*4.2. Modeling of EBDR*

The MGO enters into contracts with the customers through the EBDR program and mentions the values of incentive and penalty. The customers change their demand from *D*0*(t)* to *D(t)* according to the contract. The values of incentive and penalty for DR are given by

$$
\Delta D(t) = D(t) - D\_0(t) \tag{29}
$$

$$I(\Delta D(t)) = \text{inc}(t) \cdot [D\_0(t) - D(t)] \tag{30}$$

$$P(\Delta D(t)) = pen(t) \cdot \{ \mathcal{C}P(t) - [D\_0(t) - D(t)] \} \tag{31}$$

Equation (29) represents the shift in demand of a customer due to the contract. Equations (30) and (31) denote the total values of incentive and penalty, respectively.

The EBDR program shifts the peak demand to reduce operation costs using cross-elasticity based on market prices. The relationship between market prices and demand is represented as follows:

$$D(t\_1) = D\_0(t\_1) + \sum\_{t\_1=1}^{24} E(t\_1, t\_2) \cdot \frac{D\_0(t\_1)}{MP\_0(t\_2)} \cdot \begin{bmatrix} MP(t\_2) - MP\_0(t\_2) \end{bmatrix} \quad \text{if} \ t\_1 \neq t\_2 \tag{32}$$

Equation (32) represents the power demand at *t*<sup>1</sup> according to the market price at *t*2, considering a 24-h interval. When incentive and penalty are included in the price, the formula is modified as follows:

$$D(t\_1) = D\_0(t\_1) \left\{ 1 + \sum\_{\substack{t\_1 = 1 \\ t\_1 \neq t\_2}}^{24} E(t\_1, t\_2) \frac{\left[MP(t\_2) - MP\_0(t\_2) + inc(t\_2) + pen(t\_2)\right]}{MP\_0(t\_2)} \right\} \tag{33}$$

Equation (33) indicates only the change in demand due to the market price, without considering emission constraints. The change in demand when emission constraints are included is calculated by utilizing the concept of self-elasticity as follows:

$$\begin{cases} D\_{\varepsilon}(t\_1) = D(t\_1) + E\_{\varepsilon}(t\_1, t\_1) \cdot \frac{D(t\_1)}{cm\_{\max}(t)} \cdot [cm(t\_1) - cm\_{\max}(t)] & \text{if } cm(t\_1) > cm\_{\max}(t\_1) \\ \qquad \qquad D\_{\varepsilon}(t\_1) = D(t\_1) & \text{if } cm(t\_1) \le cm\_{\max}(t\_1) \end{cases} \tag{34}$$

Here, emission constraints are imposed when emissions are greater than the set emission limit. The above equation yields the 24-h interval consumption of customers participating in the EBDR program, in order to minimize MG operating costs while considering emission constraints.
