**2. Formulation of STHS Problem**

The STHS problem is to allocate the generation to the hydro and thermal units so that the required load demand is achieved and it reduces the net cost without affecting other constraints of Hydro and Thermal systems.

#### **2.1 Hydro-thermal scheduling (HTS)**

Since hydropower unit's fuel cost are trivial when assessed with that of thermal unit, the optimal HTS solution lessens the net coal cost of the thermal units with the maximum utilization of the accessible hydro resource. In line with this, the optimal HTS problem is formulated as the fuel cost *FC* as given in (1).

$$FC(PT\_{i\_j}) = \sum\_{i=1}^{N} \sum\_{j=1}^{Z} a\_i PT\_{i\_j}^2 + b\_i PT\_{i,j} + c\_i \tag{1}$$

Considering the VPL effect as a sinusoidal variation the Eq. (1) modifies to (2).

$$FC(PT\_{i,j}) = \sum\_{i=1}^{Ns} \sum\_{j=1}^{Z} a\_i PT\_{i,j}^2 + b\_i PT\_{i,j} + c\_i + \left| d\_i \times \sin\left(e\_i \times \left( PT\_{i,\\_\text{min}} - PT\_{i,j} \right) \right) \right| \tag{2}$$

The prime goal of HTS problem is to reduce the net fuel cost *F* of the thermal plants. Then the objective function is given in (3).

$$\text{Minimize } F = \sum\_{j=1}^{Z} \sum\_{i=1}^{N\_r} FC(PT\_{ij}) \tag{3}$$

where the symbols carry the meaning as defined earlier. The following operational restrictions are to be satisfied. *Dynamic Economic Load Dispatch of Hydrothermal System DOI: http://dx.doi.org/10.5772/intechopen.108052*

a. *Load Demand constraints*: It is defined as the balance of the net hydro and thermal generation with the load inclusive of losses in each slot of scheduling *j* as given in (4)

$$\sum\_{i=1}^{N\_l} PT\_{i,j} + \sum\_{i=1}^{N\_h} PH\_{i,j} = PD\_j + PL\_j, \text{for } j = 1, 2, \dots, Z \tag{4}$$

b. *Generation constraints of Thermal Plant*: The *ith* thermal generator must operate within the lower and upper bound *PTi* min and *PTi* max respectively as shown in (5)

$$PT\_{i\text{ min}} \le PT\_{\vec{\eta}} \le PT\_{i\text{ max}} \tag{5}$$

c. *Generation constraints of Hydro Plant*: The *ith* hydro plant generator must operate between its minimum and maximum bounds *PHi* min and *PHi* max respectively as given in (6)

$$PH\_{i\text{ min}} \le PH\_{i,j} \le PH\_{i\text{ max}} \tag{6}$$

d. *Reservoir constraint*: The *ith* reservoir volume capacity has to lie within the lowest and highest margins as expressed in (7)

$$V\_{i\min} \le V\_{i,j} \le V\_{i\max} \tag{7}$$


$$V\_{i(j+1)} = V\_{\vec{\eta}} + \sum\_{u=1}^{R\_u} \left[ q\_{u(j-\tau)} + s\_{u(j-\tau)} \right] - q\_{i(j+1)} - s\_{i(j+1)} + r\_{i(j+1)} \text{ for } j = 1, 2 \dots, Z \tag{8}$$

Where τ is the time gap for water transportation to the reservoir *i* from its upstream reservoir *u* at time slot *j* and *Ru* is the combination of the upstream hydraulic reservoirs before the hydro plant *i*

g. *The power generation of the hydro plant PHij*. It depends on water discharge rate and reservoir storage capacity. It is expressed as in (10)

$$PH\_{\vec{\eta}} = c\_{1i}V\_{\vec{\eta}}^2 + c\_{2i}q\_{\vec{\eta}}^2 + c\_{3i}\left(V\_{\vec{\eta}}q\_{\vec{\eta}}\right) + c\_{4i}V\_{\vec{\eta}} + c\_{5i}q\_{\vec{\eta}} + c\_{6i} \tag{9}$$

Where, *c*1*<sup>i</sup>* to *c*6*<sup>i</sup>* are the constants.
