Method for solving the short-term thermal unit commitment

The model calculates optimal commitment and production schedules of thermal plants by using the Augmented Lagrangian Relaxation (ALR) method. In this method, a quadratic penalty function is added to the Lagrangian function in order to “convexify”

the Lagrangian, and therefore, enhance the convergence characteristics of the solution algorithm (Luenberger and Ye, 2008). The solution algorithm usually implies scheduling of generation in two stages. At the first stage of the algorithm, also known as

the Dual Optimization stage, satisfactory commitment plans of generators are derived by using an iterative procedure of multiplier updating. After that, the Primal Solution to the generation scheduling problem is obtained in the Economic Dispatch (ED) procedure. The Primary Solution of the problem determines optimal allocation of the system demand among the committed generation sources taking into account various constraints of individual units and system-level constraints including those that were not addressed in the dual optimization stage of the algorithm (Wang et al., 1995; Shaw, 1995).

For practical application in a real-sized power system, the Indirect method for the SCUC is used. In this method, transmission network constraints can be omitted from the Dual Optimization stage for the purpose of computational simplicity (Shaw, 1995).

In order to find the best possible commitment schedule of plants that satisfies the system-level constraints (4.15) and (4.16), the model performs the iterative procedure of the Lagrange multiplier adjustment. The values of the Lagrange multipliers are modified by using the sub-gradient maximization approach. The method is selected due to its simple implementation and low computational burden (Redondo and Conejo, 1999). In accordance with this method, at each iteration, the system balance and reserve multipliers are to be modified by using the following rule:

() () ( () () () ())

 - constant parameters determined heuristically k - iteration number

A distinctive feature of the sub-gradient method is that the quality of the obtained solution depends on the choice of parameters that determine the step length (Murai and Kato, 2002). In order to reduce uncertainty caused by parameter tuning in the subgradient-based method, and thereby increase the robustness of the simulations, short-term unit commitment is performed using the simplified technique of the Lagrange multiplier update. In effect, the initial solution of the considered UC problem is found, assuming the total reserve requirements to be zero. After that, a reserve feasible schedule is obtained by further stepping of the balance constraint multipliers (Murillo-Sanchez and Thomas, 1998).

In order to minimize the total cost of electricity production at the committed thermal power plants, a procedure of economic dispatch is performed. At the ED stage, the model optimizes the power output of the plants subject to balance constraints (4.15), the plants’ production output limits (4.17), and the transmission network limitations (4.20).

The zones j are connected with the paths lwhich represent the total transmission line capacity between the model regions. The network flows are modelled using a DC approximation in which the relationships between total power injections at zones and flows on transmission paths are established by a power transfer distribution matrix (PTDF). The matrix consists of sensitivity factors PTDF_lj, which quantify the changes in the value of flow on path l in the case of additional MW of power injection at the zone j (Leuthold et al., 2012).

In contrast to direct methods, which address all model constraints in both parts of the optimization algorithm, the solution obtained by using the Indirect method may suffer from feasibility problems. Therefore, additional shutdowns and start-ups of generators may be required to maintain the feasibility of the final commitment schedule with respect to all types of imposed constraints. Since network constraints were omitted from the UC stage, the final production schedules may need to be repaired in accordance with the information about the overloaded lines in the transmission system. In this work, a congestion management scheme as proposed in Ongsakul and Petcharaks (2003) is used

to prevent excessive power flows on the transmission lines. The above-mentioned approach to the problem of optimal generators’ re-dispatch comprises the following steps. First, the simulation model re-evaluates the results of the ED stage ignoring network limitations such as they do not exist. The resulting unconstrained power flows in the network paths are compared with the maximum transfer capacity of the corresponding paths, and the difference P_l(t) is used to determine the set of overloaded transmission paths G^t during hour t. After that, the congestion indices of the uncommitted generators are calculated as follows: with power plants committed one by one in accordance with their congestion index until all congestions on transmission paths are eliminated. Owing to the regional nature of the model, coefficients PTDFli in the expression (4.23) that represent the impacts of a single plant power output on flows in the transmission system are substituted with the region-specific coefficients PTDF_lj. Despite the simplifications, the algorithm is expected to ensure the least-cost re-dispatch of generation by selecting the plants from those model regions where an increase in the power output leads to a reduction in the overload on the congested transmission paths.

The overall procedure of the generation dispatch applied in this doctoral dissertation is summarized in the following steps:

1. Solve the hydro dispatch sub-problem by using the peak-shaving method.

2. Obtain a preliminary solution of the thermal generation sub-problem. Initialize the balance constraint multipliers by solving the Economic Dispatch problem with the spinning reserves integrated into the total load requirements and the lower production limits of generators set equal to zero.

3. Find the minimum cost commitment and production schedule for each generator by solving N_t separate single-unit optimization sub-problems.

4. Update the values of the Lagrange multipliers using the sub-gradient method.

5. If the reserve feasibility is reached, perform Economic Dispatch. Otherwise, go to step 3.

6. Perform the transmission-constrained Economic Dispatch with the committed generation sources.

7. If transmission limits are violated, perform the procedure of re-dispatch using the calculated congestion indices of uncommitted generators.