Special Ordered Sets Research Articles

Optimal utilization strategy of the LiFePO[formula omitted] battery storage

The paper provides a comprehensive battery storage modeling approach, which accounts for operation- and degradation-aware characteristics and can be used in optimization problem formulations. Particularly, Mixed-Integer Linear Programming (MILP) compatible models have been developed for the lithium iron phosphate (LiFePO4) battery storage using the Special Order Sets 2 to represent the nonlinear characteristics, including efficiency, internal resistance growth, and capacity fade. Such formulation can be used in problems related to various applications, i.e., power systems, smart grid, and vehicular applications, and it allows finding the globally optimal solution using off-the-shelf academic and commercial solvers. In the numerical study, the proposed modeling approach has been applied to realistic scenarios of peak-shaving, where the importance of considering the developed models is explicitly demonstrated. Operation- and degradation-aware techno-economic analysis showed that the optimal battery capacity is driven by operating rather than service requirements. Particularly, a considerable battery over-sizing becomes economically feasible when the battery storage is used more extensively. Another finding suggests that to achieve the maximum value from battery storage, its operation strategy needs to be significantly modified during the course of its lifetime. In the scenarios considered, the charging time gradually increased from four to seven hours, while the average SoC decreased by 20%. Such an adaptable scheduling results in reduced battery degradation and a longer lifetime, which may provide as much as 12.1% of savings in the battery storage system project.

Strategic bidding in price coupled regions

With the emerging deregulated electricity markets, a part of the electricity trading takes place in day-ahead markets where producers and retailers place bids in order to maximize their profit. We present a price-maker model for strategic bidding from the perspective of a producer in Price Coupled Regions (PCR) considering a capacitated transmission network between local day-ahead markets. The aim for the bidder is to establish a production plan and set its bids taking into consideration the reaction of the market. We consider the problem as deterministic, that is, the bids of the competitors are known in advance. We are facing a bilevel optimization problem where the first level is a Unit Commitment problem, modeled as a Mixed Integer Linear Program (MILP), and the second level models a market equilibrium problem through a Linear Program. The problem is first reformulated as a single level problem. Properties of the optimal spot prices are studied to obtain an extended formulation that is linearized and tightened using new valid inequalities. Several properties of the spot prices allow to reduce significantly the number of binary variables. Two novel heuristics are proposed, the first applicable in PCR, the second for general formulations with Special Ordered Sets (SOS) of type 1. Our computational experiments highlights the risk of a loss for the bidder if some aspects usually not considered in the literature, such as Price Coupled Regions, or an accurate UC problem, are not taken into account. They also show that the reformulation techniques, combined with new valid inequalities, allow to solve much larger instances than the current state-of-the-art. Finally, our experiments also show that the proposed heuristics deliver very high quality solutions in a short computation time.

Graph computing based security constrained unit commitment in hydro-thermal power systems incorporating pumped hydro storage

This paper proposes a graph computing based mixed integer programming (MIP) framework for solving the security constrained unit commitment (SCUC) problem in hydro-thermal power systems incorporating pumped hydro storage (PHS). The proposed graph computing-based MIP framework considers the economic operations of thermal units, cascade hydropower stations and PHS stations, as well as their technical impacts towards the network security. First, the hydro-thermal power system data and unit information are stored in a graph structure with nodes and edges, which enables nodal and hierarchical parallel computing for the unit commitment (UC) solution calculation and network security analysis. A MIP model is then formulated to solve the SCUC problem with the mathematical models of thermal units, cascade hydropower stations and PHS stations. In addition, two optimization approaches including convex hull reformulation (CHR) and special ordered set (SOS) methods are introduced for speeding up the MIP calculation procedure. To ensure the system stability under the derived UC solution, a parallelized graph power flow (PGPF) algorithm is proposed for the hydro-thermal power system network security analysis. Finally, case studies of the IEEE 118-bus system and a practical 2749-bus hydro-thermal power system are introduced to demonstrate the feasibility and validity of the proposed graph computing-based MIP framework.

