An improved unit decommitment algorithm for combined heat and power systems

This paper addresses the unit commitment in multi-period combined heat and power (CHP) production planning, considering the possibility to trade power on the spot market. In CHP plants (units), generation of heat and power follows joint characteristics, which means that production planning for both heat and power must be done in coordination. We present an improved unit decommitment (IUD) algorithm that starts with an improved initial solution with less heat surplus so that the relative cost-efficiency of the plants can be determined more accurately. Then the subsequent decommitment procedures can decommit (switch off) the least cost-efficient plants properly. The improved initial solution for the committed plants is generated by a heuristic procedure. The heuristic procedure utilizes both the Lagrangian relaxation principle that relaxes the system-wide (heat and power) demand constraints and a linear relaxation of the ON/OFF states of the plants. We compare the IUD algorithm with realistic test data against a generic unit decommitment (UD) algorithm. Numerical results show that IUD is an overall improvement of UD. The solution quality of IUD is better than that of UD for almost all of tested cases. The maximum improvement is 11.3% and the maximum degradation is only 0.04% (only two sub-cases out of 216 sub-cases) with an average improvement of 0.3–0.5% for different planning horizons. Moreover, IUD is more efficient (1.1–3 times faster on average) than UD.     

Efficient algorithms for combined heat and power production planning under the deregulated electricity market

Combined heat and power (CHP) production is an important energy production technology that can yield much higher total energy efficiency than separate heat and power generation. In CHP production, the heat and power production follows a joint characteristic, which means that the production planning must be done in coordination. Cost-efficient operation of a CHP system can be planned by using an optimization model. A long-term planning model decomposes into thousands of hourly models. Earlier, in the regulated electric power market, the planning problem was symmetrically driven by heat and power demand. The liberalization of the power market has created an asymmetrical planning problem, where heat production responds to the demand and power production to the volatile market price. In this paper, we utilize this asymmetry to develop novel envelope-based dual algorithms for solving the hourly CHP models efficiently. The basic idea is to transform the three-dimensional characteristic operating region for heat and power production of each CHP plant into a two-dimensional envelope by taking the power price as a parameter. Then the envelopes of each plant are used for looking up the optimal solution rapidly. We propose two versions of the algorithm: the on-line envelope construction algorithm (ECON) where the envelopes are constructed for each hour based on the power price and the off-line envelope construction algorithm (ECOFF) where envelopes are pre-computed for all different power price ranges. We derive the theoretical time complexity of the two algorithms and compare their performance empirically with realistic test models against the ILOG CPLEX solver and the Power Simplex (PS) algorithm. PS is an extremely efficient specialized primal algorithm developed for the symmetrical CHP planning problem under the regulated market. On average, when reusing previous basic solutions, ECON is 603 times faster than CPLEX and 1.3 times faster than PS. ECOFF is 1860 times faster than CPLEX and four times faster than PS.     

An efficient envelope-based Branch and Bound algorithm for non-convex combined heat and power production planning

Combined heat and power (CHP) production is universally accepted as one of the most energy-efficient technologies to produce energy with lower fuel consumption and fewer emissions. In CHP technology, heat and power generation follow a joint characteristic. Traditional CHP production is usually applied in backpressure plants, where the joint characteristic can often be represented by a convex model. Advanced CHP production technologies such as backpressure plants with condensing and auxiliary cooling options, gas turbines, and combined gas and steam cycles may require non-convex models. Cost-efficient operation of a CHP system can be planned using an optimization model based on forecasts for heat load and power price. A long-term planning model decomposes into thousands of single-period models, which can be formulated in the convex case as linear programming (LP) problems, and in the non-convex case as mixed integer programming (MIP) problems.     

An efficient linear model and optimisation algorithm for multi-site combined heat and power production

Combined heat and power (CHP) production is an important energy production technology which can help to improve the efficiency of energy production and to reduce the emission of CO₂. Cost-efficient operation of a CHP system can be planned using an optimisation model based on hourly load forecasts. A long-term planning model decomposes into hourly models, which can be formulated as linear programming (LP) problems. Such problems can be solved efficiently using the specialized Power Simplex algorithm. However, Power Simplex can only manage one heat and one power balance. Since heat cannot be transported over long distances, Power Simplex applies only for local CHP planning.In this paper we formulate the hourly multi-site CHP planning problem with multiple heat balances as an LP model with a special structure. We then develop the Extended Power Simplex (EPS) algorithm for solving such models efficiently. Even though the problem can be quite large as the number of demand sites increases, EPS demonstrates very good efficiency. In test runs with realistic models, EPS is from 29 to 85 times faster than an efficient sparse Simplex code using the product form of inverse (PFI). Furthermore, the relative efficiency of EPS improves as the problem size grows.     

