Copyright © 2012 Hongyu Wu et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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 [1–3]. UC is often formulated as a nonlinear, large-scale, mixed integer programming (MIP) problem, and many approaches, such as dynamic programming (DP) , genetic algorithms (GAs) [5, 6], Lagrangian relaxation (LR) [1, 2, 7], Benders decomposition (BD) [8, 9], mixed integer linear programming (MILP) [10–12], and particle swarm optimization (PSO) [13–15], have been applied to solve the UC problems. LR is one of the most successful methods among them. The main advantage of applying LR is that its computational complexity of solving the dual problem is almost lineally related to the system size and therefore applicable for large-scale problem. In addition, Lagrange multipliers can be interpreted as the system shadow prices, which are important economic indicators of prices.
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 [16, 17]. In the current operation practice, a generation schedule is obtained from SCUC in day-ahead market and taken as an energy delivery schedule on an hourly basis in real time. Generators that are committed in the day-ahead scheduling have the obligation to deliver the awarded energy in real time. Generation companies (GENCOs) would be subject to real-time locational marginal prices (LMPs) and possibly incur penalties for deviating from the day-ahead schedule in the energy market [18, 19].
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 1(a)). The ramp-rate constraints are also simplified as limits on the difference of average generation levels in consecutive time periods [16, 17, 20]. The most obvious advantage of the staircase power output is its computational simplicity since the energy output at each time period is numerically equal to its average generation level.
Figure 1: The comparison of power output models.
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 staircase generation schedules are actually impossible to be implemented for GENCOs. In fact, we found that even though the ramp-rate constraints were satisfied, generation schedules with staircase generation levels might be still unrealizable in terms of energy delivery. A sufficient and necessary condition was thus established to check whether a generation schedule is deliverable in terms of energy . To our best knowledge, there is little effort in literature to further address this issue. Therefore, it is still open and pressing to obtain energy-realizable schedules for SCUC.
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  is difficult to be incorporated into SCUC for practical implementation due to the computational complexity of SCUC problem. A trade-off solution is to assume the linear variation of power output such that the energy output at each time period can be easily represented by the power output . This solution has been proven effective to treat the relationship between energy output and power output and thus it is also generalized to SCUC problem in this paper.
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 1(b)) is then formulated by introducing additional continuous variables. All individual unit constraints and systemwide constraints such as system energy balance, spinning reserve requirements and, DC transmission constraints are reformulated based on the piecewise-linear model.
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 [5, 6, 10–15] in recent years, many successful methods and important results have been obtained based on those methods. The motivation of this work, nevertheless, is not to give a full comparison between LR and other methods for solving the new SCUC problem. In this paper, we only want to suggest that one way is also valuable and important, that is, to design algorithms based on deep analysis and full utilization of the structure of SCUC. In this way, some new characteristics of the problem may be found and we may get a better understanding of the nature of SCUC problem. The algorithms designed may be still efficient since much structure information of the problem is combined with the algorithms.
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 2, in which an example is firstly given to demonstrate the deficiency of staircase power output, and the piecewise-linear formulation for SCUC is then established. The LR-based solution framework is discussed in Section 3. Numerical testing results are listed and analyzed in Section 4, and the paper is concluded in Section 5.
2. Mathematical Formulation
2.1. The Deficiency of Staircase Power Output: An Example
Following the examples in our previous work , the deficiency of staircase power output in traditional SCUC is presented in this section for self-containing. Let the minimum and maximum generation levels of a thermal unit be 100 MW and 300 MW, and the ramping limit (up or down) is 5 MW/min in an SCUC problem. Assume the generation output in the first hour is maintained at 300 MW.
Variables , represent the average generation levels (in MW) in the first two hours. If the time period is one hour, they are numerically equal to the energy delivery (in MWh) in the time period, where , are within their limits and satisfy the ramping constraints [8, 16, 17]:
It is obvious that is a feasible solution to this problem. However, if 100 MWh is taken as the scheduled energy output at the second hour based on the traditional SCUC formulation, it is unrealizable since the generation level is physically constrained and cannot change instantaneously at the beginning of hour 2 (point A in Figure 2). Even if the unit fully ramps down, the practical minimum energy output during hour 2 is 166 MWh, which is numerically equal to the area of ACDEF in Figure 2, and much greater than the scheduled energy output, the area of BDEF. The above example suggests that satisfying the ramping constraints cannot guarantee the desired energy output. In other words, the generation schedule with staircase power output obtained from traditional SCUC formulation may not be realizable in terms of energy delivery.
Figure 2: Power generation and energy delivery.
2.2. The Piecewise-Linear Formulation for SCUC
Suppose a power system with thermal units and the horizon of scheduling is partitioned into time periods. The SCUC problem is formulated as the following mixed-integer optimization problem with the objective to minimize the total operating costs. Quadratic fuel cost and piecewise linear start-up cost are generally adopted in literature, and the detailed mathematical formulations can be found in [1, 22]. All constraints are listed as follows, in which , , .
(a) Discrete State Transition
(b) Minimum Up/Down Time Limits
(c) Power Generation Limits
(d) Coincidence Constraints on Power Output
The coincidence constraint (2.6) suggests that and coincide for each two consecutive ON-state periods as illustrated in Figure 3 and implies that the power output trajectory of each unit must be continuous at the transition point of two adjacent ON-state periods. In comparison with the staircase power output model, the piecewise-linear model has better practicality since sudden changes are not allowed at this transition point.
Figure 3: Power output modeled as a piecewise-linear function.
(e) Minimum Generation at the First/Last ON-State Period
Constraints (2.8) and (2.9) are effective only for some units . As observed in Figure 3, when a unit switches ON at period , according to constraint (2.7). If constraint (2.8) is active, we have . It is seen that and do not coincide in the start-up process, neither do they coincide in a shut-down process when the unit switches OFF. The necessity of introducing additional variables can be seen at this point.
(f) Relationship between Energy Output and Power Output
As mentioned in introduction, the energy output during period (the shadow area in Figure 3) can be easily calculated based on the assumption of linear variation of power output, and it can be clearly expressed as follows: It is also seen from Figure 3 that the additional variables are needed in this paper to formulate the energy output when considering ON/OFF-state switches. With the analysis in Section 2.1, it is found that the simplified ramp-rate constraint in traditional SCUC formulation is one of the main factors leading to the unrealizability of staircase generation schedule; a ramping model with realizable energy delivery is therefore established as follows.
(g) Ramp-Rate Constraints
Энсей Танкадо - это Северная Дакота… Сьюзан попыталась расставить все фрагменты имеющейся у нее информации по своим местам. Если Танкадо - Северная Дакота, выходит, он посылал электронную почту самому себе… а это значит, что никакой Северной Дакоты не существует.
Партнер Танкадо - призрак. Северная Дакота - призрак, сказала она. Сплошная мистификация.