Nondominated sorting-based disruption in oppositional gravitational search algorithm for stochastic multiobjective short-term hydrothermal scheduling

Abstract

This article presents a novel framework of nondominated sorting-based disruption in oppositional gravitational search algorithm (NSDOGSA) for solving an inventive stochastic multiobjective short-term hydrothermal scheduling (SMSHTS) problems. An innovative SMSHTS mathematical formulation is derived in terms of stochastic objective functions subject to stochastic hydro- and thermal constraints. Also, a versatile constraint handling procedure is proposed to satisfy all the constraints. Besides, an opposition-based learning perception is assimilated in a gravitational search algorithm (GSA) to explore the excellence of the present population and disruption operator is integrated to hasten the convergence of solutions. Moreover, a nondominated sorting procedure is hybridized to attain a set of nondominated solutions for SMSHTS problems. In addition, an elite external archive is created to keep the nondominated solutions with the help of spread indicator and also to guide the search process toward global optima. Further, a fuzzy decision making is employed for selecting the best trade-off solution among the nondominated solution set. Finally, the proposed NSDOGSA approach is validated on four test systems that consist of two fixed-head and two variable-head stochastic multiobjective hydrothermal scheduling problems. Thus, the obtained simulation results are found to be better in terms of objective function values as well to satisfy constraints compared to other methods within a reasonable execution time.

Appendices

Appendices

1.1 Appendix 1: Derivation for stochastic production cost with valve point effect

Let us consider $X$ is a random variable and the expected value, also called as mean value of this random variable $X$ is defined as $E\left( X \right)$ or $\bar{X}$. If $\bar{X}$ is an expected value of $X$ and variance $\sigma^{2}$ and $X = \bar{X} + \Delta X,$ with $\overline{\Delta X}$ = 0 and $\overline{{\Delta X^{2} }} = \sigma^{2} ,$, then a nonlinear function $f\left( X \right)$ can be written using Taylor series expansion about $\bar{X}$ as

$$ f\left( X \right) = f\left( {\bar{X}} \right) + f^{\prime } \left( {\bar{X}} \right)\Delta X + f^{\prime \prime } \left( {\bar{X}} \right)\frac{{\Delta X^{2} }}{2!} + \cdots $$

(31)

and expected value of $f\left( X \right)$ is given as

$$ \overline{f\left( X \right)} \simeq f\left( {\bar{X}} \right) + f^{\prime \prime } \left( {\bar{X}} \right)\frac{{\sigma^{2} }}{2!} $$

(32)

The deterministic non-convex production cost function of thermal generating units with valve point loading effect is usually represented as:

$$ \begin{aligned} EPC\,\,{\text{or}}\,\,f_{1} & = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {a_{i} \left( {P_{i}^{t} } \right)^{2} + b_{i} P_{i}^{t} + c_{i} } } \\ & \quad + \left| {d_{i} \times \sin \left( {e_{i} \times \left( {P_{i}^{\hbox{min} } - P_{i}^{t} } \right)} \right)} \right| \\ \end{aligned} $$

(33)

where $EPC$ is economical production cost, $Ng$ is number of thermal plants, $T$ is total scheduled intervals, $P_{i}^{t}$ is ith unit thermal generation at tth time interval, $P_{i}^{\hbox{min} }$ is lower thermal generation capacity limit of ith plant, and $a_{i} ,b_{i} ,c_{i} ,d_{i} ,e_{i}$ are ith thermal plant cost coefficients.

The stochastic non-convex production cost function $\bar{f}_{1} \,\,\,{\text{or}}\,\,\,\,\overline{EPC}$ of thermal plants in terms of $f_{1} \left( {\bar{P}_{i}^{t} } \right)$ and $f_{1}^{\prime \prime } \left( {\bar{P}_{i}^{t} } \right)$, using Eqs. (32) and (33), can be expressed as follows:

$$ \begin{aligned} f_{1} \left( {\bar{P}_{i}^{t} } \right) & = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {\bar{a}_{i} \left( {\bar{P}_{i}^{t} } \right)^{2} + \bar{b}_{i} \bar{P}_{i}^{t} + \bar{c}_{i} } } \\ & \quad + \left| {\bar{d}_{i} \times \sin \left( {\bar{e}_{i} \times \left( {\bar{P}_{i}^{\hbox{min} } - \bar{P}_{i}^{t} } \right)} \right)} \right| \\ \end{aligned} $$

