Security-constrained unit commitment (SCUC) is an important tool for independent system operators in the day-ahead electric power market. A serious issue arises that the energy realizability of the staircase generation schedules obtained in traditional SCUC cannot be guaranteed. This paper focuses on addressing this issue, and the basic idea is to formulate the power output of thermal units as piecewise-linear function. All individual unit constraints and systemwide constraints are then reformulated. The new SCUC formulation is solved within the Lagrangian relaxation (LR) framework, in which a double dynamic programming method is developed to solve individual unit subproblems. Numerical testing is performed for a 6-bus system and an IEEE 118-bus system on Microsoft Visual C# .NET platform. It is shown that the energy realizability of generation schedules obtained from the new formulation is guaranteed. Comparative case study is conducted between LR and mixed integer linear programming (MILP) in solving the new formulation. Numerical results show that the near-optimal solution can be obtained efficiently by the proposed LR-based method.

Unit commitment (UC) is an important tool for independent system operators (ISOs) to obtain economical generation schedules in the day-ahead or week-ahead electric power market. The objective of UC is to determine the commitment states and generation levels of all generators over the scheduling horizon to minimize the total generation cost while meeting all systemwide constraints, such as system load balance and spinning reserve requirements, and individual unit operating constraints [

Since the power grid is being driven to operate more and more close to its security margin, considering security-related transmission constraints in UC problem; that is, security-constrained UC (SCUC) becomes indispensable in the newly deregulated power market [

In the literature on SCUC, the power output of a unit in each time period is represented by its average generation level such that the power output is formulated as a staircase function (see Figure

The comparison of power output models.

Staircase model

Piecewise-linear model

However, a serious issue arises that the energy realizability of the staircase generation schedules obtained in traditional SCUC cannot be guaranteed as stated in our previous work [

The cause of this issue lies in the fact that the energy output is distinguished from the power output especially when the ramping characteristics of generators are considered. If the energy output is to be accurately represented, it must be formulated as an integration of power output over a time period. However, such formulation with integral constraints as proposed in [

In this paper we focus on addressing the energy unrealizable issue of traditional SCUC. First of all, this issue is demonstrated and analyzed through an example of SCUC problem. The piecewise-linear power output (see Figure

The SCUC formulation established in this paper is then solved within the LR framework with all coupling constraints on different units relaxed by the Lagrange multipliers. A double dynamic programming method is used to obtain the exact optimal solution to each individual unit subproblem, and a modified subgradient algorithm is employed to update the multipliers. After the convergence of the Lagrange multipliers, a systematic method is developed for obtaining feasible solutions based on the dual solution.

Numerical testing is performed for a 6-bus system and a modified IEEE 118-bus system. It is proved that the formulation established in this paper overcomes the unrealizable issue of traditional SCUC formulations in terms of energy delivery. Numerical testing results demonstrate that the energy realizability of generation schedules is guaranteed and the near-optimal generation schedule can be also obtained efficiently by the proposed LR-based method.

The energy-realizable schedules obtained by the proposed LR method are also compared with those obtained by MILP-based method in IEEE 118-bus system. It is found that MILP-based method outperforms the LR-based method on small-size instances, but LR method is superior to the MILP method for solving larger-scale problems in term of computational efficiency. This feature is very important for solving large-scale SCUC problems.

It should be noted that additional continuous variables are necessarily introduced in this paper to formulate the piecewise-linear power output and the energy output. The increase of the variables in our formulation has low impact on the computational complexity under LR-based solution method since they could be eliminated in the procedure of solving unit subproblems with all systemwide constraints relaxed.

With great advances in theory and algorithms associated with other techniques [

The main contributions of this paper are as follows: (1) an energy-realizable SCUC formulation is presented by modeling power output as piecewise-linear function as well as reformulating individual unit constraints and systemwide constraints; (2) a double dynamic programming algorithm is developed to solve the hard unit subproblem under the new formulation.

This paper is organized as follows. The mathematical formulation is presented in Section

Following the examples in our previous work [

Variables

It is obvious that

Power generation and energy delivery.

