Levenberg–Marquardt neural network to estimate UPFC-coordinated PSS parameters to enhance power system stability

Abstract

Due to the presence of weak tie line interconnections, small signal oscillations are created in power system networks. Damping out these oscillations is one of the most crucial issues to be settled down for the stability of power system industry. The employment of flexible AC transmission systems (FACTS) may suppress these oscillations effectively in addition to the enhancement of power transfer capability. Unified power flow controller (UPFC) is one of those FACTS devices which are installed in the powers grids, which ensures proper functionality of high-voltage transmission lines. To select the proper parameters of power system stabilizer (PSS) when applied with UPFC is a challenge in this field which can be represented as a multi-objective optimization problem. This work aims to optimize the PSS parameters of power network incorporating UPFC using the artificial neural network (ANN) in real time to damp out the small signal oscillations with a view to enhancing the stability of the power system where the Levenberg–Marquardt (LM) algorithm is used as the training algorithm. System eigenvalues obtained from ANN-tuned PSS coordinated with UPFC and the fixed gain conventional PSS with UPFC are compared to investigate the efficiency of the proposed technique for different loading conditions. Additionally, the comparison has been made in time domain simulation results which prove the superiority of the proposed technique over conventional technique. Moreover, the satisfactory values of statistical performance measures validate the efficacy of the prediction capability of the proposed LM-NN approach.

Appendix

System parameters used are given below:

• Transmission line and generator:

• M = 8 MJ/MVA, D = 0, x_d = 1.0 pu, x_q = 0.6 pu, T′_d0 = 5.044 s, x′_d = 0.3 pu, ω_b = 377 rad/s and X_L = 0.1 pu

• Machine excitation system: K_A = 100, T_A = 0.01 s

• Transformer: X_ET = 0.1 pu, X_BT = 0.1 pu, X_T = 0.1 pu

• DC link capacitor:

• V_DC = 2 pu and C_DC = 1.2 pu

• PSS parameters (fixed gain):

• PSS: K = 15.71, T₁ = 0.3, T₂ = 0.3, T₃ = 0.39, T₄ = 0.6623

• UPFC: δ_E = 68.113°, δ_B = 41.12°, m_B = 0.96, m_E = 0.7667

• PSS parameters (LM-NN):

• 1 ≤ K ≤ 50, 0.01 ≤ T₁ ≤ 1.0

$$ {\text{Eigenvalues}}{:}\lambda = \sigma + j\omega $$

(A.1)

$$ {\text{Damping}}\;{\text{ratio}}{:}\zeta = - \frac{\sigma }{{\sqrt {\sigma^{2} + \omega^{2} } }} $$

(A.2)

Performance of the proposed technique was tested with different well-known error measures including MSE, RMSE, MAPE and R².

For total n data samples, actual values y_a and predicted values y_p mathematical formulas for the error measures are presented below:

$$ {\rm MSE} = \frac{1}{n}\sum\limits_{i = 1}^{n} {((y_{{\hbox{a}}} )_{i} - } (y_{{\hbox{p}}} )_{i} )^{2} $$

(A.3)

$$ {\rm RMSE} = \sqrt {\frac{{\sum\nolimits_{i = 1}^{n} {((y_{{\hbox{a}}} )_{i} - } (y_{{\hbox{p}}} )_{i} )^{2} }}{n}} $$

(A.4)

$$ {\rm MAPE} = \frac{1}{n}\sum\limits_{i = 1}^{n} {\left| {\frac{{(y_{{\hbox{a}}} )_{i} - (y_{{\hbox{p}}} )_{i} }}{{(y_{{\hbox{a}}} )_{i} }}} \right| \times 100} $$

(A.5)

$$ R^{2} = 1 - \sum\limits_{i = 1}^{n} {\frac{{((y_{{\hbox{a}}} )_{i} - (y_{{\hbox{p}}} )_{i} )^{2} }}{{((y_{{\hbox{a}}} )_{i} - \bar{y}_{{\hbox{a}}} )^{2} }}} $$

(A.6)

where $\bar{y}_{\text{a}}$ is the mean of actual value.