A hierarchical transactive energy management framework for optimizing the reserve profile in district energy systems

District energy systems (DESs) have become a popular form of satisfying comprehensive energy demands for different types of loads in multiple local buildings. For DESs, the operational flexibility could be maintained by energy conversion and storage facilities. This paper proposes a hierarchical optimization framework for leveraging and aggregating the DES flexibility to provide contingency reserves. To characterize and quantify the flexibility in individual DESs, the concept of available reserve profile, which is measured by a set of indices, is established. A two-stage robust optimization (RO) model is developed for calculating the indices, which considers the uncertainties associated with wind power and ambient temperature. The lower stage of the two-stage model is managed by district energy system operators (DESOs) which submit reserve profiles to the district energy system coordinator (DESC) at the upper stage, which is responsible for the coordination process. Correspondingly, information privacy is preserved using a coordinated data-sharing strategy. Using reserve profiles submitted by multiple DESOs, the DESC applies the proposed coordination model to provide a certain reserve capacity schedule to DESs, which satisfies the stated objectives. The coordination model is formulated and solved based on the special ordered set (SOS) technique and particle swarm optimization (PSO) algorithm. A test system is developed to illustrate the technical viability and economic feasibility of the proposed technique.

A Risk-averse Inventory-based Supply Chain Protection Problem with Adapted Stochastic Measures under Intentional Facility Disruptions: Decomposition and Hybrid Algorithms

Owing to rising intentional events, supply chain disruptions have been considered by setting up a game between two players, namely, a designer and an interdictor contesting on minimizing and maximizing total cost, respectively. The previous studies have found the equilibrium solution by taking transportation, penalty and restoration cost into account. To contribute further, we examine how incorporation of inventory cost influences the players’ strategies. Assuming risk-averse feature of the designer and fully optimizing property of the interdictor with limited budget, the conditional-value-at-risk is employed to be involved in total cost. Using special order sets of type two and duality role, the linearized tri-level problem is solved by column-and-constraint generation and benders decomposition algorithms in terms of small-sized instances. In terms of larger-sized instances, we also contribute to prior studies by hybridizing corresponding algorithms with bio-geography based optimization method. Another non-trivial extension of our work is to define adapted stochastic measures based on the proposed mean-risk tri-level formulation. Borrowing instances from prior papers, the computational results indicate the managerial insights on players’ decisions, the model’s efficiency and performance of the algorithms.

A Mixed Integer Linear Programming Method for Optimizing Layout of Irrigated Pumping Well in Oasis

Optimization of irrigation well layout plays a vital role in the rational utilization of groundwater and to balance the water–energy nexus, especially in arid irrigation districts. This study proposes the mixed integer linear programming model (MILP) for well layout optimization with minimum well irrigation costs. This model efficiently establishes a link between irrigation area and wells to express the constraints of ensuring that irrigation area can be covered with optimal wells by using grid points to represent the irrigation area. It also uses the special ordered sets (SOS) modeling tool to decompose the mixed integer nonlinear programming into a mixed integer linear programming by assigning SOS-constrained weights to discrete points of a nonlinear function. This method was used in Cele Oasis of the Tarim Basin of the Xinjiang Province, an arid region in northwestern China. Since the original well layout was already established, different economic criteria like implicit cost and explicit cost were considered and two optimization results were yielded. The results showed that (1) the implicit cost optimization (ICO) and explicit cost optimization (ECO) reduced total costs by 7.64% and 3.56% compared with the condition of without optimization; and (2) the ICO and ECO reduced the optimal number of wells by 52.89% and 10.74% compared with the existing number of wells. Based on the analysis of the results, it is suggested that the manager should close uneconomical wells after determining the economic criteria. This method for well layout optimization can assist managers to make more rational plans for irrigation systems to exploit groundwater more efficiently, economically, and in a more environmentally friendly manner.

