Multi-objective optimal power flow considering wind power cost functions using enhanced PSO with chaotic mutation and stochastic weights

Abstract

A multi-objective optimal power flow (OPF) solution using an enhanced NSPSO, incorporating chaotic mutation and stochastic weight trade-off features, is proposed here. The objective functions considered embody the different aspects of the power system, viz. financial, reliability and operational efficiency. The proposed OPF algorithm balances the exploration of global and the utilization of local bests with the stochastic weight and dynamic coefficient trade-off techniques, thus enhancing the searching capability. Also, for countering the premature convergence issue and to improve diversity, the feature of chaotic mutation is incorporated. The pareto-optimal front is provided by the combination of crowding distance approach and non-dominated sorting principle. The best solution can be obtained from a dual-stage process, by selecting from the collective of local best compromises using a fuzzy function. To assess the performance, the proposed method is tested on a standard IEEE 30-bus test system and is compared with popular multi-objective algorithms. IEEE 118-bus system is also used for testing applicability to large systems. Based on the obtained results, the proposed method gives a more reliable operating solutions, better optimal front and hence an improved solution providing a better trade-off.

Appendix: The incomplete gamma function

The function of overestimation cost:

$$\begin{aligned} C_{Rwi} \left( {W_{si} -W_{ai} } \right)&=d_{Rwi} \times \left( {W_{si} -W_{ai} } \right) =d_{Rwi} \nonumber \\&\quad \times E\left( {Y_{oe,i} } \right) \end{aligned}$$

(42)

$$\begin{aligned}&E\left( {Y_{oe,i} } \right) =s_1 +s_2 \end{aligned}$$

(43)

where

$$\begin{aligned} s_1= & {} W_{si} f_p \left( {W=0} \right) \nonumber \\= & {} W_{si} \left\{ {1-\exp \left[ {-\left( {\frac{v_{in,i} }{C_i }} \right) ^{k_i }} \right] +\exp \left[ {-\left( {\frac{v_{\mathrm{out},i} }{C_i }} \right) ^{k_i }} \right] } \right\} \nonumber \\ \end{aligned}$$

(44)

$$\begin{aligned} s_2= & {} \mathop {\int }\nolimits _0^{W_{si} } \left( {W_{si} -W} \right) f_W \left( W \right) {\hbox {d}}W \nonumber \\= & {} W_{si} \mathop {\int }\nolimits _0^{W_{si} } f_W \left( W \right) {\hbox {d}}W-\mathop {\int }\nolimits _0^{W_{si} } Wf_W \left( W \right) {\hbox {d}}W \nonumber \\= & {} W_{si} \left[ {F_W \left( {W_{si} } \right) -F_W \left( 0 \right) } \right] -H_{31} \end{aligned}$$

(45)

where

$$\begin{aligned} H_{31}&=\mathop {\int }\nolimits _0^{W_{si} } Wf_W \left( W \right) {\hbox {d}}W=\frac{K_i h_i v_{in,i} }{W_{ri} }\nonumber \\&\quad \times \mathop {\int }\nolimits _0^{W_{si} } W\left[ {\frac{\left( {1+\frac{h_i W}{W_{ri} }} \right) v_{in,i} }{C_i }} \right] ^{K_i -1}\nonumber \\&\quad \times \exp \left\{ {-\left[ {\frac{\left( {1+\frac{h_i W}{W_{ri} }} \right) v_{in,i} }{C_i }} \right] ^{K_i }} \right\} {\hbox {d}}W \end{aligned}$$

(46)

where $h_i =\left( {\frac{v_{ri} }{v_{in,i} }} \right) -1$

With variable substitution, $t=\left( {1+\frac{h_i W_{si} }{W_{ri} }} \right) vin,i.$H$_{31}$ is converted as follows.

