## 2 Mathematical Formulation for Multiyear Network Expansion Planning
### Objective Function  
In this study, the objective function is defined in a way that the cost of investment, operation and repair, along with the reliability cost are minimized in the presence of the DGs and its impact on the network. The main goal is to provide the best schedule for installing the DGs and network development within a specified time. The first phase of the objective function is the cost of investment including the installation cost of DGs and reinforcement of the feeder. The second phase is the operating cost, which includes the cost of purchasing power from the upstream grid, the operating cost of the DGs, the maintenance cost, and the cost of losses. The third phase of the objective function is the reliability value consisting of the unsupplied energy of the consumer. Finally, the mathematical model of the objective function is shown as follows:  
\[Min\ F=IC+OC+RC \tag{1}\]  
where \(IC\) and \(OC\) are the investment cost and operation cost, respectively. The mathematical formulations are as follows:  
\[IC=\sum\limits_{i=1}^{N_{y}}\sum\limits_{j=1}^{N_{b}}\gamma(i)\times\left\{ \begin{array}{l}\left[Inv^{DG}\times IC_{j}\times k_{g,j}\times\left(u\left(i -Y_{j}^{DG}+1\right)-u\left(t-Y_{j}^{DG}\right)\right)\right]+\\ \left[FR_{j}\times FL_{j}\times k_{r,j}\times\left(u\left(i-Y_{j}^{reni}+1 \right)-u\left(t-Y_{j}^{reni}\right)\right)\right]+\\ \end{array} \right\} \tag{2}\]  
\[OC=\sum\limits_{i=1}^{N_{y}}\sum\limits_{j=1}^{N_{b}}\gamma(i)\left\{\begin{array} []{l}\sum\limits_{l=1}^{N_{b}}\left[\pi_{i,ls}\times PUL_{l,ls}\times TD_{ls} \right]+\\ \times \begin{array}{l}\sum\limits_{N_{y}}N_{bs}\\ \sum\limits_{i=1}^{N_{b}}\left[\mathcal{O}pe^{DG}\times PG_{i,j,ls}\times TD_{ ls}\right]\end{array} \right\} \tag{3}\]  
Moreover, the converting function \(\gamma(t)\) to return the net present value of future cost is computed as follows:  
\[\gamma(t)=\frac{1}{\left(1+Dr\right)^{t}} \tag{4}\]  
where \(Dr\) is the discount rate, \(Inv^{DG}\) is the investment cost of the DG, and IC is the installation capacity. \(K_{g,i}\) and \(k_{r,i}\) are binary variables to represent the installation decision of DG at bus \(i\) and the decision for reinforcement of the feeder section before bus \(i\), respectively. \(U(t)\) is the unit step function ( if \(t>0\to U(t)=1\), O otherwise \(U(t)=0\)). \(FR\) is the reinforcement cost of the feeder section. \(\pi_{i,ls}\) is the electricity price for purchasing power from the upstream grid at load level \(ls\) of year \(i\). \(PUL_{l,ls}\) is the generated active power from the upstream grid at load level \(ls\) of year \(i\). \(TD_{ls}\) is the time duration of load level \(ls\). \(\mathcal{O}pe^{DG}\) is the operation and maintenance cost of the DG. \(PG_{i,j,ls}\) is the generated active power of DG in bus \(j\) and load level \(ls\) of year \(i\). \(N_{y}\), \(N_{b}\), and \(N_{ls}\) are the number of years of the planning period, number of load buses, and number of load levels, respectively. \[\pi_{i,ls}=\left(1+IR\right)^{i}\times\pi_{0,ls} \tag{5}\]  
where \(\pi_{0,ls}\) is the electricity price for purchasing power from the upstream grid at load level \(ls\) of year 0. \[PUL_{l,ls}=Demand_{i,ls}+P_{i,ls}^{loss}-\sum\limits_{j=1}^{N_{b}}PG_{j,j,ls} \tag{6}\]  
where \(Demand_{i,ls}\) is the total load demand of the distribution network at load level \(ls\) of year \(i\). The reliability cost (RC) is calculated through Equation (7) [26]:  
\[RC=\sum\limits_{i=1}^{N_{y}}\gamma(i)\times VOL_{i}\times EENS_{i} \tag{7}\]  
\[VOL_{i}=\left(1+IR\right)^{i}\times VOL_{0} \tag{8}\]  
\[EENS_{i}=\sum\limits_{ls=1}^{N_{b}}\frac{TD_{ls}}{8760}\times\left\{\sum \limits_{j=1}^{N_{b}}\sigma_{j}\times FLj\times\left[SP_{j,i,ls}+TPP_{j,i,ls} \right]\times T_{rs}+UP_{j,i,ls}\times T_{rp}\right\} \tag{9}\]  
where \(EENS_{i}\) is the expected energy not supplied of the distribution network in year \(i\). \(VOL_{0}\) is the value of lost load in the base year. \(\sigma_{j}\) is the average failure rate of the feeder section located before bus \(i\).