Design of UPFC controller using modified bilinear equation for Improving transient stability

American Journal of Applied Sciences, Dec, 2008 by Majid Nayeripour, Taher Niknam

INTRODUCTION

A Unified Power Controller (UPFC) is a versatile FACTS device that is used to regulate the active and reactive power flow and the line voltage at it's both sides. It can also improve the dynamic or transient stability of power system (1), (2).

For the safe and good operation of PWM inverters of UPFC, the DC capacitor voltage of inverters should be constant (3) For this aim, a PI controller is used to balance the input and output power of dc capacitor via determining the d-axis current reference in shunt inverter. In deed, since the speed of d-axis current tracking of shunt inverter from it's reference is high, the interaction between series and shunt inverters via dc link capacitor is not considered and therefore, the series and shunt controller in UPFC are designed individually (4), (5).

For example, in (6) in order to improve transient stability, the series controller is designed by setting the variation of line power equal to zero and two conventional PI controllers is used for the shunt controller. In that paper, the shunt inverter is connected to a voltage regulated bus and it does not control the bus voltage and the interaction of shunt and series inverter via dc link is not considered.

In spite of individually design of shunt and series controllers, under certain condition and operating point, the interaction between series and shunt inverters may be cause to instability as discussed in (6), (7).

For better performance of UPFC in transient state, It is required that the variation of dc capacitor voltage to be considered in UPFC modeling. Design of simultaneous shunt and series controllers under one control law with considering the variation of dc capacitor voltage is the main aim of this paper. In this design, the output voltage of each inverter is considered as the dc capacitor voltage multiplying by input controller (Eq. 2). So, similar the bilinear equation, we have bilinear term that is exposed by the multiplying the state variable and input in the state equation.

If the variation of dc capacitor voltage is ignored, these nonlinear terms are considered as the constant inputs.

In this research, based on bilinear equation, the structure and mathematical model of UPFC is derived. Using this model, the controllers of shunt and series inverters are designed simultaneously. To reduce the total energy of system, an adaptive scheme for gain-scheduling is used under supervisory control.

UPFC STRUCTURE AND MODELING

Figure 1 shows the block diagram of a UPFC and equivalent circuit of its shunt and series inverter. The dc sides of these inverters are connected to a dc link capacitor. The ac side of series inverter is in series with the power line network via a three phase transformer. The ac side of shunt inverter is also connected to the input bus of UPFC through a three phase transformer.

The series inverter voltage can regulate the active and reactive power of transmission line in dynamic and steady state conditions. The shunt inverter regulates the dc bus voltage required for series inverter via the balance of input and output active power of capacitor. It can also regulate the input line voltage by generation or consumption of reactive power (8).

According to Fig. 1, there are the following equations:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (2)

In above equation [V.sub.E] and [V.sub.B] are the voltages of shunt and series inverter outputs, ([M.sub.E], [[delta].sub.E]) and ([M.sub.B], [[delta].sub.B])are modulation index and phase angle of shunt and series inverters respectively. [K.sub.E] and [K.sub.B] are the coefficients including transformers ratios and relating the dc to ac voltage of the shunt and series inverter respectively. These coefficients are usually greater than one. The input and output current of dc capacitor is:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (3)

After some manipulations, the state equations of the UPFC in per- unit will be represented as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

The state variables and the inputs of the UPFC are:

X = [[I.sub.Bd] [I.sub.Bq] [I,sub.Ed] [I.sub.Eq] [V.sub.C]] [U.sub.1] = [K.sub.B] [m.sub.B] cos[[delta].sub.B] [U.sub.2] = [K.sub.B] [m.sub.B] sin[[delta].sub.B] [U.sub.3] = [K.sub.E] [m.sub.E] cos[[delta].sub.E] [U.sub.4] = [K.sub.E] [m.sub.E] cos[[delta].sub.E] (5)

Equation (4) can be rewritten in the bilinear form of (6) as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (6)

The above state equation is represented in global d-q components. In design of controller, first we assume the input bus voltage of UPFC as the reference:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (7)

Then the other states and voltages are represented in new local D-Q components as:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (8)

So, the controller is designed in local D-Q components and then the variables are converted to global d-q components and are inserted to inverters.

UPFC CONTROLLER DESIGN USING LIAPUNOV ENERGY FUNCTION