Multi-band power oscillation damping controller for power system supported by static VAR compensator

Abstract

Low-frequency electromechanical oscillations damping is powerfully crucial in power system operation. In order to fulfill this requirement, power oscillation damping (POD) controllers are installed in synchronous generators and flexible alternating current transmission system devices. These controllers can have either a conventional fixed structure composed by stages of gain and phase compensation or a multi-band modern structure (PSS4B) composed by three bands that correspond to a specific frequency range (low, intermediate and high frequency). In the PSS4B structure, each band consists of two branches based on differential filters (with a gain, lead–lag blocks and a hybrid block). This paper investigates the application of PSS4B as a POD controller for the static VAR compensator (PSS4B-SVC-POD) to damp oscillations in multi-machine power systems. The proposed PSS4B-SVC-POD and power system stabilizers (fitted on generators) are simultaneously designed through an optimization approach aiming at maximizing the closed-loop damping ratio. Modal analysis, frequency responses and time simulations for the well-known two-area four-generator power system show the good performance of the proposed controller fitted on static VAR compensator.

Appendices

Appendix A

This appendix presents the parameters of the multi-band stabilizer obtained from central frequencies and their associated gains. According to [16], from central frequencies $\left[ {\begin{array}{*{20}c} {F_{Li} } & {F_{Ii} } & {F_{Hi} } \\ \end{array} } \right]$ and gains $\left[ {\begin{array}{*{20}c} {K_{Li} } & {K_{Ii} } & {\begin{array}{*{20}c} {K_{Hi} } & {K_{Gi} } \\ \end{array} } \\ \end{array} } \right]$ in Fig. 8, it is possible to define the simplified control structure depicted in Fig. 9 by using Eqs. (A.1) to (A.5):

$$ T_{{L2{\text{i}}}} = T_{{L7{\text{i}}}} = \frac{1}{{2 \cdot \pi \cdot F_{Li} \cdot \sqrt R }} $$

(A.1)

$$ T_{{L1{\text{i}}}} = T_{{L2{\text{i}}}} /R $$

(A.2)

$$ T_{{L8{\text{i}}}} = T_{{L7{\text{i}}}} \cdot R $$

(A.3)

$$ K_{{L1{\text{i}}}} = K_{{L2{\text{i}}}} = \frac{{R^{2} + R}}{{R^{2} - 2R + 1}} $$

(A.4)

$$ K_{{L11{\text{i}}}} = K_{{L17{\text{i}}}} = 1, $$

(A.5)

where the constant R is set to 1.2 and it controls the bandwidth. These equations are associated with the low band, and they can be directly extended for other bands.

Appendix B

This appendix presents the two-area four-generator multi-machine system data used in the case study on 100 MVA and 60 Hz base. The base case without SVC is obtained from [28].

Transmission line data are presented in Table 7, in which R is the series resistance (pu), X is the series reactance (pu) and B is the total shunt susceptance (pu).

Table 7 Transmission line data

Table 8 presents the load-flow results without the SVC taken from [28]. For the load-flow solution, node 4 is chosen as the swing bus. Voltages and powers are given in (pu) and phase angles are given in (degrees).

Table 8 Load-flow results for the system—without SVC

In order to control the voltage at node 11 and to improve the power system dynamic performance, one SVC is allocated in the system. The goal is to find the steady-state value for b_SVC so that $V_{11} = 1.01 {\text{pu}}$. By applying the power flow approach described in section 2.1 [29], the obtained solution is $b_{\text{SVC}} = 1.00 {\text{pu}}$. Also, it is one of the initial conditions required for power system linearization. After including the SVC at node 11, the new load-flow results are presented in Table 9.

Table 9 Load-flow results for the system—with SVC

The dynamic machine data, excitation system data and SVC dynamic data are given in Tables 10, 11 and 12. Reactances are given in (pu), inertia constants (H) are given in (s), damping constants are given in (pu), gains and time constants are given in (pu) and (s), respectively.