Suppose a power system with

Power output modeled as a piecewise-linear function.

As mentioned in introduction, the energy output during period

The individual unit operating constraints are listed in (

Combining (

The basic idea of the LR-based method is to relax systemwide constraints and convert the original problem into a two-level optimization structure [

The framework of the Lagrangian relaxation.

It is seen that the individual unit subproblem (

Suppose

The new state transition diagram.

Based on the new state transition diagram, the entire schedule horizon is divided into several consecutive ON-state periods and several consecutive OFF-state periods. The continuous and discrete decision variables are therefore decoupled, and the solution to subproblem (

The framework of double dynamic programming.

The start-up cost during the consecutive OFF-state periods can be easily obtained when the piecewise linear formulation of

Based on the above analysis, the continuous optimization problem (

As shown in Figure

Let

Let

If the optimal generation levels and costs related to all consecutive ON-state periods and the start-up costs associated with all consecutive OFF-state periods are obtained, the optimal switch-ON decisions across the schedule horizon can be easily determined by using a discrete DP algorithm in high level, as seen in Figure

The principal advantage of double DP is that the exact optimal solution to the subproblem (

A modified subgradient method with adaptive step size is employed to update the Lagrange multipliers. Letting

A systematic method including heuristics is developed for constructing feasible solutions since the dual solution obtained usually does not satisfy the relaxed constraints (

Determine the feasibility (satisfying constraints (

Heuristics combined with “opportunity-cost” based criterion presented in [

Security-constrained economic dispatch (SCED) is performed for

The new SCUC formulation and LR-based solution method presented in this paper are tested with two cases consisting of a six-bus system and a modified IEEE 118-bus system. The amount of spinning reserve requirement at each time period is set to 2% of the hourly load, reserve responsive time is 10 minutes, and

The six-bus system, as shown in Figure

Generator data for example 1.

Unit | Bus no. | Pmax (MW) | Pmin (MW) | Initial status (h) | Min down (h) | Min up (h) | Ramp (MW/h) |
---|---|---|---|---|---|---|---|

G1 | 1 | 220 | 100 | 4 | 4 | 4 | 30 |

G2 | 2 | 100 | 10 | −3 | 3 | 2 | 50 |

G3 | 6 | 20 | 10 | −1 | 1 | 1 | 20 |

Transmission line data for example 1.

Line no. | From bus | To bus | X (pu) | Flow limit (MW) |
---|---|---|---|---|

1 | 1 | 2 | 0.170 | 200 |

2 | 1 | 4 | 0.258 | 100 |

3 | 2 | 3 | 0.037 | 100 |

4 | 2 | 4 | 0.197 | 100 |

5 | 3 | 6 | 0.018 | 100 |

6 | 4 | 5 | 0.037 | 100 |

7 | 5 | 6 | 0.140 | 100 |

Hourly load data for example 1.

H | Load (MWh) | H | Load (MWh) | H | Load (MWh) | H | Load (MWh) |
---|---|---|---|---|---|---|---|

1 | 175.19 | 7 | 168.39 | 13 | 242.18 | 19 | 245.97 |

2 | 165.15 | 8 | 177.60 | 14 | 243.60 | 20 | 237.35 |

3 | 158.67 | 9 | 186.81 | 15 | 248.86 | 21 | 237.31 |

4 | 154.73 | 10 | 206.96 | 16 | 255.79 | 22 | 215.67 |

5 | 155.06 | 11 | 228.61 | 17 | 256.00 | 23 | 185.93 |

6 | 160.48 | 12 | 236.10 | 18 | 246.74 | 24 | 195.60 |

The one-line diagram of six-bus system.

Commitment states in TF and NF are listed in Tables

Numerical testing results for example 1: unit commitment in TF.

Unit | Hours (0–24) |
---|---|

1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |

2 | 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 |

3 | 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 |

Numerical testing results for example 1: unit commitment in NF.

Unit | Hours (0–24) |
---|---|

1 | 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |

2 | 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 |

3 | 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 |

Figure

The power output curve of unit 1.

