This paper aims at minimizing the active power loss in the transmission lines by determining the optimal solutions to the ORPD problem. The proposed JAYA algorithm helps in determining the optimal values of the control variables while simultaneously satisfying all the constraints in the system. The objective function of the ORPD problem is shown below [25]:

$$ f_{n} = {\text{min }}\left( {P_{{{\text{loss}}}} } \right){ } = { }\mathop \sum \limits_{k = 1}^{Nl} G_{k} \left( {V_{i}^{2} + V_{j}^{2} - 2V_{i} V_{j} \cos \delta_{ij} } \right) $$

(1)

where, *Nl* represent the total number of transmission lines, the conductance of the *k*th branch is shown as \(G_{k}\), *V*_{i} and *V*_{j} represent the magnitudes of the bus voltage for the buses *i* and *j*, respectively, and \(\delta\)_{ij} stand for the phase difference between *V*_{i} and *V*_{j}.

### Constraints

The following shows the different constraints of the objective function:

#### Equality constraints

$$ P_{gi} - P_{di} - V_{i} \mathop \sum \limits_{j = 1}^{Nb} V_{j} \left( {G_{ij} \cos \theta_{ij} + B_{ij} \sin \theta_{ij} } \right) = 0 $$

(2)

$$ Q_{gi} - Q_{di} - V_{i} \mathop \sum \limits_{j = 1}^{Nb} V_{j} \left( {G_{ij} \cos \theta_{ij} + B_{ij} \sin \theta_{ij} } \right) = 0 $$

(3)

The above constraints depict the load flow equations, where \(Nb\) represent the total number of buses, \(P_{gi}\) and \(Q_{gi}\) represent the active and reactive power generation and \(P_{di}\) and \(Q_{di}\) are the active and reactive power load demands at the *i*th bus, respectively. \(G_{ij}\) and \(B_{ij}\) represent the conductance and susceptance between two different buses (i.e., *i*th and *j*th), respectively, and \(\theta_{ij}\) is the angle between the *i*th and *j*th bus.

#### Inequality constraints

• Generator constraints

The generator active power, reactive power and voltage magnitudes are restricted within their limits and must not be violated during solving the problem. The limits are shown below:

$$ V_{gi}^{{{\text{min}}}} \le V_{gi} \le V_{gi}^{{{\text{max}}}} ,\quad i = 1, \ldots , N_{g} $$

(4)

$$ P_{gi}^{{{\text{min}}}} \le P_{gi} \le P_{gi}^{{{\text{max}}}} ,\quad i = 1, \ldots , N_{g} $$

(5)

$$ Q_{gi}^{{{\text{min}}}} \le Q_{gi} \le Q_{gi}^{{{\text{max}}}} ,\quad i = 1, \ldots , N_{g} $$

(6)

where, \(N_{g}\) represent the total number of generator buses, \(V_{gi}^{{{\text{min}}}}\), \(P_{gi}^{{{\text{min}}}}\) and \(Q_{gi}^{{{\text{min}}}}\) are the minimum limits and \(V_{gi}^{{{\text{max}}}}\), \(P_{gi}^{{{\text{max}}}}\) and \(Q_{gi}^{{{\text{max}}}}\) are the maximum limits of the generator bus voltages, active and reactive power, respectively. \(V_{gi}\), \(P_{gi}\) and \(Q_{gi}\) are the amount of voltage, active and reactive power generation at the *i*th bus.

• Transformer constraints

The minimum and maximum limits of the settings of the tap-changing transformer are given by:

$$ T_{i}^{{{\text{min}}}} \le T_{i} \le T_{i}^{{{\text{max}}}} ,\quad i = 1, \ldots , N_{T} $$

(7)

where, \(N_{T}\) shows the number of tap-changing transformers in the system. \(T_{i}\) is the tap-setting position value of the tap-changing transformer at the *i*th bus and \(T_{i}^{{{\text{min}}}}\), \(T_{i}^{{{\text{max}}}}\) are its minimum and maximum limits.

• VAR compensator constraints

The limits of the reactive power to be injected by the VAR compensators are given as:

$$ Q_{ci}^{{{\text{min}}}} \le Q_{ci} \le Q_{ci}^{{{\text{max}}}} ,\quad i = 1, \ldots , N_{C} $$

(8)

where, \(N_{C}\) represent the total number of shunt compensators at the buses and \(Q_{ci}^{{{\text{min}}}}\), \(Q_{ci}^{{{\text{max}}}}\) are the minimum and maximum limits of the reactive power injection \(Q_{ci}\), respectively.

• Operating constraints

The voltage at the load buses and the apparent power at the branches must remain within a specified limit. Their limits are shown below:

$$ V_{Li}^{{{\text{min}}}} \le V_{Li} \le V_{Li}^{{{\text{max}}}} ,\quad i = 1, \ldots , N_{PQ} $$

(9)

$$ S_{Li} \le S_{Li}^{{{\text{max}}}} ,\quad i = 1, \ldots , NL $$

(10)

where, \(N_{PQ}\) depict the total number of load buses, and \(S_{Li}^{{{\text{max}}}}\) is the maximum value of the apparent power flow at the *i*th bus where \(S_{Li}\) is the apparent power at that branch. \(V_{Li}\) is the magnitude of the voltage at the *i*th load bus and \(V_{Li}^{{{\text{min}}}}\), \(V_{Li}^{{{\text{max}}}}\) are its minimum and maximum limits.

Among all the mentioned variables, the load bus voltages, the reactive power generation and apparent power flow are the dependent variables considered. These variables are constrained using penalty coefficients to the objective function in Eq. (1). Thus, the objective function modified as,

$$ f = P_{{{\text{loss}}}} + \lambda_{V} \mathop \sum \limits_{i = 1}^{{{N_{V}{{\text{lim}}}} }} \left( {V_{i} - V_{i}^{{{\text{lim}}}} } \right)^{2} + \lambda_{Q} \mathop \sum \limits_{i = 1}^{{{N_{Q}{{\text{lim}}}} }} \left( {Q_{gi} - Q_{gi}^{{{\text{lim}}}} } \right)^{2} $$

(11)

The limits of \(V_{i}^{{{\text{lim}}}}\) and \(Q_{gi}^{{{\text{lim}}}}\) are:

$$ V_{i}^{{{\text{lim}}}} = \left\{ {\begin{array}{*{20}c} {V_{i}^{{{\text{min}}}} ,} & {{\text{if}}} & {V_{i} < V_{i}^{{{\text{min}}}} } \\ {V_{i}^{{{\text{max}}}} ,} & {{\text{if}}} & {V_{i} > V_{i}^{{{\text{max}}}} } \\ \end{array} } \right. $$

(12)

$$ Q_{gi}^{{{\text{lim}}}} = \left\{ {\begin{array}{*{20}c} {Q_{gi}^{{{\text{min}}}} ,} & {{\text{if}}} & {Q_{gi} < Q_{gi}^{{{\text{min}}}} } \\ {Q_{gi}^{{{\text{max}}}} ,} & {{\text{if}}} & {Q_{gi} > Q_{gi}^{{{\text{max}}}} } \\ \end{array} } \right. $$

(13)

where, \(\lambda_{V}\) and \(\lambda_{Q}\) are penalty coefficients, \(N_{V}^{{{\text{lim}}}}\) is the number of buses on which the voltages are outside limits and \(N_{Q}^{{{\text{lim}}}}\) is the number of buses on which the reactive power generations are outside limits.