$$\begin{aligned} H_{31}&=\frac{K_i W_{ri} }{h_i v_{in,i} }\mathop {\int }\nolimits _{v_{in,i} }^{v_{1,i} } \left( {\frac{t_i }{C_i }} \right) ^{K_i }\exp \left( {-\frac{t_i^{K_i } }{C_i^{K_i } }} \right) {\hbox {d}}t_i \nonumber \\&-\frac{K_i W_{ri} }{h_i C_i }\mathop {\int }\nolimits _{v_{in,i} }^{v_{1,i} } \left( {\frac{t_i }{C_i }} \right) ^{K_i -1}\exp \left( {-\frac{t_i^{K_i } }{C_i^{K_i } }} \right) {\hbox {d}}t_i \nonumber \\&=G_1 -\frac{K_i W_{ri} }{h_i C_i }\mathop {\int }\nolimits _{v_{in,i} }^{v_{1,i} } \left( {\frac{t_i }{C_i }} \right) ^{K_i -1}\exp \left( {-\frac{t_i^{K_i } }{C_i^{K_i } }} \right) {\hbox {d}}t_i \nonumber \\&=G_1 -\left\{ {\frac{W_{ri} }{h_i }\left[ {\exp \left( {-\frac{v_{in,i}^{K_i } }{C_i^{K_i } }} \right) } \right] -\exp \left( {-\frac{v_{1,i}^{K_i } }{C_i^{K_i } }} \right) } \right\} \nonumber \\\end{aligned}$$

(47)

$$\begin{aligned} v_{1,i}&=v_{in,i} +\frac{\left( {v_{ri} -v_{in,i} } \right) W_{si} }{W_{ri} } \end{aligned}$$

(48)

$$\begin{aligned}&{F}_{{W}} \left( {{W}_{{si}} } \right) -{F}_{{W}} \left( 0 \right) =\exp \left[ {-\left( {\frac{{v}_{{in},{i}} }{{C}_{i} }} \right) ^{{K}_{i} }} \right] \nonumber \\&-\exp \left[ {-\left( {\frac{{v}_{1,{i}} }{{C}_{i} }} \right) ^{{K}_{i} }} \right] \end{aligned}$$

(49)

With variable substitution, $y=\left( {\frac{t_i }{C_i }} \right) ^{k},G_1$ is converted as followed.

$$\begin{aligned} G_1&=\frac{K_i W_{ri} }{h_i v_{in,i} }\mathop {\int }\nolimits _{v_{in,i} }^{v_{1,i} } \left( {\frac{t_i }{C_i }} \right) ^{K_i }\exp \left( {-\frac{t_i^{K_i } }{C_i^{K_i } }} \right) {\hbox {d}}t\nonumber \\&=\frac{W_{ri} C_i }{h_i v_{in,i} }\mathop {\int }\nolimits _{\left( {v_{in,i} /C_i } \right) ^{K_i }}^{\left( {v_{1,i} /C_i } \right) ^{K_i }} y^{1/K_i }\exp \left( {-y} \right) {\hbox {d}}y \end{aligned}$$

(50)

According to the definition of the incomplete gamma function (IGF).

$$\begin{aligned} \Gamma \left( {\alpha ,x} \right) =\mathop {\int }\nolimits _x^\infty y^{\alpha -1}\exp \left( {-y} \right) {\hbox {d}}y \end{aligned}$$

(51)

Apply IGF into G$_{1}$;

$$\begin{aligned} G_1&=\frac{W_{ri} C_i }{h_i v_{in,i} }\Gamma \left[ {1+\frac{1}{K_i },\left( {\frac{v_{in,i} }{C_i }} \right) ^{K_i }} \right] \nonumber \\&\quad -\frac{W_{ri} C_i }{h_i v_{in,i} }\Gamma \left[ {1+\frac{1}{K_i },\left( {\frac{v_{1,i} }{C_i }} \right) ^{K_i }} \right] \end{aligned}$$

(52)

