The unit commitment (UC) problem which is an important subject in power system engineering is solved by using Lagragian relaxation (LR), penalty function (PF), and augmented Lagrangian penalty function (ALPF) methods due to their higher solution quality and faster computational time than metaheuristic approaches. This problem is considered to be a nonlinear programming-(NP-) hard problem because it is nonlinear, mixed-integer, and nonconvex. These three methods used for solving the problem are based on dual optimization techniques. ALPF method which combines the algorithmic aspects of both LR and PF methods is firstly used for solving the UC problem. These methods are compared to each other based on feasible schedule for each stage, feasible cost, dual cost, duality gap, duration time, and number of iterations. The numerical results show that the ALPF method gives the best duality gap, feasible and dual cost instead of worse duration time and the number of iterations. The four-unit Tuncbilek thermal plant which is located in Kutahya region in Turkey is chosen as a test system in this study. The programs used for all the analyses are coded and implemented using general algebraic modeling system (GAMS).

The UC problem decides to which electricity generation units should be running in each stage to satisfy a predictably varying demand for electricity. The solution of the UC problem is a complex optimization problem, and UC problem can be considered as two linked optimization problems: the first is a combinatorial optimization problem, and the second is a nonlinear programming problem. Due to important startup and shutdown costs, the problem is in general very hard to solve, as it is not possible to perform a separate optimization for each time interval. The exact solution of the UC can be obtained by a complete enumeration of all feasible combination of generating units, which could be very huge number [

The problem is structured binary variables making it clearly nonconvex. The binary variables cause a great deal of trouble and reason for the difficulty in solving the UC. Load balance constraint and spinning reserve constraint are coupling constraints across the unit so that one unit affects what will happen on other units if the coupling constraints are met. There have been various methods which are based on mathematical programming and metaheuristic-based for solving the thermal and hydrothermal UC problem in literature. They are based on mathematical programming and metaheuristic based approaches. These major methods are priority list [

The LR which is one of the most successful approaches for UC is dual optimization technique. This method obtains an appropriate condition to generate feasible solution for UC. One of the most obvious advantages of the LR method is its quantitative measure of the solution quality since the cost of the dual function is a lower bound on the cost of the primal problem [

The LR, PF, and ALPF methods which present the solving approach based on dual optimization technique are used for solving the UC problem known as an important and hard-solving problem in power system engineering, and the results from the programs coded and implemented using GAMS for these methods are compared to each other. The UC problem for the four-unit Tuncbilek thermal plant is considered for this analysis. A 24-hours day is subdivided into eight discrete stages, and the predicted load of the system can be considered constant over each interval. In this case, the UC problem is an economic combinatorial problem of which solution will yield the minimum operating cost for the scheduling of startup and shutdown of units for each stage in a day.

The remaining sections are outlined as follows. Section

UC has been used to plan over a given time horizon the most economical schedule of committing and dispatching generating units to meet forecasted demand levels and spinning reserve requirements while all generating unit constraints are satisfied. The objective function of the UC problem can be formulated as

The minimization of the objective function is provided to the following constraints.

In this paper, three different methods based on dual optimization techniques, LR, PF, and ALPF are used for solving the UC problem. Duality is particularly important in optimization theory. The convergence of the dual optimization methods can be measured by the relative size of the duality gap between the primal and dual solutions. As illustrated in Figure

A simplified graphic representation of dual optimization [

General constrained minimization problem is as follows:

The dual procedure attempts to reach the constrained optimum by maximizing the Lagrangian with respect to Lagragian multipliers, while minimizing with respect to the other variables in the problem.

LR method can eliminate the dimensionality problem. The LR method is based on the dual optimization theory. The solution of the LR method greatly depends on the Lagragian multipliers. Therefore, setting the initial Lagragian multipliers and updating them are significant to the optimality of the solution. Inappropriate method of updating the Lagrangain multipliers may cause the solution adjustment to oscillate around the global optimum [

The Lagrangian equation is

Dual function is

The algorithm of LR method is

Choose starting

Find a value for each

Assuming that the

PF method is one of the widely used methods for obtaining optimal solutions nonconvex problems. It is motivated by the desire to use unconstrained optimization techniques to solve constrained problems. It uses a mathematical function that will increase the objective for any given constrained violation. General transformation of constrained problem into an unconstrained problem [

The algorithm of PF method is

Choose

Choose penalty factor

Find the following values:

If

The ALPF method is an approach for solving nonlinear programming problems by using the ALPF in a manner that combines the algorithmic aspects of both LR and PF methods [

Lagragian equation is

The algorithm of ALPF method is as follows.

Select some initial Lagrangian multipliers. Lagrangian multipliers are

Let _{n}

Measure the level of the violation by using (

Choose

Solve minimum of the Lagrangian subject to

Replace

The LR method solves the UC problem by relaxing or temporarily ignoring the coupling constraints. This is done through the dual optimization procedure attempting to reach the constrained optimum by maximizing the Lagrangian. In this section, the solving algorithm of LR method for the UC problem is explained as a sample. The Lagrangian equation can be written as follows [

Goal of separating the units from one another has been achieved. The term inside the outer brackets

Equation (

This is solved as a one-variable DP problem. This can be visualized in Figure

Only two possible states for unit

At the

The minimum of this function is

The solution to this equation is

There are three cases in this situation.

