**2. Problem Formulation**

The main objective that is been focused is to reduce the congestion cost of the system taken into consideration.

$$\text{Minimize } \sum\_{k=1}^{N\_k} \mathbb{C}\_k^n \{ \Delta P\_k^n \} \Delta P\_k^n \tag{1}$$

where,

*Cn k*: Rescheduling cost of power by generators as per increase and decrease price bids at interval *n*.

Δ *Pn k*: Incremental change in active power adjustment of the generator at interval *n*.

*Pmin k*& *Pmax k*: Minimum and maximum limits of generation. Subject to constraints mentioned below.

$$P\_{gj} - P\_{dj} = \sum\_{k=1}^{n} \left| V\_j \parallel V\_k \parallel Y\_{jk} \right| \cos \left( \delta\_i - \delta\_k - \Theta\_{jk} \right) \tag{2}$$

$$Q\_{\mathbf{g}j} - Q\_{dj} = \sum\_{k=1}^{n} \left| V\_j \parallel V\_k \parallel Y\_{j\mathbf{k}} \right| \cos \left( \delta\_j - \delta\_\mathbf{k} - \theta\_{j\mathbf{k}} \right) j = 1, \ 2, \ \dots, n \tag{3}$$

$$P\_{\mathcal{g}k}^{\rm min} \le P\_{\mathcal{g}k} \le P\_{\mathcal{g}k}^{\rm max} \tag{4}$$

$$\mathbf{Q}\_{\mathcal{g}k}^{\min} \le \mathbf{Q}\_{\mathcal{g}k} \le \mathbf{Q}\_{\mathcal{g}k}^{\max} \,\, k = 1, \, 2, \, \dots, \, \mathbf{N}\_{\mathcal{g}} \tag{5}$$

As the pumped storage units are connected on to the bus to reduce the congestion cost of the system, the additional constraints considered are as follows

$$e^n = e^{initial} \text{ } n = 0 \text{, } e^n = e^{final} \text{ } n = 24 \text{.} \tag{6}$$

$$\epsilon^{n+1} = \epsilon^n + t \left( \eta\_{\text{Ps}} \ P^{\text{\tiny n}}\_{\text{ps}} - \frac{P^{\text{\tiny n}}\_{\text{Hs}}}{\eta\_{\text{Hs}}} \right) \tag{7}$$

$$P\_{Ps}^{\min} \le P\_{Ps}^{\text{tr}} \le P\_{Ps}^{\max} \tag{8}$$

$$P\_{Hss}^{\rm min} \le P\_{Hss}^{n} \le P\_{Hss}^{\rm max} \tag{9}$$

$$e^l \le e^{\mu} \le e^{\mu} \tag{10}$$