Therefore;

$$\begin{aligned}&E\left( {Y_{{oe,i}} } \right) = s_{1} + s_{2} \nonumber \\&\quad = W_{{si}} \left[ {1 - \exp \left( { - \frac{{v_{{in,i}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) + \exp \left( { - \frac{{v_{{out,i}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) } \right] \nonumber \\&\quad + \left( {\frac{{W_{{ri}} v_{{in,i}} }}{{v_{{ri}} {-} v_{{in,i}} }} {+} W_{{si}} } \right) \left[ {\exp \left( { - \frac{{v_{{in,i}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) {-} \exp \left( { - \frac{{v_{{1,i}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) } \right] \nonumber \\&\quad + \left( {\frac{{W_{{ri}} C_{i} }}{{v_{{ri}} - v_{{in,i}} }}} \right) \left[ \Gamma \left( {1 + \frac{1}{{K_{i} }},\left( {\frac{{v_{{1,i}} }}{{C_{i} }}} \right) ^{{K_{i} }} } \right) \right. \nonumber \\&\quad -\left. \Gamma \left( {1 + \frac{1}{{K_{i} }},\left( {\frac{{v_{{in,i}} }}{{C_{i} }}} \right) ^{{K_{i} }} } \right) \right] \end{aligned}$$

(53)

The function of underestimation cost:

$$\begin{aligned}&C_{Pwi} \left( {W_{ai} -W_{si} } \right) =d_{Pwi} \times \left( {W_{ai} -W_{si} } \right) =d_{Pwi} \nonumber \\&\quad \times E\left( {Y_{ue,i} } \right) \end{aligned}$$

(54)

$$\begin{aligned}&E\left( {Y_{ue,i} } \right) =s_3 +s_4 \end{aligned}$$

(55)

where

$$\begin{aligned} s_3= & {} \left( {W_{ri} -W_{si} } \right) f_p \left( {W_{si} =W_{ri} } \right) \nonumber \\ {}= & {} W_{si} \left\{ {\exp \left[ {-\left( {\frac{v_{ri} }{C_i }} \right) ^{K_i }} \right] -\exp \left[ {-\left( {\frac{v_{\mathrm{out},i} }{C_i }} \right) ^{K_i }} \right] } \right\} \nonumber \\\end{aligned}$$

(56)

$$\begin{aligned} s_4= & {} \mathop {\int }\nolimits _{W_{si} }^{W_{ri} } \left( {W-W_{si} } \right) f_W \left( W \right) {\hbox {d}}W \end{aligned}$$

(57)

The underestimation analysis is done in a similar method of overestimation cost.

$$\begin{aligned}&E\left( {Y_{{ue,i}} } \right) {=} \left( {W_{{ri}} {-} W_{{si}} } \right) \left[ {\exp \left( { - \frac{{v_{{ri}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) {-} \exp \left( { - \frac{{v_{{out,i}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) } \right] \nonumber \\&\quad + \left( {\frac{{W_{{ri}} v_{{in,i}} }}{{v_{{ri}} {-} v_{{in,i}} }} {+} W_{{si}} } \right) \left[ {\exp \left( { - \frac{{v_{{ri}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) {-} \exp \left( { - \frac{{v_{{1,i}}^{{K_{i} }} }}{{C_{i}^{{K_{i} }} }}} \right) } \right] \nonumber \\&\quad + \left( {\frac{{W_{{ri}} C_{i} }}{{v_{{ri}} - v_{{in,i}} }}} \right) \left[ \Gamma \left( {1 + \frac{1}{{K_{i} }},\left( {\frac{{v_{{1,i}} }}{{C_{i} }}} \right) ^{{K_{i} }} } \right) \right. \nonumber \\&\quad \left. - \Gamma \left( {1 + \frac{1}{{K_{i} }},\left( {\frac{{v_{{ri}} }}{{C_{i} }}} \right) ^{{K_{i} }} } \right) \right] \end{aligned}$$

(58)