A much more complicated system is used to show the overall performance of our proposed LR-based method. The system has 54 units, 186 transmission lines, and 91 demand sides. The detailed parameters on generators, transmission network, and system load are given in

To investigate the computational efficiency of the LR-based solution method proposed in this paper, we extend the schedule horizon from one day to one week by duplicating the daily system load to every day in the horizon. Figure

Convergence behavior of LR for 1-day and 7-day horizons (normalized values are obtained by dividing its values by the obtained maximum value).

The results with respect to B&B algorithm used in constructing feasible solutions under LR are listed in Table

Numerical testing results for example 2: general results with different horizons.

LR | MILP | |||||

Day | CPU time^{1} (s) | Feasible cost ($) | Duality gap (%) | Iteration number (B&B) | CPU time^{2} (s) | Feasible cost ($) |

1 | 8.1 | 1984081 | 0.75 | 124 | 6.8 | 1979625 |

2 | 18.1 | 3749990 | 0.73 | 248 | 13.9 | 3746033 |

3 | 27.3 | 5713873 | 0.82 | 372 | 25.1 | 5712852 |

4 | 45.7 | 7591807 | 0.76 | 496 | 63.2 | 7591460 |

5 | 72.1 | 9420439 | 0.89 | 620 | 93.6 | 9421239 |

6 | 92.7 | 10850249 | 0.84 | 744 | 138.9 | 10852089 |

7 | 130.8 | 12235715 | 0.91 | 868 | 199.2 | 12236687 |

In order to evaluate the overall performance of the LR-based method, comparative cases are studied between the LR-based method and MIP-based method in this system. The MILP-based method employs branch-and-cut method that combines branch-and-bound and cutting plane technique. Once the model is formulated and represented in the MILP format, the solution is sought by engaging a general-purpose software, that is, CPLEX [

Table

The execution times with different schedule horizons under LR and MILP are also listed in Table ^{1}” and “CPU time^{2}” report the computing times of LR and MILP for solving the new SCUC formulation, respectively. Note that the stopping criteria described in (

It is also seen in Table

The realizability of generation schedule is very important to power system operation. Traditional SCUC formulations adopted in literature have a serious issue that the solution may be unrealizable in terms of energy delivery. This issue is analyzed through an example in this paper and a new SCUC formulation is established by modeling power outputs of units with piecewise linear functions. An LR-based method is developed to solve the problem, and the schedules obtained are near optimal, energy-realizable, and closer to practical operation of the thermal unit. Numerical testing results show the validation of the formulation and the effectiveness of the LR-based solution method. The energy-realizable schedules obtained by LR are also compared with those obtained by MILP. It is shown that the proposed LR-based method proposed is still competitive with those based on the general-purpose MILP solvers and even outperforms them for solving large-scale SCUC problems.

Total number of transmission lines

Total number of buses

The time span in each period, usually in hour

Minimum ON time of unit

Minimum OFF time of unit

Reserve responsive time for unit, usually set to 10 min or 30 min

Minimum power generation of unit

Maximum power generation of unit

System load at period

System reserve requirement at period

Load demand at bus

Limit of DC power flow in transmission line

Matrix of network sensitivity coefficient associated with units

Matrix of network sensitivity coefficient associated with demands

Coefficient between energy output of unit

System emission limits at period

Maximum ramp rate of unit

Fuel cost of unit

Emission of unit

Start-up cost of unit

Energy output of unit

Power generation level of unit

Power generation level of unit

Number of periods that unit

Discrete decision variable,

The Lagrange multiplier corresponding to energy balance constraint at period

The Lagrange multiplier corresponding to reserve requirements at period

The Lagrange multiplier corresponding to emission limits at period

The Lagrange multiplier associated with inequality (

The Lagrange multiplier associated with inequality (

The Lagrange multiplier associated with inequality (

The Lagrange multiplier associated with inequality (

This work is supported in part by the National Natural Science Foundation (60921003, 60736027, 60974101, 61174146), in part by the Program for New Century Talents of Education Ministry (NCET-08-0432), Foundation for Authors of National Outstanding Doctoral Dissertation (201047), and in part by the 111 International Collaboration Program of China.