An MILP-Based Optimal Power and Gas Flow in Electricity-gas Coupled Networks

Abstract This paper focuses on the optimal power and gas flow (OPGF) in the electricity-gas coupled networks (EGCN) and proposes two mixed-integer linear programming (MILP) formulations via piecewise linear approximations (PWL) for solving the OPGF problem. The direct current power flow formulation is employed to model the electricity network. The special order sets (SOSs) and a two-stage PWL strategy combining incremental model and SOSs are developed to address the non-convex Weymouth gas flow equations. The two MILP formulations obtained are tractable and identify the global solutions. Numerical results investigate the performances of proposed two MILP formulations.

Integer programming for urban design

We present an integer program (IP) for urban design that (1) maximizes the floor area; (2) fills building volume with room templates; (3) encodes translational symmetry in urban layout; and (4) constructs economical urban routes. Regardless that integer programming is intensively studied in operational research (OR), its role in solving geometrical and topological problems in urban design was overlooked. Based on a regular grid, our 0–1 IP formulates the sunlight-gain rules, which give urban sites their shapes, especially for residential projects. With predefined plot templates, the gross floor area (volume) within a given site can be maximized under various sunlight requirements. Subsequently, the IP fills each building volume with 2D/3D room templates. Finally, an IP-based algorithm constructs routes that connect all plots and the site’s entrances to public transportation. Both the classical Steiner tree model and the latest coverage network model are extended to create reasonable routes. In addition, this work extends the concept of special ordered sets (SOS) to encode translational symmetry in urban layouts. Encoding layout symmetry can benefit from the solvers’ SOS2 mechanism in the Branch-and-Bound search algorithm. The results indicate that traditional decision making for cities could be partially automated by IP and an abundance of valid solutions are available for designers.

Optimized Energy Management System to Reduce Fuel Consumption in Remote Military Microgrids

This paper presents an optimized energy management system (OEMS) to control the microgrid of a remote temporary military base (FOB) featuring diesel generators, a battery energy storage system (BESS), and photovoltaic (PV) panels. The information of the expected electric demand is suitably used to improve the sizing and management of the BESS, according to the days of operation. The OEMS includes power electronics to charge the batteries from either the PV source or the diesel generators, and it can function as a current source when it is supplementing the power from one of the generators or as a voltage source when it is the sole source of power for the loads. The new contribution of this paper includes the optimization of a FOB's microgrid, where critical loads must be serviced at all times. The proposed optimization, which uses Special Order Sets for the semicontinuous function handling, also integrates economic evaluations by properly taking into account how the size of BESS affects its charge/discharge cycle; thus, the FOBs’ battery lifetime, in addition to its fuel consumption. Results from optimization are employed by the OEMS to coordinate the energy sources, and match the critical and noncritical loads with the available supply. Fuel savings of $\approx {\text{30}}\%$ (and $\approx {\text{50}}\%$ adding the PV source) can be achieved with respect to the already improved, but not optimal, solution of a previous work.

Modeling of non-linear CHP efficiency curves in distributed energy systems

Abstract Distributed energy resources gain an increased importance in commercial and industrial building design. Combined heat and power (CHP) units are considered as one of the key technologies for cost and emission reduction in buildings. In order to make optimal decisions on investment and operation for these technologies, detailed system models are needed. These models are often formulated as linear programming problems to keep computational costs and complexity in a reasonable range. However, CHP systems involve variations of the efficiency for large nameplate capacity ranges and in case of part load operation, which can be even of non-linear nature. Since considering these characteristics would turn the models into non-linear problems, in most cases only constant efficiencies are assumed. This paper proposes possible solutions to address this issue. For a mixed integer linear programming problem two approaches are formulated using binary and Special-Ordered-Set (SOS) variables. Both suggestions have been implemented into the optimization model DER–CAM to simulate investment decisions of CHP micro-turbines and CHP fuel cells with variable efficiencies. The approaches have further been applied successfully in a case study with four different commercial buildings. Comparison of the results between the standard version and the new approaches indicate that total annual system costs remain almost unchanged. System performance is subject to change and storage technologies become more important. Part load operation has mainly been found important for fuel cell units. The micro-turbine is found almost exclusively in full load, thus rendering the application of the new approaches for this technology unnecessary for the considered unit sizes and building types. The approach using binary variables was the most promising method to model variable efficiencies in terms of computational costs and results. It should especially be considered for specific fuel cell technologies. Further investigation on the impacts of this approach on the prediction of fuel cell and micro-turbine performance is suggested.