DP is used to determine the optimal schedule of each unit over the scheduled time period. To minimize

In the LR method, the value obtained by (

The LR, PF, and ALPF methods are applied for solving the UC problem for four-unit Tuncbilek thermal plant separately, and the numerical results are given in tables. These methods are compared to each other based on feasible schedule for each stage, feasible cost, dual cost, duality gap, duration time, and number of iterations using the table in the comparative results part. The spinning reserve demand is set as 10% of the system demand. All these methods are coded in GAMS which is a high-level modeling system for mathematical programming problems [

Unit characteristics for four-unit Tuncbilek thermal plant.

Unit | SC ($) | a ($/h) | b ($/MWh) | c ($/MW^{2}h) | SD ($) | ||||
---|---|---|---|---|---|---|---|---|---|

1 | 8 | 32 | 60 | 0.515 | 10.86 | 149.9 | 120 | 1 | 1 |

2 | 17 | 65 | 240 | 0.227 | 8.341 | 284.6 | 480 | 2 | 2 |

3 | 35 | 150 | 550 | 0.082 | 9.9441 | 495.8 | 1100 | 3 | 3 |

4 | 30 | 150 | 550 | 0.074 | 12.44 | 388.9 | 1100 | 3 | 3 |

In this study, a 24-hours day is subdivided into 8 discrete stages. The load demands for the stages are given in Table

Load data (MW).

Stage | Load | Stage | Load |
---|---|---|---|

1 | 168 | 5 | 313 |

2 | 150 | 6 | 347 |

3 | 260 | 7 | 308 |

4 | 275 | 8 | 231 |

LR method is applied to UC problem. The feasible unit combination for the LR method is given in Table

Feasible unit combination for LR method.

Stage | Unit combination |
---|---|

1 | 0 0 1 1 |

2 | 0 0 1 1 |

3 | 0 1 1 1 |

4 | 1 1 1 1 |

5 | 1 1 1 1 |

6 | 1 1 1 1 |

7 | 1 1 1 1 |

8 | 0 1 1 1 |

PF method is applied to UC problem. The feasible unit combination for the PF method is given in Table

Feasible unit combination for PF method.

Stage | Unit combination |
---|---|

1 | 0 0 1 1 |

2 | 0 0 1 1 |

3 | 0 1 1 1 |

4 | 0 1 1 1 |

5 | 1 1 1 1 |

6 | 1 1 1 1 |

7 | 0 1 1 1 |

8 | 0 1 1 1 |

ALPF method is applied to UC problem. The feasible unit combination for the ALPF method is given in Table

Feasible unit combination for ALPF method.

Stage | Unit combination |
---|---|

1 | 0 0 1 1 |

2 | 0 0 1 1 |

3 | 0 1 1 1 |

4 | 0 1 1 1 |

5 | 0 1 1 1 |

6 | 1 1 1 1 |

7 | 0 1 1 1 |

8 | 0 1 1 1 |

Feasible cost, dual cost and duality gap values, duration time, and number of iterations are found for each method. These numerical results are given in Table

The numerical results for LR, PF, and ALPF methods.

Compared items | LR | PF | ALPF |
---|---|---|---|

Feasible cost ($) | 56381.931 | 55801.477 | 55233.897 |

Dual cost ($) | 55428.014 | 55305.332 | 54894.266 |

Duality gap (%) | 1.1721 | 0.8971 | 0.6187 |

Number of iterations | 19 | 36 | 67 |

Duration time ( | 5.32 | 3.56 | 14.63 |

It can be seen from Table

Three different dual approach-based methods, LR, PF and ALPF methods, are used for solving the UC problem known as an important and hard-solving problem in power system engineering, and the results from the programs coded and implemented using GAMS for these methods are compared to each other according to feasible cost, dual cost, duality gap, duration time, and number of iterations. The numerical results show that the ALPF method which is firstly used for solving the UC problem in literature gives the best duality gap, feasible and dual cost values instead of worse duration time and the number of iterations. It is seen that ALPF method yields the minimum duality gap value with minimum cost. On the other hand, LR method reaches the solution with the minimum number of iterations. In addition to this, PF method yields the fastest solution. Since the most important factor in the solution of UC problems is feasible cost value, it can be said that ALPF is the most suitable method among others.

Objective function

Upper bound of the dual function

Equality constraint

Inequality constraint

Multiplier for PF method

Multiplier for PF and ALPF method

Number of generating units

Primal problem

Dual problem

Nominal demand at hour

Generation output of unit

Maximum available capacity of unit

Minimum available capacity of unit

System spinning reserve at hour

Ramp up rate of unit

Ramp down rate of unit

Start up cost of unit

Shut down cost of unit

Status value of unit

Time horizon for UC (

Minimum down-time of unit

Continuously off-time of unit

Continuously on-time of unit

Minimum up-time of unit

Total cost (

Multiplier for ALPF.