*2.1. DG Model*

The power generation costs of a diesel generator are generally expressed as the sum of a quadratic and sinusoidal function.

$$\mathbb{C}\_{\mathcal{G},i}(t) = a\_i P\_{\mathcal{G},i}^2(t) + b\_i P\_{\mathcal{G},i}(t) + c\_i + \left| d\_i \sin[P\_{\mathcal{G},i}^{\min}(t) - P\_{\mathcal{G},i}(t)] \right| \tag{1}$$

Here, the sinusoidal function represents a valve point effect to practically account for the generation cost function. Figure 2 illustrates the valve point effect for a conventional generator. It represents a sharp increase in losses due to the wire drawing effect caused by the opening of each steam admission valve [30].

**Figure 2.** Valve point effect.

#### *2.2. PV Model*

The output power of a PV is dependent on solar irradiation. Forecasted solar irradiation is commonly determined using the beta probability distribution function (PDF) expressed as follows [31].

$$PDF\_B(si) = \begin{cases} \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)} \cdot si^{(a-1)} \cdot \left(1 - si\right)^{(b-1)} & 0 \le si \le 1 \text{ } , a \ge 0 \text{ }, b \ge 0\\\ 0 & \text{otherwise} \end{cases} \tag{2}$$

$$a = \frac{\mu\_s \times b}{1 - \mu\_s} \tag{3}$$

$$b = (1 - \mu\_s) \times \left(\frac{\mu\_s \times (1 + \mu\_s)}{\sigma\_s^2} - 1\right) \tag{4}$$

The shape parameters *a* and *b* are determined according to the mean (μ*s*) and standard deviation (σ*s*) of solar irradiation data.

Mathematically, the generated power from the PV array is represented as

$$P\_s(t) = \eta\_s \times A\_s \times SI(1 + \beta(T\_t - 25))\tag{5}$$

The generated solar power is represented by the product of PV panel efficiency, size of the PV array, and solar irradiation. β denotes temperature coefficients of the maximum power of the PV array.