Toward Searching Possible Oscillatory Region in Order Space for Nonlinear Fractional-Order Systems

Finding the oscillatory region in the order space is one of the most challenging problems in nonlinear fractional-order systems. This paper proposes a method to find the possible oscillatory region in the order space for a nonlinear fractional-order system. The effectiveness of the proposed method in finding the oscillatory region and special order sets placed in its boundary is confirmed by presenting some examples.

Bid Optimization for Internet Graphical Ad Auction Systems via Special Ordered Sets

This paper describes an optimization model for setting bid levels for certain types of advertisements on web pages. This model is non-convex, but we are able to obtain optimal or near-optimal solutions rapidly using branch and cut open- source software. The financial benefits obtained using the prototype system have been substantial.

Piecewise linear relaxation of bilinear programs using bivariate partitioning

AbstractSeveral operational and synthesis problems of practical interest involve bilinear terms. Commercial global solvers such as BARON appear ineffective at solving some of these problems. Although recent literature has shown the potential of piecewise linear relaxation via ab initio partitioning of variables for such problems, several issues such as how many and which variables to partition, which partitioning scheme(s) and relaxation model(s) to use, placement of grid points, etc., need detailed investigation. To this end, we present a detailed numerical comparison of univariate and bivariate partitioning schemes. We compare several models for the two schemes based on different formulations such as incremental cost (IC), convex combination (CC), and special ordered sets (SOS). Our evaluation using four process synthesis problems shows a formulation using SOS1 variables to perform the best for both partitioning schemes. It also points to the potential usefulness of a 2‐segment bivariate partitioning scheme for the global optimization of bilinear programs. We also prove some simple results on the number and selection of partitioned variables and the advantage of uniform placement of grid points (identical segment lengths for partitioning). © 2009 American Institute of Chemical Engineers AIChE J, 2010

A special ordered set approach for optimizing a discontinuous separable piecewise linear function

We give a special ordered set (SOS) approach that optimizes a discontinuous separable piecewise linear function, even when a mixed-integer programming (MIP) model is not available for it. When a MIP model is available, our SOS model gives a linear programming relaxation bound that is as good as the MIPs.

A multiobjective optimization model for processing and distributing biosolids to reuse fields

Advanced wastewater treatment plants remove deleterious nutrients, chemicals, and microorganisms from wastewater and produce biosolids products to be reused at farms and other sites. These biosolids are carefully regulated by environmental restrictions but still may be malodorous to the local populations. In this paper, we develop a multiobjective optimization model to simultaneously minimize the biosolids odours as well as processing and distribution costs. The model employs a linear odour function and bilinear costs; the latter being approximated via Schur's decomposition and special ordered set (SOS) type 2 variables resulting in a mixed integer linear multiobjective optimization problem. Such a model can be used proactively by these plants to produce the least malodorous product at minimal costs. We demonstrate use of the model with a case study for the Blue Plains advanced wastewater treatment plant run by the DC Water and Sewer Authority in Washington, DC.

A practical approach to approximate bilinear functions in mathematical programming problems by using Schur's decomposition and SOS type 2 variables

This paper provides a new methodology to solve bilinear, non-convex mathematical programming problems by a suitable transformation of variables. Schur's decomposition and special ordered sets (SOS) type 2 constraints are used resulting in a mixed integer linear or quadratic program in the two applications shown. While Beale, Tomlin and others developed the use of SOS type 2 variables to handle non-convexities, our approach is novel in two aspects. First, the use of Schur's decomposition as an integral part of the approximation step is new and leads to a numerically viable method to separate the variables. Second, the combination of our approach for handling bilinear side constraints in a complementarity or equilibrium problem setting is also new and opens the way to many interesting and realistic modifications to such models. We contrast our approach with other methods for solving bilinear problems also known as indefinite quadratic programs. From a practical point of view our methodology is helpful since no specialized procedures need to be created so that existing solvers can be used. The approach is illustrated with two engineering examples and the mathematical analysis appears in the Appendices.

