Inter-area and intra-area oscillation damping for UPFC in a multi-machine power system based on tuned fractional PI controllers

Abstract

The low-frequency oscillations such as inter-area and intra-area modes of oscillations are difficult to avoid their occurrence and control in a weakly connected power system. It is essential to damp these multi-modal oscillations because these oscillations may lead to many instability issues. In this study, the application of one of the flexible AC transmission systems (FACTS) compensating device, a unified power flow controller (UPFC) is focused on its small-signal stability issues referring to inter-area and intra-area modes of operation in a multi-machine power system. Factional order Proportional Integral (FOPI) controller is applied for the control strategy against the conventional PI controller due to its flexibility and higher controllability for its extra degree of freedom to control. FOPI controller performance isn’t free from parameter dependency. An adaptive differential evolution (ADE) algorithm is suggested to optimally tune the FOPI parameters for enhancing the performance. Apart from that, the dynamic characteristics of UPFC are extensively presented through modeling in the d–q axis rotating synchronous frame of reference. To justify the feasibility and effectiveness of the proposed approach, comparative results are presented under a wide variety of disturbances in a standard multi-machine power system. It is found from the non-linear simulations that UPFC based power system is well capable to damp out these oscillations effectively to enhance the stability of small-signal disturbance substantially. The result analysis with PI and FOPI control procedures employing ADE justifies the enhanced control performance of the UPFC.

Appendix 1

The ith machine model as considered in the study is given as follows:

$$ \frac{{d\delta_{{}} }}{dt} = \omega_{b} (\omega_{{}} - 1) $$

(A1)

$$ \frac{{d\omega_{{}} }}{dt} = \frac{{T_{m} - T_{e} - D_{{}} (\omega_{{}} - 1)}}{{M_{{}} }} $$

(A2)

$$ \frac{{dE^{\prime}_{q} }}{dt} = \frac{{E_{fd} - (x_{d} - x^{\prime}_{d} )i_{d} - E^{\prime}_{q} }}{{T^{\prime}_{do} }} $$

(A3)

$$ \frac{{dE^{\prime}_{fd} }}{dt} = \frac{{K_{a} (V_{ref} - V_{i} + U_{i} ) - E_{fd} }}{{\tau_{a} }} $$

(A4)

$$ T_{e} = E^{\prime}_{q} i_{q} - (x^{\prime}_{q} - x^{\prime}_{d} )i_{q} i_{d} $$

(A5)

d and q denotes the direct and quadrature axis, respectively; δ and ω denotes the rotor angle and rotor speed, respectively; $\omega_{b}$ denotes the synchronous speed; $T_{m}$ and $T_{e}$ denotes the input and output powers of the generator, respectively; M and D denotes the inertia constant and damping coefficient, respectively; $E_{fd}$ denotes the field voltage; $T_{do}^{^{\prime}}$ denotes the open circuit field time constant; $x_{d}$ and $x^{\prime}_{d}$ denotes the d –axis reactance and d –axis transient reactance of the generator, respectively; $K_{a}$ and $\tau_{a}$ denotes the gain and time constant of exciter, respectively; $V_{ref}$ and $V_{i}$ denotes the reference voltage and terminal voltage of exciter respectively; $U_{i}$ denotes the input voltage of PSS, respectively; $x_{d}$ and $x^{\prime}_{d}$ denotes the generator’s d –axis reactance and d –axis transient reactance, respectively; $x_{q}$ and $x^{\prime}_{q}$ denotes the generator’s q –axis reactance and q –axis transient reactance, respectively; $i_{d}$ and $i_{q}$ denotes the machine's current d-axis and q-axis components, respectively;

1.1 Generator, exciter, and PSS DATA

$$ x_{d} = 1.7;x_{q} = 1.64;x_{d}^{^{\prime}} = 0.254;T_{do}^{^{\prime}} = 5.91{\text{s}} ;\,\omega_{o} = 120\pi \,{\text{rad}}/{\text{s}} ;\, D = 0;M = 4.74\,{\text{s}} $$

