Exploiting Identical Generators in Unit Commitment

A novel projected two-binary-variable formulation for unit commitment in power systems

The short-term unit commitment (UC) problem in hydrothermal power generation is a large-scale, mixed-integer nonlinear program, which is difficult to solve efficiently, especially for large-scale instances. It is possible to approximate the nonlinear objective function of the problem by means of piecewise-linear functions, so that UC can be approximated by an mixed-integer linear program (MILP); applying the available efficient general-purpose MILP solvers to the resulting formulations, good quality solutions can be obtained in a relatively short amount of time. We build on this approach, presenting a novel way to approximating the nonlinear objective function based on a recently developed class of valid inequalities for the problem, called ldquoperspective cuts.rdquo At least for many realistic instances of a general basic formulation of UC, an MILP-based heuristic obtains comparable or slightly better solutions in less time when employing the new approach rather than the standard piecewise linearizations, while being not more difficult to implement and use. Furthermore, ldquodynamicrdquo formulations, whereby the approximation is iteratively improved, provide even better results if the approximation is appropriately controlled.

In recent years, high penetration of variable generating sources, such as wind power, has challenged independent system operators (ISO) in keeping a cheap and reliable power system operation. Any deviation between expected and real wind production must be absorbed by the power system resources (reserves), which must be available and ready to be deployed in real time. To guarantee this resource availability, the system resources must be committed in advance, usually the day-ahead, by solving the so-called unit commitment (UC) problem. If the quantity of committed resources is extremely low, there will be devastating and costly consequences in the system, such as significant load shedding. On the other hand, if this quantity is extremely high, the system operation will be excessively expensive, mainly because facilities will not be fully exploited. This thesis proposes computationally efficient models for optimal day-ahead planning in (thermal) power systems to adequately face the stochastic nature of wind production in the real-time system operation. The models can support ISOs to face the new challenges in short-term planning as uncertainty increases dramatically due to the integration of variable generating resources. This thesis then tackles the UC problem in the following aspects: Power system representation: This thesis identifies drawbacks of the traditional energy-block scheduling approach, which make it unable to adequately prepare the power system to face deterministic and perfectly known events. To overcome those drawbacks, we propose the ramp-based scheduling approach that more accurately describes the system operation, thus better exploiting the system flexibility. UC computational performance: Developing more accurate models would be pointless if these models considerably increase the computational burden of the UC problem, which is already a complex integer and non-convex problem. We then devise simultaneously tight and compact formulations under the mixed-integer programming (MIP) approach. This simultaneous characteristic reinforces the convergence speed by reducing the search space (tightness) and simultaneously increasing the searching speed (compactness) with which solvers explore that reduced space. Uncertainty management in UC: By putting together the improvements in the previous two aspects, this thesis contributes to a better management of wind uncertainty in UC, even though these two aspects are in conflict and improving one often means harming the other. If compared with a traditional energy-block UC model under the stochastic (deterministic) paradigm, a stochastic (deterministic) ramp-based UC model: 1) leads to more economic operation, due to a better and more detailed system representation, while 2) being solved significantly faster, because the core of the model is built upon simultaneously tight and compact MIP formulations. To further improve the uncertainty management in the proposed ramp-based UC, we extend the formulation to a network-constrained UC with robust reserve modelling. Based on robust optimization insights, the UC solution guarantees feasibility for any realization of the uncertain wind production, within the considered uncertainty ranges. This final model remains as a pure linear MIP problem whose size does not depend on the uncertainty representation, thus avoiding the inherent computational complications of the stochastic and robust UCs commonly found in the literature.

There is increasing evidence of shortage of ramping resources in the real-time operation of power systems. To explain and remedy this problem systematically, in this paper we take a novel look at the way the day-ahead unit commitment (UC) problem represents the information about load, generation and ramping constraints. We specifically investigate the approximation error made in mapping of the original problem, that would decide the continuous-time generation and ramping trajectories of the committed generating units, onto the discrete-time problem that is solved in practice. We first show that current practice amounts to approximating the trajectories with linear splines. We then offer a different representation through cubic splines that provides physically feasible schedules and increases the accuracy of the continuous-time generation and ramping trajectories by capturing sub-hourly variations and ramping of load in the day-ahead power system operation. The corresponding day-ahead UC model is formulated as an instance of mixed-integer linear programming (MILP), with the same number of binary variables as the traditional UC formulation. Numerical simulation over real load data from the California ISO demonstrate that the proposed UC model reduces the total day-ahead and real-time operation cost, and the number of events of ramping scarcity in the real-time operations.