Table 10 Machine data

Table 11 Excitation system data

Table 12 SVC data

In order to evaluate the performance of POD controllers, an additional operating condition which has not been used in the project is taken into account. In this new operating condition, a simple generation redispatch has been conducted. For this purpose, P_g1 and P_g2 (area 1) were set to 600 MW and P_g3 (area 2) was set to 800 MW. The fourth generator is the swing one. It is important to emphasize the need for applying the methodology proposed in [29] to calculate the new value of b_SVC so that $V_{11} = 1.01 {\text{pu}}$. The obtained value for this additional operating condition is $b_{\text{SVC}} = 0.8548 {\text{pu}}$. Table 13 presents the new power flow solution.

Table 13 Load-flow results for the system—with SVC—generation redispatch

Appendix C

This appendix presents the control parameters optimally adjusted. Gains are given in (pu), time constants in (s) and frequencies in (Hz). The optimized values are presented in Tables 14 and 15.

Table 14 Conventional SVC-POD

Table 15 Proposed PSS4B-SVC-POD

Appendix D

In order to obtain the linearized system of equations (see Sect. 2.10), the numerical perturbation approach [35] has been employed in this paper. However, Taylor series expansion can also be used to obtain the analytical partial derivatives, as presented in this appendix.

4.1 Linearization of P _Gk and Q _Gk

$$ \begin{aligned} & \Delta P_{Gk} = \left[ {\frac{{\partial P_{Gk} }}{{\partial \delta_{k} }}} \right] \cdot \Delta \delta_{k} + \left[ {\frac{{\partial P_{Gk} }}{{\partial \theta_{k} }}} \right] \cdot \Delta \theta_{k} + \left[ {\frac{{\partial P_{Gk} }}{{\partial E_{qk}^{'} }}} \right] \cdot \Delta E_{qk}^{'} + \left[ {\frac{{\partial P_{Gk} }}{{\partial V_{k} }}} \right] \cdot \Delta V_{k} \\ & \left[ {\frac{{\partial P_{Gk} }}{{\partial \delta_{k} }}} \right] = \left[ { - \frac{{\partial P_{Gk} }}{{\partial \theta_{k} }}} \right] = \frac{{E_{qk}^{'} \cdot V_{k} \cdot { \cos }\left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} + V_{k}^{2} \cdot \left( {\frac{1}{{X_{qk} }} - \frac{1}{{X_{dk}^{'} }}} \right) \cdot { \cos }2\left( {\delta_{k} - \theta_{k} } \right) \\ & \left[ {\frac{{\partial P_{Gk} }}{{\partial E_{qk}^{'} }}} \right] = \frac{{V_{k} \cdot {\text{sen}}\left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} \\ & \left[ {\frac{{\partial P_{Gk} }}{{\partial V_{k} }}} \right] = \frac{{E_{qk}^{'} \cdot { \sin }\left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} + V_{k} \cdot \left( {\frac{1}{{X_{qk} }} - \frac{1}{{X_{dk}^{'} }}} \right) \cdot { \sin }2\left( {\delta_{k} - \theta_{k} } \right) \\ \end{aligned} $$

(D.1)

$$ \begin{aligned} & \Delta Q_{Gk} = \left[ {\frac{{\partial Q_{Gk} }}{{\partial \delta_{k} }}} \right] \cdot \Delta \delta_{k} + \left[ {\frac{{\partial Q_{Gk} }}{{\partial \theta_{k} }}} \right] \cdot \Delta \theta_{k} + \left[ {\frac{{\partial Q_{Gk} }}{{\partial E_{qk}^{'} }}} \right] \cdot \Delta E_{qk}^{'} + \left[ {\frac{{\partial Q_{Gk} }}{{\partial V_{k} }}} \right] \cdot \Delta V_{k} \\ & \left[ {\frac{{\partial Q_{Gk} }}{{\partial \delta_{k} }}} \right] = \left[ { - \frac{{\partial Q_{Gk} }}{{\partial \theta_{k} }}} \right] = - \frac{{E_{qk}^{'} \cdot V_{k} \cdot {\text{sen}}\left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} - V_{k}^{2} \cdot \left( {\frac{1}{{X_{qk} }} - \frac{1}{{X_{dk}^{'} }}} \right) \cdot {\text{sen}}2\left( {\delta_{k} - \theta_{k} } \right) \\ & \left[ {\frac{{\partial Q_{Gk} }}{{\partial E_{qk}^{'} }}} \right] = \frac{{V_{k} \cdot { \cos }\left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} \\ & \left[ {\frac{{\partial Q_{Gk} }}{{\partial V_{k} }}} \right] = \frac{{E_{qk}^{'} \cdot { \cos }\left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} - \frac{{2V_{k} }}{{X_{dk}^{'} }} - V_{k} \cdot \left( {\frac{1}{{X_{qk} }} - \frac{1}{{X_{dk}^{'} }}} \right) \cdot \left[ {1 - { \cos }2\left( {\delta_{k} - \theta_{k} } \right)} \right] \\ \end{aligned} $$