The transfer function of the exciter and stabilizer are as follows:

$$ V_{i} (s) = K_{a} K_{e} K_{g} \frac{{(1 + s\tau_{f} )}}{{(1 + s\tau_{a} )(1 + s\tau_{e} )(1 + s\tau_{g} )(1 + s\tau_{f} ) + K_{a} K_{e} K_{g} K_{f} }} $$

$$ U_{i} (s) = K_{i} \frac{{s\tau_{\omega } )}}{{(1 + s\tau_{\omega } )}}\left[ {\frac{{(1 + s\tau_{1} )(1 + s\tau_{3} )}}{{(1 + s\tau_{2} )(1 + s\tau_{4} )}}} \right]\Delta w(s) $$

$K_{a} ,K_{e} ,K_{f} ,and_{{}} K_{g}$ denotes amplifier gains, exciter, sensor, and generator, respectively; $\tau_{a} ,\tau_{a} ,\tau_{e} ,\tau_{f} ,and \, \tau_{g}$ denotes the amplifier time constants, exciter, sensor, and generator, respectively; $K_{i}$ denotes gain constant of the stabilizer; $\tau_{\omega } ,\tau_{1} ,\tau_{2} ,\tau_{3} ,and \, \tau_{4}$ denotes the time constants of stabilizer, respectively;

$$ K_{a} = 300;\tau_{a} = 0.05;K_{e} = 35;\tau_{e} = 0.95 \, {\text{s}} ;K_{f} = 0.025;\tau_{f} = 1 \, {\text{s}} ;K_{g} = 1;\tau_{g} = 1 \, {\text{s}} ;\,\,\tau_{\omega } = 10 \, {\text{s}} ;\,\,\tau_{1} = 0.05,\tau_{3} = 3,\tau_{2} = \tau_{4} = 0.02 $$

1.2 UPFC data

$V_{d\_c}$ denotes the dc-link capacitor voltage; $C_{d\_c}$ denotes the dc-link capacitor;$X_{E}$ and $X_{B}$ denotes the shunt converter transformer and series converter transformer, respectively; $R_{shc}$ and $L_{shc}$ denotes the shunt converter resistance and inductance, respectively; $R_{\sec }$ and $L_{\sec }$ denotes the series converter resistance and inductance, respectively;

$$ V_{d\_c} = 33.11 \, {\text{kV}};_{{}} C_{d\_c} = 3500 \, \upmu F;_{{}} X_{E} = 160 \, MVA, \, 0.85/0.185 \, {\text{kV}};_{{}} X_{B} = 160 \, MVA, \, 0.185/0.05 \, {\text{kV}}; $$

$$ R_{shc} = 0.003p.u.,_{{}} L_{shc} = 0.1p.u.,_{{}} R_{\sec } = 0.004p.u.,_{{}} L_{\sec } = 0.05p.u. $$

1.3 ADE data

$$ NP = 20;\,F = 0.5;\,iter_{\max } = 100;\,C_{ri} = 0.1 - 0.9 $$

1.4 PSO tuned PI controller data data

$$ \begin{gathered} k_{p1} = 0.5;k_{i1} = 6.73;k_{p2} = 1.54;k_{i2} = 2.98; \hfill \\ k_{p3} = 1.110;k_{i3} = 3.21;k_{p4} = 0.92;k_{i4} = 8.96; \hfill \\ \end{gathered} $$

1.5 ADE tuned PI controller data

$$ \begin{gathered} k_{p1} = 0.95;k_{i1} = 3.54;k_{p2} = 0.54;k_{i2} = 1.78; \hfill \\ k_{p3} = 2.50;k_{i3} = 1.53;k_{p4} = 0.34;k_{i4} = 5.76; \hfill \\ \end{gathered} $$

1.6 ADE tuned FOPI controller data

$$ \begin{gathered} k_{p1} = 0.94;k_{i1} = 3.16;k_{p2} = 0.91;k_{i2} = 1.89; \hfill \\ k_{p3} = 1.77;k_{i3} = 2.86;k_{p4} = 0.84;k_{i4} = 5.56; \hfill \\ \lambda_{1} = 0.87;\lambda_{2} = 0.56;\lambda_{3} = 0.72;\lambda_{4} = 0.97 \hfill \\ \end{gathered} $$