In the literature, e.g. [10], one can find the so-called basic UAV mission target assignment in which m UAVs each with a capacity limit q visit n targets in a cooperative manner (and return to their departure points) such that the cost incurred by each UAV’s travel is minimized. In [10], we proposed a mixed integer linear program (MILP) formulation which exactly solves the problem, as well as four alternative MILP formulations which are computationally less intensive (and therefore suited for real-time purposes) yet yield a theoretically guaranteed sub-optimal solution. In this chapter, we further consider timing constraints imposed on some p of the targets, so-called prime targets. This consideration is often required for scenarios in which prime targets must be visited in a pre-defined time interval, and mathematically results in the addition of several integer linear constraints to the previous MILP formulation making the problem computationally intractable. We propose a novel procedure of adding these cumbersome timing constraints to the previous MILP formulation, in order to avoid increasing too much computational cost under practically valid assumptions. We first show that the proposed procedure still guarantees the previously claimed theoretical solution quality associated with the basic mission. We then show through extensive numerical simulations that under certain conditions, our algorithms return solutions which are still computationally manageable.

Optimal operation of water-energy microgrids; a mixed integer linear programming formulation

This paper proposes an innovative genetic algorithm (GA) approach to solve the thermal unit commitment (UC) problem in power generation industry through a constraint satisfaction technique. Due to a large variety of constraints to be satisfied, the solution space of the UC problem is highly nonconvex, and therefore the UC problem can not be solved efficiently by the standard GA. To effectively deal with the constraints of the problem and greatly reduce the search space of the GA, the minimum up- and down-time constraints are embedded in the binary strings that are coded to represent the on-off states of the generating units. The violations of the other constraints are handled by integrating penalty factors into the cost function. Numerical results on the practical Taiwan Power (Taipower) system of 38 thermal units over a 24-hour period show that the features of easy implementation, fast convergence, and highly near-optimal solution in solving the UC problem can be achieved by the proposed GA approach. >

Integrating Demand Response (DR) programs into the Unit Commitment (UC) problem is a promising method to enhance the efficiency and reliability of power systems. This work introduces a new formulation that incorporates price elasticity into the UC problem using a relaxed optimization approach. Our objective is to maximize overall system performance by reducing generation costs and maximizing the utility function while accounting for how demand changes in response to electricity prices, i.e., price elasticity-based DR. The proposed model employs Mixed-Integer Linear Programming (MILP) techniques to efficiently solve the UC problem, using a linear function to model price elasticity-based DR. Our approach has demonstrated its effectiveness in achieving substantial cost reductions and improved load management, as shown through numerical simulations.

The redundancy allocation problem (RAP) aims to find an allocation of redundant components that maximizes system reliability under resource constraints. In this paper, we consider the RAP for complex systems with heterogeneous components, which may have bridges or interconnecting subsystems and are generally more complicated in structure than commonly studied series-parallel systems. A major challenge of the RAP is that the reliability function is nonlinear. To tackle this, we develop six exact mixed integer linear programming (MILP) formulations, which, to our knowledge, are the first of such formulations for the problem. The key idea is to use binary variables and big-M constraints to successively linearize each product term in the reliability function. In addition, we propose a novel tree representation of the reliability function to establish the interconnections between different product terms, which is then used to reduce the number of decision variables and constraints needed in the linearization. We also propose an alternative form of the reliability function that only involves the failure probabilities of subsystems, and leverage this to derive better variable bounds, smaller big-M constants, and simpler formulations. We compare the six MILP formulations by solving them directly using an MILP solver based on three sets of test instances of varying complexity. Results indicate that all formulations achieve optimal solutions quickly for the first set and within reasonable computation times for the second set. For the third set, two formulations can always be solved within a given time limit. Overall, the MILP formulation derived based on the alternative form of the reliability function and a specific type of tree representation scheme performs the best. We also demonstrate that our approach is more robust and scalable than a benchmark branch-and-bound method. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: This work was supported by National Natural Science Foundation of China [Grants 72310107003, 72471189, 71901176]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2024.0842 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2024.0842 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

This paper presents the solving unit commitment (UC) problem using Modified Subgradient Method (MSG) method combined with Simulated Annealing (SA) algorithm. UC problem is one of the important power system engineering hard-solving problems. The Lagrangian relaxation (LR) based methods are commonly used to solve the UC problem. The main disadvantage of this group of methods is the difference between the dual and the primal solution which gives some significant problems on the quality of the feasible solution. In this paper, MSG method which does not require any convexity and differentiability assumptions is used for solving the UC problem. MSG method depending on the initial value reaches zero duality gap. SA algorithm is used in order to assign the appropriate initial value for MSG method. The major advantage of the proposed approach is that it guarantees the zero duality gap independently from the size of the problem. In order to show the advantages of this proposed approach, the four-unit Tuncbilek thermal plant and ten-unit thermal plant which is usually used in literature are chosen as test systems. Penalty function (PF) method is also used to compare with our proposed method in terms of total cost and UC schedule.

