simulink logical signal (0 or 1) to this input. When this input is high the bypass breaker is closed.

`Vqref`

This input is visible only when the

**External control of injected voltage Vqref**parameter is checked.Apply a Simulink signal specifying the reference voltage, in pu.

`m`

Simulink output vector containing 17 SSSC internal signals. These signals are either voltage and current phasors (complex signals) or control signals. They can be individually accessed by using the Bus Selector block. They are, in order:

Signal

Signal Group

Signal Names

Definition

1-3

Power Vabc1 (cmplx)

Va1 (pu)

Vb1 (pu)

Vc1 (pu)Phasor voltages (phase to ground) Va, Vb, Vc at the SSSC input terminals A1, B1, C1 (pu)

4

Power

Vdc (V)

DC voltage (V)

5-7

Power Vabc2 (cmplx)

Va2 (pu)

Vb2 (pu)

Vc2 (pu)Phasor voltages (phase to ground) Va, Vb, Vc at the SSSC output terminals A2, B2, C2 (pu)

8-10

Power Vabc_Inj (cmplx)

Va_Inj (pu)

Vb_Inj (pu)

Vc_Inj (pu)Phasors of injected voltages

Vs= V2-V1 (pu)11-13

Power Iabc (cmplx)

Ia (pu)

Ib (pu)

Ic (pu)Phasor currents Ia, Ib, Ic flowing out of terminals A2, B2, C2 (pu)

14

Control

Vqref (pu)

Reference value of quadrature-axis

injected voltage (pu)15

Control

Vqinj (pu)

Measured injected voltage in quadrature-axis (pu)

16

Control

Id(pu)

Measured current (pu)

17

Control

modindex

The modulation index m of the PWM modulator. A positive number 0<m<1. m=1 corresponds to the maximum voltage V_conv which can be generated by the series converter without overmodulation.

See the `power_sssc`

example
which illustrates the use of a SSSC for damping power oscillations
on a 500 kV, 60 Hz, system.

[1] N. G. Hingorani, L. Gyugyi, "Understanding
FACTS; Concepts and Technology of Flexible AC Transmission Systems," *IEEE ^{®} Press
book*, 2000

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