Practical Robust Optimization Method for Unit Commitment of a System with Integrated Wind Resource
Profit based unit commitment: A parallel ABC approach using a workstation of optimal unit commitment with probabilistic reserve definition. Unit commitment (UC) is a critically important function in operation scheduling, the correlation relationship between the interrelated random variables and its . Here, we define as the start-up costs for the entire commitment. The unit commitment problem (UC) in electrical power production is a large family of . The relationship is nonlinear and nonconvex, making the problem.
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 [ 1819 ].
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 [ 161720 ]. 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.
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 [ 18 ].
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 [ 18 ].
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 [ 21 ] 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 [ 3 ]. 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.
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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 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.
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 [ 5610 — 15 ] in recent years, many successful methods and important results have been obtained based on those methods.
Unit commitment problem in electrical power production
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: This paper is organized as follows.
The significant penetration of wind generation in the grid raises a complex challenge to system operators caused by the highly uncertain and variable wind power generation pattern which is difficult to predict.
Furthermore, the spatial correlations between wind farms governed by atmospheric conditions complicate the difficulty on system operation and security; thus it is important to quantify the uncertainties and correlations of the wind power outputs at the geographically distributed wind sites.
Many research efforts have been made to deal with wind power uncertainty in UC [ 4 — 7 ]. Much of the work has focused on the solution of the optimization problem with the explicit representation of uncertainty. There are two principal approaches in the literature: The first approach explicitly makes use of probability distributions to represent various sources of the uncertainties and solves the problem in terms of expectation with discrete scenarios of future uncertain events [ 4 — 6 ].
This approach is simple and effective but has to address the challenges including determining the probabilities of the scenarios that can adequately capture a broad range of uncertainties and developing tractable numerical methods for dealing with huge number of scenarios with application to large-scale systems.
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The robust optimization techniques recently gained attentions for optimization problems under parameter uncertainties [ 89 ]. There are also some technical publications introducing the robust optimization methods into the realm for power system operation [ 79 — 11 ].
Le Cadre et al. A two-stage adaptive robust optimization model is proposed for unit commitment problem with load and wind forecasting uncertainties [ 79 ]. Since the wind power uncertainty is rather difficult to characterize in terms of analytic probability distributions, a common way to describe the forecasted wind power output is to give its intervals an associated confidence.
In general, the robust optimization approach can provide a more reasonable way to quantify the uncertain information in power systems, especially the renewable resources, such as the wind; it also requires less information than the scenario tree approach and thus can give a schedule results for system operators in a more robust and reliable manner. One of the main challenges of this approach currently is that it needs major computational effort for the decomposition and coordination process to converge the above two-stage robust model.
Furthermore, in the above publications, the spatial correlation between nodal injections caused by distributed wind sites, which would affect nodal wind power injections and thus real power flow through transmission lines, is not explicitly considered in the UC problem formulation. To effectively address the above challenges, we propose a new formulation and method for solving the UC problem for a system integrated with wind resource based on robust optimization schemes.
The wind uncertainty is modeled as an interval with an associated confidence depending on the individual risk tolerance [ 12 — 14 ].
However, it does not mean the wind farm power output can be independently selected within the above intervals; in other words, for example, if wind farms 1 and 2 have a positive correlation between their power output, then wind farm 2 will have a very low probability to create a small power output under the fact that wind farm 1 has a large power output.
To consider the grid line power flow constraints, we explicitly represent the correlations between the uncertain wind power outputs based on the principal component analysis PCA techniques.
The PCA techniques can capture effectively the correlation relationship between the interrelated random variables and its based transformation allows us to convert a large number of interrelated variables into uncorrelated principal components PCs [ 1516 ]. In this way, the multivariate statistical wind power interdependency can be represented by a series of uncorrelated variables, PCs; this will make it easier to devise a solution approach for the extended UC problem. If PCA-based transformation is not used, the data we collected from the wind power forecasting cannot be directly used in the day-ahead scheduling UC problem since only the data of intervals does not completely represent the information for the wind power output.
Only after transforming the correlated wind power outputs by their corresponding uncorrelated principal components, we can use the proposed solution approach to obtain the final scheduling results.
To conclude, the PCA technique is the key step for preprocessing the data for the correlated wind power outputs. Finally, it is worth noting that the PCA technique can only perform an orthogonal transformation so as to obtain an uncorrelated data. If the distribution considered is multivariate Gaussian, resulting PCs will be independent; otherwise, PCs will be uncorrelated but still dependent. In fact, wind power output is a non-Gaussian stochastic process and thus the diagonal covariance matrix of PCs only implies that they are uncorrelated.
In this paper, we adopt the preprocessing and transformation techniques which have been proposed in the previous publications [ 1718 ] to obtain approximately Gaussian datasets of wind power output.
After that, we can thus treat the PCs as approximately mutually independent.