One of the main obstacles in using mixed integer linear programming (MILP) formulations for large-scale open pit production scheduling is the size of the problem. The objective of this work is to develop, implement, and verify deterministic MILP formulations for long-term large-scale open pit production scheduling problems. The objective of the model is to maximize the net present value, while meeting grade blending, mining and processing capacities, and the precedence of block extraction constraints. We present four MILP formulations; the first two models are modifications of available models; we also propose, test and validate two new MILP formulations. To reduce the number of binary integer variables in the formulation, we aggregate blocks into larger units referred to as mining-cuts. We compare the performances of the proposed models based on net present value generated, practical mining production constraints, size of the mathematical programming formulations, the number of integer variables required in formulation, and the computational time required for convergence. An iron ore mine case study is represented to illustrate the practicality of the models as well.

This paper presents centralized control for a wind farm using model predictive control (MPC) method. In order to solve intra-day unit commitment (UC) problem, the UC problem is solved by using the MPC method with short-term wind power forecasting. We introduce a new dynamics model for the UC with some constraints to utilize the benefits of the MPC method. The objective function considering the operation and maintenance costs is formulated by adding a new variable in order to average the operating time of each wind turbine (WT) within the whole time. The proposed method could solve the UC problems on-line using the prediction time horizon that could be selected flexibly considering time horizon based on wind power forecasting errors. From the simulation study using 10 WTs, we observed that the proposed dynamics model of UC effectively provided the optimal solution to each scenario. Numerical study will validate that the proposed method can be applied to solving the UC problem of a large scale wind farm by aggregating WTs.

An MILP formulation for optimizing detailed schedules of a multiproduct pipeline network

To solve joint optimization scheduling problem of thermal power plants based on the electricity market, the constraints and characteristics of optimal operation of thermal power plants are analyzed. From the perspective of economics and environmental science, we propose and establish an electricity market based joint optimization scheduling model of thermal power plants. Mutation particle swarm optimization algorithm (MPSO) is proposed to solve the model. Using an authentication instance, we compare and analyze the performances of the MPSO method and the mixed integer programming (MIP) method in solving the model. Further, considering a commonly used system, we show the efficiency of the MPSO method in solving the model; we apply two case studies to illustrate the performance of the MPSO method in solving the thermal unit commitment (UC) problem. To demonstrate the effectiveness of the MPSO method in solving the UC problem, we use MPSO method, PSO-LR (Lagrangian relaxation) [H. H. Balci and J. Valenzuela, Int. J. Appl. Math. Comput. Sci. 14(3), 411–421 (1989)], improved PSO [B. Zhao et al., Elect. Power Energy Syst. 28, 482–490 (2006)], mimetic algorithm [J. Valenzuela and A. Smith, J. Heuristics 8(2), 173–195 (2002)], adaptive LR [W. Ongsakul and N. Petcharaks, IEEE Trans. Power Syst. 19(1), 620–628 (2004)], stochastic priority list (PL) [T. Senjyu et al., Electr. Power Syst. Res. 76, 283–292 (2006)], PL based evolutionary algorithm [D. Srinivasan and J. Chazelas, in International Conference on Power System Technology (2004), Vol. 90, pp. 1746–1751], enhanced adaptive LR [W. Ongsakul and N. Petcharaks, IEEE Trans. Power Syst. 19(1), 620–628 (2004)], mixed-integer linear programming (MILP) [M. Carrin and J. M. Arroyo, IEEE Trans. Power Syst. 21(1), 321–332 (2006)], MILP [G. Morales-Espana et al., IEEE Trans. Power Syst. 28, 4897 (2013)], NBD [T. Niknam et al., Appl. Energy 86, 1667–1674 (2009)], generalized new benders decomposition approach (NBD) [S. Rahimi et al., World Appl. Sci. J. 6(12), 1665–1672 (2009)], LR and genetic algorithm[C. P. Cheng et al., IEEE Trans. Power Syst. 15(2), 707–714 (2000)], and seeded memetic algorithm (SMA) [J. Valenzuela and A. Smith, J. Heuristics 8(2), 173–195 (2002)] to solve the UC problem, simulation results are compared with the ones obtained by other methods. It is shown that the optimal operation model is right, the MPSO method is better than MIP method in solving the model, and they have universality. The effectiveness of the MPSO method in solving the UC problem is better than other methods. The MPSO method has better solutions in comparison with other methods, especially for systems with a large number of units.

This paper presents an algorithm for integrating and managing electrochemical energy storage systems (ESSs) on unit commitment (UC) problem. As some elements required for integrating electrochemical ESS, such as the power converter, have non-linear characteristics, its corresponding linear modeling could be difficult to be developed and included on the UC problem, which could lead to unfeasible solutions or unexpected results. In order to incorporate full models of ESS and its interface with the power system, in this paper an algorithm to incorporate electrochemical ESS management on the UC problem is presented. An insular power system of 10-units is analyzed, and conclusions are duly drawn.