(34)

$$ \begin{aligned} f_{1}^{\prime \prime } \left( {\bar{P}_{i}^{t} } \right) & = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {2\bar{a}_{i} } } \\ & \quad + \left| {\bar{d}_{i} \times \left( {\bar{e}_{i} } \right)^{2} \times \sin \left( {\bar{e}_{i} \times \left( {\bar{P}_{i}^{\hbox{min} } - \bar{P}_{i}^{t} } \right)} \right)} \right| \\ \end{aligned} $$

(35)

Thus, by using the above Eqs. (34) and (35), the expected production cost with valve point loading effect is expressed as follows:

$$ \begin{aligned} \bar{f}_{1} & = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {\bar{a}_{i} \left( {\bar{P}_{i}^{t} } \right)^{2} + \bar{b}_{i} \bar{P}_{i}^{t} + \bar{c}_{i} + \bar{a}_{i} \text{var} \left( {P_{i}^{t} } \right)} } \\ & \quad + \left| {\bar{d}_{i} \times \sin \left( {\bar{e}_{i} \times \left( {\bar{P}_{i}^{\hbox{min} } - \bar{P}_{i}^{t} } \right)} \right)} \right| \\ & \quad + \left( {\left| {\bar{d}_{i} \times \left( {\bar{e}_{i} } \right)^{2} \times \sin \left( {\bar{e}_{i} \times \left( {\bar{P}_{i}^{\hbox{min} } - \bar{P}_{i}^{t} } \right)} \right)} \right| \times \frac{{\text{var} \left( {P_{i}^{t} } \right)}}{2}} \right) \\ \end{aligned} $$

(36)

1.2 Appendix 2: Derivation for stochastic environmental emission pollution function

The deterministic emission pollution function of thermal generating units is usually represented as:

$$ \begin{aligned} EEP\,\,\,{\text{or}}\,\,f_{2} & = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {\alpha_{i} + \beta_{i} P_{i}^{t} + \gamma_{i} \left( {P_{i}^{t} } \right)^{2} } } \\ & \quad + \eta_{i} \times \exp \left( {\delta_{i} \times P_{i}^{t} } \right) \\ \end{aligned} $$

(37)

where $EEP$ is environmental emission pollution, and $\alpha_{i} ,\beta_{i} ,\gamma_{i} ,\eta_{i} ,\delta_{i}$ represents ith thermal plant of emission pollution coefficients. The stochastic emission function $\bar{f}_{2} \,\,\,{\text{or}}\,\,\,\overline{EEP}$ of thermal plants in terms of $f_{2} \left( {\bar{P}_{i}^{t} } \right)$ and $f_{2}^{\prime \prime } \left( {\bar{P}_{i}^{t} } \right)$, using Eqs. (37) and (38) can be expressed as follows:

$$ f_{2} \left( {\bar{P}_{i}^{t} } \right) = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {\bar{\alpha }_{i} + \bar{\beta }_{i} \bar{P}_{i}^{t} + \bar{\gamma }_{i} \left( {\bar{P}_{i}^{t} } \right)^{2} + \bar{\eta }_{i} \times exp\left( {\bar{\delta }_{i} \times \bar{P}_{i}^{t} } \right)} } $$

(38)

$$ f_{2}^{''} \left( {\bar{P}_{i}^{t} } \right) = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {2\bar{\gamma }_{i} + \bar{\eta }_{i} \left( {\bar{\delta }_{i} } \right)^{2} exp\left( {\bar{\delta }_{i} \times \bar{P}_{i}^{t} } \right)} } $$

(39)

Thus, by using the above Eqs. (38) and (39), the expected emission function is expressed as follows:

$$ \begin{aligned} \bar{f}_{2} & = \sum\limits_{t = 1}^{T} {\sum\limits_{i = 1}^{Ng} {\bar{\alpha }_{i} + \bar{\beta }_{i} \bar{P}_{i}^{t} + \bar{\gamma }_{i} \left( {\bar{P}_{i}^{t} } \right)^{2} + \bar{\eta }_{i} \times exp\left( {\bar{\delta }_{i} \times \bar{P}_{i}^{t} } \right)} } \hfill \\ & \quad + \bar{\gamma }_{i} \text{var} \left( {P_{i}^{t} } \right) + \bar{\eta }_{i} \left( {\bar{\delta }_{i} } \right)^{2} exp\left( {\bar{\delta }_{i} \times \bar{P}_{i}^{t} } \right)\frac{{\text{var} \left( {P_{i}^{t} } \right)}}{2} \hfill \\ \end{aligned} $$

(40)

1.3 Appendix 3: Input data of fixed-head stochastic multiobjective short-term hydrothermal scheduling for Test system-I and Test system-II

See Tables 12, 13 and 14.

