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Implement phasor model of three-phase static var compensator

The Static Var Compensator (SVC) is a shunt device of the Flexible AC Transmission Systems (FACTS) family using power electronics to control power flow and improve transient stability on power grids [1]. The SVC regulates voltage at its terminals by controlling the amount of reactive power injected into or absorbed from the power system. When system voltage is low, the SVC generates reactive power (SVC capacitive). When system voltage is high, it absorbs reactive power (SVC inductive). The variation of reactive power is performed by switching three-phase capacitor banks and inductor banks connected on the secondary side of a coupling transformer. Each capacitor bank is switched on and off by three thyristor switches (Thyristor Switched Capacitor or TSC). Reactors are either switched on-off (Thyristor Switched Reactor or TSR) or phase-controlled (Thyristor Controlled Reactor or TCR).

The figure below shows a single-line diagram of a static var compensator and a simplified block diagram of its control system.

**Single-line Diagram of an SVC and Its Control
System Block Diagram**

The control system consists of

A measurement system measuring the positive-sequence voltage to be controlled. A Fourier-based measurement system using a one-cycle running average is used.

A voltage regulator that uses the voltage error (difference between the measured voltage Vm and the reference voltage Vref) to determine the SVC susceptance B needed to keep the system voltage constant

A distribution unit that determines the TSCs (and eventually TSRs) that must be switched in and out, and computes the firing angle α of TCRs

A synchronizing system using a phase-locked loop (PLL) synchronized on the secondary voltages and a pulse generator that send appropriate pulses to the thyristors

The SVC (Phasor Type) block is a phasor model, and 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, and dynamic loads to perform transient stability studies and observe impact of the SVC on electromechanical oscillations and transmission capacity. This model does not include detailed representations of the power electronics, the measurement system, or the synchronization system. These systems are approximated rather by simple transfer functions that yield a correct representation at the system's fundamental frequency.

A detailed model of a SVC using three TSCs and one TCR is provided
in the `power_svc_1tcr3tsc` example.

The SVC can be operated in two different modes:

In voltage regulation mode (the voltage is regulated within limits as explained below)

In var control mode (the SVC susceptance is kept constant)

When the SVC is operated in voltage regulation mode, it implements the following V-I characteristic.

**SVC V-I characteristic**

As long as the SVC susceptance B stays within the maximum and
minimum susceptance values imposed by the total reactive power of
capacitor banks (Bc_{max}) and reactor banks (Bl_{max}),
the voltage is regulated at the reference voltage Vref. However, a
voltage droop is normally used (usually between 1% and 4% at maximum
reactive power output), and the V-I characteristic has the slope indicated
in the figure. The V-I characteristic is described by the following
three equations:

where

V | Positive sequence voltage (pu) |

I | Reactive current (pu/Pbase) (I > 0 indicates an inductive current) |

Xs | Slope or droop reactance (pu/Pbase) |

Bcmax | Maximum capacitive susceptance (pu/Pbase) with all TSCs in service, no TSR or TCR |

Blmax | Maximum inductive susceptance (pu/Pbase) with all TSRs in service or TCRs at full conduction, no TSC |

Pbase | Three-phase base power specified in the block dialog box |

When the SVC is operating in voltage regulation mode, its response speed to a change of system voltage depends on the voltage regulator gains (proportional gain Kp and integral gain Ki), the droop reactance Xs, and the system strength (short-circuit level).

For an integral-type voltage regulator (Kp = 0), if the voltage measurement time constant Tm and the average time delay Td due to valve firing are neglected, the closed-loop system consisting of the SVC and the power system can be approximated by a first-order system having the following closed-loop time constant:

*T _{c}* =
1 / (

where

| Closed loop time constant |

| Proportional gain of the voltage regulator (pu_B/pu_V/s) |

| Slope reactance pu/Pbase |

| Equivalent power system reactance (pu/Pbase) |

This equation demonstrates that you obtain a faster response speed when the regulator gain is increased or when the system short-circuit level decreases (higher Xn values). If you take into account the time delays due to voltage measurement system and valve firing, you obtain an oscillatory response and, eventually, an instability with too weak a system or too large a regulator gain.

The SVC 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.

**SVC modeled using positive-sequence only**The SVC is modeled by a three-wire system using two current sources. The SVC does not generate any zero-sequence current, but it can generate negative-sequence currents during unbalanced system operation. The negative-sequence susceptance of the SVC is assumed to be identical to its positive-sequence value, as determined by the B value computed by the voltage regulator.

Check this box to ignore negative-sequence current.

**System nominal voltage and frequency [Vrms f]**The nominal line-to-line voltage in Vrms and the nominal system frequency in hertz.