Connecting special ordered inequalities and transformation and reformulation technique in multiple choice programming

An article entitled: “A Note on Modeling Multiple Choice Requirements for Simple Mixed Integer Programming Solvers” was published by Ogryczak ( Comput. Oper. Res. 23 (1996) 199). In this article, Ogryczak proposed a reformulation technique called special ordered inequalities (SOI) to model the non-convex programming problems with special ordered sets (SOS) of variables. The SOI technique appears to be analogous to the reformulation technique introduced by Bricker (AIIE Trans. 9 (1977) 105) and is related to the reformulation and transformation technique (RTT) developed by Lin and Bricker (Eur. J. Oper. Res. 55(2) (1991) 228); Lin and Bricker (Eur. J. Oper. Res. 88 (1996) 182). Since none of this literature was cited in the references of Ogryczak ( Comput. Oper. Res. 23 (1996) 199), we would like to use this note to differentiate SOI and RTT and to elaborate their connection. Scope and purpose In the context of non-convex programming, two major types of special ordered sets (SOS) of variables have been identified and studied by researchers. SOS1 are sets of non-negative variables where, for each set, at most one of the variables can be non-zero in the final solution. The most common application of SOS1 is multiple choice programming (MCP) which can be found in the modeling of many integer programming problems in location, distribution, scheduling, etc. SOS2 requires that, for each set, at most two of the variables can be non-zero in the final solution and, if they are, they must be adjacent. SOS2 has been widely used in separable programming to model non-linear functions using sets of piece-wise linear functions. Bricker introduced an explicit reformulation technique for SOS in 1977. Lin and Bricker developed a reformulation and transformation technique (RTT) to implicitly compose the optimal Simplex tableau for MCP in 1991. They also elaborated upon it with a computational report in 1996. Without citing the work by Bricker, or that by Lin and Bricker, Ogryczak proposed an analogous reformulation technique called special ordered inequalities (SOI) for SOS in 1996. This note aims to elaborate upon the connection between SOI and RTT as the supplementary information for the future research in SOS.

Reformulation of the set partitioning problem as apure network with special order set constraints

In this paper, the set partitioning problem is reformulated as a network flow problemwith special order sets. The reformulation is based on identifying network structure that ishidden in the element-set incidence matrix. The special order sets and the revealed networkin the reformulation together define an effective structure for solution by enumeration. Theresulting enumeration procedure for the solution of the set partitioning problem is computationallyadvantageous since it uses a pure network relaxation that is solved using reoptimization,allows a large number of variables to be fixed in a subproblem, and is defined overa relatively small enumeration tree. Computational experience is included.

A note on modeling multiple choice requirements for simple mixed integer programming solvers

This note introduces the Special Ordered Inequalities (SOI) as a new technique for integer programming modeling of multiple choice requirements and piecewise linear functions. The standard modeling technique based on the use of Special Ordered Set (SOS) structure requires a branch and bound solver to be armed with the special tools for SOS processing. Otherwise, the standard branching rule applied on individual SOS variables may lead to enormously long computation process thus making impossible to solve quite small problems in a reasonable time. The proposed SOI modeling technique allows us to eliminate the necessity of the special branching rule for handling SOS structure. The standard branching on individual variables of SOI is equivalent to the special SOS branching. Thus the proposed modeling technique seems to be very attractive for users of simple mixed integer programming solvers as well as for those who want to build solver independent mathematical programming models.

Southern California Gas Company Uses Special Ordered Sets to Model Regulatory Guidelines

Modeling rule-based decisions often conflicts with the mathematics of optimization because the rules may conditionally force actions that are not economically optimal. In order to simulate the reality of the public utility business, Southern California Gas Company (SoCalGas) includes in their operations planning models rule-based guidelines regulating the treatment of customers and suppliers. One approach to modeling such conditional guidelines is to use mixed integer programming (MIP); however, MIP is often avoided if possible because of its reputation for poor performance. SoCalGas uses an advanced mixed integer programming solution technique, special ordered sets (SOS) of type 2, to model the guidelines effectively and efficiently.