Table 12 Stochastic hydro-system data for fixed-head SMSHTS of Test system-I & II

Table 13 Stochastic thermal generator data for fixed-head SMSHTS of Test system-I & II

Table 14 Stochastic system load demands for fixed-head SMSHTS of Test system-I & II

The transmission loss formula coefficients for fixed-head SMSHTS of Test system-I are

$$ \overline{B} = \left[ {\begin{array}{llll} {0.00014} & {0.00001} & {0.000015} & {0.000015} \\ {0.00001} & {0.00006} & {0.000010} & {0.000013} \\ {0.00001} & {0.00001} & {0.000068} & {0.000065} \\ {0.000015} & {0.000013} & {0.000065} & {0.00007} \\ \end{array} } \right] $$

The transmission loss formula coefficients for fixed-head SMSHTS of Test system-II are

$$ \overline{B} = \left[ {\begin{array}{llllll} {0.000049} & {0.000014} & {0.000015} & {0.000015} & {0.000020} & {0.000017} \\ {0.000014} & {0.000045} & {0.000016} & {0.000020} & {0.000018} & {0.000015} \\ {0.000015} & {0.000016} & {0.000039} & {0.000010} & {0.000012} & {0.000012} \\ {0.000015} & {0.000020} & {0.000010} & {0.000040} & {0.000014} & {0.000010} \\ {0.000020} & {0.000018} & {0.000012} & {0.000014} & {0.000035} & {0.000011} \\ {0.000017} & {0.000015} & {0.000012} & {0.000010} & {0.000011} & {0.000036} \\ \end{array} } \right] $$

1.4 Appendix 4: Input data of variable-head stochastic multiobjective short-term hydrothermal scheduling for Test system-III and Test system-IV

See Tables 15, 16, 17 and 18.

Table 15 Stochastic hydropower generation coefficients, reservoir storage volume limits, reservoir end storage conditions, water discharge rate limits, reservoir inflows $\left( { \times 10^{3} m^{3} } \right),$ number of upstream plants, water transportation delays and hydro-plant generation limits (MW) for variable-head SMSHTS of Test system-III & IV

Table 16 Stochastic reservoir inflows $\left( { \times 10^{4} m^{3} } \right)$ of Test system-III & IV

Table 17 Stochastic system load demands of Test system-III & IV

Table 18 Stochastic thermal generator data of Test system-III & IV

The transmission loss coefficients of Test system-IV

$$ \begin{aligned} \overline{B} & = \left[ {\begin{array}{lllllll} {0.34} & {0.13} & {0.09} & { - 0.01} & { - 0.08} & { - 0.01} & { - 0.02} \\ {0.13} & {0.14} & {0.10} & {0.01} & { - 0.05} & { - 0.02} & { - 0.01} \\ {0.09} & {0.10} & {0.31} & 0 & { - 0.11} & { - 0.07} & { - 0.05} \\ { - 0.01} & {0.01} & 0 & {0.24} & { - 0.08} & { - 0.04} & { - 0.07} \\ { - 0.08} & { - 0.05} & { - 0.01} & { - 0.08} & {1.92} & {0.27} & { - 0.02} \\ { - 0.01} & { - 0.02} & { - 0.07} & { - 0.04} & {0.27} & {0.32} & 0 \\ { - 0.02} & { - 0.01} & { - 0.05} & { - 0.07} & { - 0.02} & 0 & {1.35} \\ \end{array} } \right] \times 10^{ - 4} \hfill \\ \overline{B}_{0} & = \left[ {\begin{array}{lllllll} { - 0.75} & {0.06} & {0.70} & {0.03} & {0.27} & {0.77} & {0.01} \\ \end{array} } \right] \times 10^{ - 6} \hfill \\ \overline{B}_{00} &= 0.55\,\,{\text{MW}} \hfill \\ \end{aligned} $$