# A Multi-Period Optimal Reactive Power Dispatch Approach Considering Multiple Operative Goals
## 2 Proposed Mathematical Modeling
### Equality Constraints  
In the proposed MP-ORPD model, a set of equality constraints are taken into account regarding the power flow in branches and power balance constraints in buses. Equations (5) and (6) define the power flow in branches. In this case, \(f^{p}_{ijct}\) and \(f^{p}_{jict}\) represent the active power flow from bus \(i\) to bus \(j\) and vice versa; while \(f^{Q}_{ijct}\) and \(f^{Q}_{jict}\) in Equations (7) and (8) represent the reactive power flows from bus \(i\) to bus \(j\) and vice versa, respectively. Note that the power flow equations are different at each end of the line; this is because the tap ratio transformer \(\pi_{ijct}\) is taken account. \(G^{\mathcal{L}}_{ijc}\), \(B^{\mathcal{L}}_{ijc}\), \(B^{\mathcal{C}}_{ijc}\), and \(\phi_{ijc}\) are the conductance, susceptance, branch charging susceptance, and the angle on line \(ijc\in\mathcal{L}\), respectively.  
Equations (9) and (10) respectively represent the active and reactive power balance constraints derived from [PERSON]'s laws. Here, \(P^{\mathcal{G}}_{gt}\) and \(p^{\mathcal{G}}_{gt}\) are the fixed and variable active power of generator \(g\) at time \(t\); \(D^{p}_{it}\) and \(D^{\mathcal{Q}}_{it}\) are the active and reactive power demand at bus \(i\) for period \(t\); \(G^{\mathcal{E}}_{i}\) and \(B^{\mathcal{E}}_{i}\) are the shunt conductance and susceptance at bus \(i\in\mathcal{N}\), respectively; finally, \(B^{\mathcal{S}}_{k}\) is the shunt susceptance of shunt element \(k\in\mathcal{S}\).  
\[f^{p}_{ijct}=\frac{1}{\pi^{2}_{ijct}}G^{\mathcal{L}}_{ijc}v^{2}_{it}-\frac{1}{ \alpha_{ijct}}v_{it}v_{jt}\Bigl{(}G^{\mathcal{L}}_{ijc}\cos\bigl{(}\theta_{it} -\theta_{it}-\phi_{ijc}\bigr{)}+B^{\mathcal{L}}_{ijc}\sin\bigl{(}\theta_{it}- \theta_{it}-\phi_{ijc}\bigr{)}\Bigr{)},\forall ijc\in\mathcal{L};t\in\mathcal{T} \tag{5}\]  
\[f^{p}_{jict}=G^{\mathcal{L}}_{ijc}v^{2}_{it}-\frac{1}{\alpha_{ijct}}v_{it}v_{jt }\Bigl{(}G^{\mathcal{L}}_{ijc}\cos\bigl{(}\theta_{it}-\theta_{it}+\phi_{ijc} \bigr{)}+B^{\mathcal{L}}_{ijc}\sin\bigl{(}\theta_{it}-\theta_{it}+\phi_{ijc} \bigr{)}\Bigr{)},\forall ijc\in\mathcal{L};t\in\mathcal{T} \tag{6}\]  
\[f^{Q}_{ijct}=-\frac{1}{\alpha^{2}_{ijct}}\Biggl{(}B^{\mathcal{L}}_{ijc}+\frac {B^{\mathcal{C}}_{ijc}}{2}\Biggr{)}v^{2}_{it}-\frac{1}{\alpha_{ijct}}v_{ji}v_{jt }\Bigl{(}G^{\mathcal{L}}_{ijc}\cos\bigl{(}\theta_{it}-\theta_{it}-\phi_{ijc} \bigr{)}-B^{\mathcal{L}}_{ijc}\sin\bigl{(}\theta_{it}-\theta_{it}-\phi_{ijc} \bigr{)}\Bigr{)},\forall ijc\in\mathcal{L};t\in\mathcal{T} \tag{7}\]  
\[f^{Q}_{ijct}=-\Biggl{(}B^{\mathcal{L}}_{ijc}+\frac{B^{\mathcal{C}}_{ijc}}{2} \Biggr{)}v^{2}_{it}-\frac{1}{\alpha_{ijct}}v_{it}v_{jt}\Bigl{(}G^{\mathcal{L}}_ {ijc}\cos\bigl{(}\theta_{it}-\theta_{it}+\phi_{ijc}\bigr{)}-B^{\mathcal{L}}_{ ijc}\sin\bigl{(}\theta_{it}-\theta_{it}+\phi_{ijc}\bigr{)}\Bigr{)},\forall ijc \in\mathcal{L};t\in\mathcal{T} \tag{8}\]  
\[\sum_{g\in\mathcal{G}_{i(j+\
eq{\rm{ad}})}}P^{\mathcal{G}}_{gt}+\sum_{g\in \mathcal{G}_{iack}}p^{\mathcal{G}}_{gt}-\sum_{(jc):ijc\in\mathcal{L}}f^{p}_{jict }-\sum_{(jc):jic\in\mathcal{L}}f^{p}_{jict}-D^{p}_{it}-{v_{it}}^{2}G^{\mathcal{ E}}_{i}=0,\forall i\in\mathcal{N};t\in\mathcal{T} \tag{9}\]  
\[\sum_{g\in\mathcal{G}_{i}}q^{\mathcal{G}}_{gt}+{v_{it}}^{2}\sum_{k\in\mathcal{ S}_{i}}B^{\mathcal{S}}_{k}n^{\mathcal{S}}_{kt}-\sum_{(jc):ijc\in\mathcal{L}}f^{Q}_{ ijct}-\sum_{(jc):jic\in\mathcal{L}}f^{Q}_{jict}-D^{Q}_{it}+{v_{it}}^{2}B^{ \mathcal{E}}_{i}=0,\forall i\in\mathcal{N};t\in\mathcal{T} \tag{10}\]