(D.2)

4.2 Linearization of $f_{1k} = \dot{\delta }_{k}$

$$ \Delta \dot{\delta }_{k} = \left[ {\omega_{s} } \right] \cdot \Delta \omega_{puk} $$

(D.3)

4.3 Linearization of $f_{2k} = \Delta \dot{\omega }_{puk}$

$$ \Delta \dot{\omega }_{puk} = \left[ {\frac{1}{{2H_{k} }}} \right] \cdot \Delta P_{mk} + \left[ { - \frac{1}{{2H_{k} }}} \right] \cdot \Delta P_{Gk} + \left[ { - \frac{D}{{2H_{k} }}} \right] \cdot \Delta \omega_{puk} $$

(D.4)

where $\Delta P_{Gk}$ is given by equation (D.1).

4.4 Linearization of I _dk

$$ \begin{aligned} & \Delta I_{dk} = \left[ {\frac{{\partial I_{dk} }}{{\partial \delta_{k} }}} \right] \cdot \Delta \delta_{k} + \left[ {\frac{{\partial I_{dk} }}{{\partial \theta_{k} }}} \right] \cdot \Delta \theta_{k} + \left[ {\frac{{\partial I_{dk} }}{{\partial E_{qk}^{'} }}} \right] \cdot \Delta E_{qk}^{'} + \left[ {\frac{{\partial I_{dk} }}{{\partial V_{k} }}} \right] \cdot \Delta V_{k} \\ & \left[ {\frac{{\partial I_{dk} }}{{\partial \delta_{k} }}} \right] = \left[ { - \frac{{\partial I_{dk} }}{{\partial \theta_{k} }}} \right] = \frac{{V_{k} \cdot {\text{sen}}\left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} \\ & \left[ {\frac{{\partial I_{dk} }}{{\partial E_{qk}^{'} }}} \right] = \frac{1}{{X_{dk}^{'} }} \\ & \left[ {\frac{{\partial I_{dk} }}{{\partial V_{k} }}} \right] = - \frac{{\cos \left( {\delta_{k} - \theta_{k} } \right)}}{{X_{dk}^{'} }} \\ \end{aligned} $$

(D.5)

4.5 Linearization of $f_{3k} = \dot{E}_{qk}^{'}$

$$ \Delta \dot{E}_{qk}^{'} = \left[ { - \frac{1}{{T_{d0k}^{'} }}} \right] \cdot \Delta E_{qk}^{'} + \left[ { - \frac{{\left( {X_{dk} - X_{dk}^{'} } \right)}}{{T_{d0k}^{'} }}} \right] \cdot \Delta I_{dk} + \left[ {\frac{1}{{T_{d0k}^{'} }}} \right] \cdot \Delta E_{FDk} , $$

(D.6)

where ΔI_dk is given by Eq. (D.5).

4.6 Linearization of $f_{4k} = \dot{E}_{FDk}$

$$ \Delta \dot{E}_{FDk} = \left[ { - \frac{1}{{T_{Ak} }}} \right] \cdot \Delta E_{FDk} + \left[ {\frac{{K_{Ak} }}{{T_{Ak} }}} \right] \cdot \left[ {\Delta V_{{{\text{REF}}1k}} - \Delta V_{k} + \Delta V_{{{\text{PSS}}k}} } \right] $$

(D.7)