**Three-phase base power Pbase**Three-phase base power, in VA, used to specify the following parameters in pu: droop reactance Xs, gains Kp and Ki of the voltage PI regulator, and reference susceptance Bref. This base power is also used to normalize the output B susceptance signal.

**Reactive power limits [Qc Ql]**The maximum SVC reactive powers at 1 pu voltage, in vars. Enter a positive value for the capacitive reactive power Qc (vars generated by the SVC) and a negative value for the inductive reactive power Ql (vars absorbed by the SVC).

**Average time delay due to thyristor valves firing Td**Average time delay simulating the non instantaneous variation of thyristor fundamental current when the distribution unit sends a switching order to the pulse generator. Because pulses have to be synchronized with thyristor commutation voltages, this delay normally varies between 0 and 1/2 cycle. The suggested average value is 4 ms.

**Mode of operation**Specifies the SVC mode of operation. Select either

`Voltage regulation`or`Var control (Fixed susceptance Bref)`.**External control of reference voltage Vref**If this parameter is checked, a Simulink

^{®}input named Vref appears on the block, allowing to control the reference voltage from an external signal (in pu). Otherwise a fixed reference voltage is used, as specified by the parameter below.**Reference voltage Vref**This parameter is not visible when the

**Mode of operation**parameter is set to`Var Control`or when the**External control of reference voltage Vref**parameter is checkedReference voltage, in pu, used by the voltage regulator.

**Droop Xs**This parameter is not visible when the

**Mode of operation**parameter is set to`Var Control`.Droop reactance, in pu/Pbase, defining the slope of the V-I characteristic.

**Voltage regulator [Kp Ki]**This parameter is not visible when the

**Mode of operation**parameter is set to`Var Control`.Proportional gain, in (pu of B)/(pu of V), and integral gain, in pu_B/pu_V/s, of the voltage regulator.

**Bref for var control mode**This parameter is not visible when the

**Mode of operation**parameter is set to`Voltage regulation`.Reference susceptance, in pu/Pbase, when the SVC is operating in var control mode.

`A B C`The three terminals of the SVC.

`Vref`Simulink input of the reference voltage signal.

This input is visible only when the

**External control of reference voltage Vref**parameter is checked.`m`Simulink output vector containing six SVC 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 Iabc (cmplx)

Ia(pu) Ib(pu)

Ic(pu)Phasor currents Ia, Ib, Ic flowing into the SVC (pu)

4

Control

Vm (pu)

Positive-sequence value of measured voltage (pu)

5

Control

B (pu)

SVC susceptance output of the voltage regulator (pu). A positive value indicates that the SVC is capacitive.

6

Control

Q (pu)

SVC reactive power output (pu). A positive value indicates inductive operation.

The `power_svc``power_svc` example
illustrates the steady-state and dynamic performance of a +200 Mvar/-
100 Mvar SVC regulating voltage on a 500 kV, 60 Hz, system.

Open the SVC block menu and look at its parameters. The SVC
is set to `Voltage regulation` mode with a reference
voltage Vref = 1.0 pu. The voltage droop reactance is 0.03 pu/200
MVA, so that the voltage varies from 0.97 pu to 1.015 pu when the
SVC current goes from fully capacitive to fully inductive. Double-click
the blue block to display the SVC V-I characteristic.

The Three-Phase Programmable Voltage Source is used to vary the system voltage and observe the SVC performance. Initially the source is generating its nominal voltage (500 kV). Then, voltage is successively decreased (0.97 pu at t = 0.1 s), increased (1.03 pu at t = 0.4 s) and finally returned to nominal voltage (1 pu at t = 0.7 s).

Start the simulation and observe the SVC dynamic response to voltage steps on the Scope. Waveforms are reproduced on the figure below. Trace 1 shows the actual positive-sequence susceptance B1 and control signal output B of the voltage regulator. Trace 2 shows the actual system positive-sequence voltage V1 and output Vm of the SVC measurement system.

The SVC response speed depends on the voltage regulator integral gain Ki (proportional gain Kp is set to zero), system strength (reactance Xn), and droop (reactance Xs).

As mentioned above, neglecting the voltage measurement time constant Tm and the average time delay Td due to valve firing, the system can be approximated by a first-order system having a closed-loop time constant:

*T _{c}* =
1 / (

With given system parameters (Ki = 300; Xn = 0.0667 pu/200 MVA; Xs = 0.03 pu/200 MVA), the closed-loop time constant is Tc = 0.0345 s.

If you increase the regulator gain or decrease the system strength, Tm and Td are no longer negligible, and you instead observe an oscillatory response and eventually instability. The figure below compares the SVC susceptance (B output of the voltage regulator) for two different short-circuit levels: 3000 VA and 600 MVA.

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