Optimal Scheduling of Hydrothermal System:
The previous sections have dealt with the problem of optimal scheduling of a power system with thermal plants only. Optimal operating policy in this case can be completely determined at any instant without reference to operation at other times. This, indeed, is the static optimization problem. Operation of a system having both hydro and thermal plants is, however, far more complex as hydro plants have negligible operating cost, but are required to operate under constraints of water available for hydro generation in a given period of time. The problem thus belongs to the realm of dynamic optimization. The problem of minimizing the operating cost of a hydrothermal system can be viewed as one of minimizing the fuel cost of thermal plants under the constraint of water availability (storage and inflow) for hydro generation over a given period of operation.
Theory of Hydrothermal Systems:
For the sake of simplicity and understanding, the problem formulation and solution technique are illustrated through a simplified hydrothermal system of Fig. 7.12. This Hydrothermal System consists of one hydro and one thermal plant supplying power to a centralized load and is referred to as a Fundamental Hydrothermal System. Optimization will be carried out with real power generation as control variable, with transmission loss accounted for by the loss formula of Eq. (7.26).
For a certain period of operation T (one year, one month or one day, depending upon the requirement), it is assumed that (i) storage of hydro reservoir at the beginning and the end of the period are specified, and (ii) water inflow to reservoir (after accounting for irrigation use) and load demand on the system are known as functions of time with complete certainty (deterministic case). The problem is to determine q(t), the water discharge (rate) so as to minimize the cost of thermal generation.
under the following constraints:
- Meeting the load demand
This is called the power balance equation.
- Water availability
where J(t) is the water inflow (rate), X′(t) water storage, and X′(0), X′(T) are specified water storages at the beginning and at the end of the optimization interval.
- The hydro generation PGH(t) is a function of hydro discharge and water storage (or head), i.e.
The problem can be handled conveniently by discretization. The optimization interval T is subdivided into M subintervals each of time length ΔT. Over each subinterval it is assumed that all the variables remain fixed in value. The problem is now posed as
under the following constraints:
- Water continuity equation
The above equation can be written as
In Eqs. (7.73), X° and XM are the specified storage’s at the beginning and end of the optimization interval.
- Hydro generation in any subinterval can be expressed as
In the above problem formulation, it is convenient to choose water discharges in all subintervals except one as independent variables, while hydro generations, thermal generations and water storages in all subintervals are treated as dependent variables. The fact, that water discharge in one of the subintervals is a dependent variable, is shown below:
Adding Eq. (7.73) for m = 1, 2, …, M leads to the following equation, known as water availability equation
Because of this equation, only (M — 1) qs can be specified independently and the remaining one can then be determined from this equation and is, therefore, a dependent variable. For convenience, q1 is chosen as a dependent variable, for which we can write
The problem is solved here using non-linear programming technique in conjunction with the first order gradient method. The Lagrangian L is formulated by augmenting the cost function of Eq. (7.71) with equality constraints of Eqs. (7.72)— (7.74) through Lagrange multipliers (dual variables) λ1m, λ2m and λ3m. Thus,
The dual variables are obtained by equating to zero the partial derivatives of the Lagrangian with respect to the dependent variables yielding the following equations
and using Eq. (7.73) in Eq. (7.77), we get
The dual variables for any subinterval may be obtained as follows:
- Obtain λ1m from Eq. (7.78).
- Obtain λ3m from Eq. (7.79).
- Obtain λ21 from Eq. (7.81) and other values of λ2m(m ≠ 1) from Eq.
The gradient vector is given by the partial derivatives of the Lagrangian with respect to the independent variables. Thus
- Assume an initial set of independent variables qm (m≠1) for all subintervals except the first.
- Obtain the values of dependent variables Xm,PGHm,PGTm,q1 using Eqs.(7.73), (7.74), (7.72) and (7.76).
- Obtain the dual variables λ1m,λ3m,λ2m(m ≠ 1) and λ21 using Eqs. (7.78), (7.79), (7.80) and (7.81).
- Obtain the gradient vector using Eq. (7.82) and check if all its elements are equal to zero within a specified accuracy. If so, optimum is reached. If not, go to step 5.
- Obtain new values of control variables using the first order gradient method, i.e.
where α is a positive scalar. Repeat from step 2.
In the solution technique presented above, if some of the control variables (water discharges) cross the upper or lower bounds, these are made equal to their respective bounded values. For these control variables, step 4 above is checked in accordance with the Kuhn-Tucker conditions (7.61) given in Sec. 7.6.
The inequality constraints on the dependent variables are treated conveniently by augmenting the cost function with penalty functions as discussed in Sec. 7.6.
The Hydrothermal System method outlined above is quite general and can be directly extended to a system having multi-hydro and multi-thermal plants. The method, however, has the disadvantage of large memory requirement, since the independent variables, dependent variables and gradients need to be stored simultaneously. A modified technique known as decomposition overcomes this difficulty. In this technique optimization is carried out over each subinterval and the complete cycle of iteration is repeated, if the water availability equation does not check at the end of the cycle.