4.7 Linearization of $f_{5i} = \dot{b}_{{{\text{SVC}}i}}$

$$ \Delta \dot{b}_{{{\text{SVC}}i}} = \left[ { - \frac{1}{{T_{{{\text{SVC}}i}} }}} \right] \cdot \Delta b_{{{\text{SVC}}i}} + \left[ {\frac{{K_{{{\text{SVC}}i}} }}{{T_{{{\text{SVC}}i}} }}} \right] \cdot \left[ {\Delta V_{{{\text{REF}}2i}} - \Delta V_{i} + \Delta V_{{{\text{POD}}i}} } \right] $$

(D.8)

4.8 Linearization of algebraic equations associated with power system (Sect. 2.7)

$$ \begin{aligned} & g_{1j} = P_{Gj} - P_{Lj} - \mathop \sum \limits_{{m \in \varOmega_{j} }} P_{jm} = 0 \\ & \Delta P_{Gj} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} P_{jm} }}{{\partial \theta_{j} }}} \right] \cdot \Delta \theta_{j} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} P_{jm} }}{{\partial \theta_{m} }}} \right] \cdot \Delta \theta_{m} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} P_{jm} }}{{\partial V_{j} }}} \right] \cdot \Delta V_{j} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} P_{jm} }}{{\partial V_{m} }}} \right] \cdot \Delta V_{m} = 0 \\ \end{aligned} $$

(D.9)

where $\Delta P_{Gj} \ne 0$ for generator busses, and it is given by equation (D.1). The other derivatives are employed in power flow tools, and, for the sake of brevity, they can be obtained from the literature [28]:

$$ \begin{aligned} & g_{2j} = Q_{Gj} - Q_{Lj} - \mathop \sum \limits_{{m \in \varOmega_{j} }} Q_{jm} = 0 \\ & \Delta Q_{Gj} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} Q_{jm} }}{{\partial \theta_{j} }}} \right] \cdot \Delta \theta_{j} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} Q_{jm} }}{{\partial \theta_{m} }}} \right] \cdot \Delta \theta_{m} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} Q_{jm} }}{{\partial V_{j} }}} \right] \cdot \Delta V_{j} + \left[ { - \frac{{\partial \mathop \sum \nolimits_{{m \in \varOmega_{j} }} Q_{jm} }}{{\partial V_{m} }}} \right] \cdot \Delta V_{m} + \Delta Q_{{{\text{SVC}}j}} = 0, \\ \end{aligned} $$

(D.10)

where $\Delta Q_{Gj} \ne 0$ for generator busses, and it is given by equation (D.2). The other derivatives are employed in power flow tools, and, for the sake of brevity, they can be obtained from the literature [28]. The term $\Delta Q_{{{\text{SVC}}j}}$ is nonzero for busses with SVC and is given by equation (D.11).

$$ \Delta Q_{{{\text{SVC}}j}} = \left[ {2 \cdot V_{j} \cdot b_{{{\text{SVC}}i}} } \right] \cdot \Delta V_{j} + \left[ {V_{j}^{2} } \right] \cdot \Delta b_{{{\text{SVC}}i}} $$

(D.11)

4.9 Linearization of equations associated with the controllers

The linearization of the differential–algebraic system of equations associated with the controllers (see Sects. 2.8, 2.9 and 3.2) is very simple. For instance, Eq. (15), repeated in Eq. (D.12), can be easily linearized as presented in Eq. (D.13):

$$ f_{6k} = \frac{{{\text{d}}V_{1ck}^{'} }}{{{\text{d}}t}} = - \frac{{V_{1ck}^{'} }}{{T_{wk} }} + \frac{{K_{{{\text{PSS}}k}} }}{{T_{wk} }} \cdot \Delta \omega_{puk} $$

(D.12)

$$ \Delta \dot{V}_{1ck}^{'} = \left[ { - \frac{1}{{T_{wk} }}} \right] \cdot \Delta V_{1ck}^{'} + \left[ {\frac{{K_{{{\text{PSS}}k}} }}{{T_{wk} }}} \right] \cdot \Delta \omega_{puk} . $$

(D.13)

As the reader can see, these equations are already linear. The nonlinearity can appear in equations that are a function of the input signal Δy (like power flows). In this case, the partial derivatives of Δy can be obtained from the literature [3, 28, 35].