Solving the Unit Commitment Problem by a Unit Decommitment Method

In this paper, we present a unified decommitment method to solve the unit commitment problem. This method starts with a solution having all available units online at all hours in the planning horizon and determines an optimal strategy for decommitting units one at a time. We show that the proposed method may be viewed as an approximate implementation of the Lagrangian relaxation approach and that the number of iterations is bounded by the number of units. Numerical tests suggest that the proposed method is a reliable, efficient, and robust approach for solving the unit commitment problem.     

CO2 emissions trading planning in combined heat and power production via multi-period stochastic optimization

The EU emissions trading scheme (ETS) taking effect in 2005 covers CO₂ emissions from specific large-scale industrial activities and combustion installations. A large number of existing and potential future combined heat and power (CHP) installations are subject to ETS and targeted for emissions reduction. CHP production is an important technology for efficient and clean provision of energy because of its superior carbon efficiency. The proper planning of emissions trading can help its potential into full play, making it become a true “winning technology” under ETS. Fuel mix or fuel switch will be the reasonable choices for fossil fuel based CHP producers to achieve their emissions targets at the lowest possible cost. In this paper we formulate CO₂ emissions trading planning of a CHP producer as a multi-period stochastic optimization problem and propose a stochastic simulation and coordination approach for considering the risk attitude of the producer, penalty for excessive emissions, and the confidence interval for emission estimates. In test runs with a realistic CHP production model, the proposed solution approach demonstrates good trading efficiency in terms of profit-to-turnover ratio. Considering the confidence interval for emission estimates can help the producer to reduce the transaction costs in emissions trading. Comparisons between fuel switch and fuel mix strategies show that fuel mix can provide good tradeoff between profit-making and emissions reduction.     

Lagrangian relaxation based algorithm for trigeneration planning with storages

Trigeneration is a booming power production technology where three energy commodities are simultaneously produced in a single integrated process. Electric power, heat (e.g. hot water) and cooling (e.g. chilled water) are three typical energy commodities in the trigeneration system. The production of three energy commodities follows a joint characteristic. This paper presents a Lagrangian relaxation (LR) based algorithm for trigeneration planning with storages based on deflected subgradient optimization method. The trigeneration planning problem is modeled as a linear programming (LP) problem. The linear cost function poses the convergence challenge to the LR algorithm and the joint characteristic of trigeneration plants makes the operating region of trigeneration system more complicated than that of power-only generation system and that of combined heat and power (CHP) system. We develop an effective method for the long-term planning problem based on the proper strategy to form Lagrangian subproblems and solve the Lagrangian dual (LD) problem based on deflected subgradient optimization method. We also develop a heuristic for restoring feasibility from the LD solution. Numerical results based on realistic production models show that the algorithm is efficient and near-optimal solutions are obtained.     

Scheduling unit — time tasks on flow — shops under resource constraints

We consider the problem of scheduling tasks on flow shops when each task may also require the use of additional resources. It is assumed that all operations have unit lengths, the resource requirements are of 0–1 type and there is one type of the additional resource in the system. It is proved that when the number of machines is arbitrary, the problem of minimizing schedule length is NP-hard, even when only one unit of the additional resource is available in the system. On the other hand, when the number of machines is fixed, then the problem is solvable in polynomial time, even for an arbitrary number of resource units available. For the two machine case anO(n log ₂ ² n) algorithm minimizing maximum lateness is also given. The presented results are also of importance in some message transmission systems.     

An adaptive algorithm for system level diagnosis

Consider a system of n units, at most t of which are faulty. An adaptive diagnosis algorithm is presented which uses a sequence of tests to identify a fault-free unit. The algorithm requires at most 2t − ν(t) tests, where ν(t) is the number of 1's in the binary representation of t. Moreover, many of the tests can be performed simultaneously. The previously best algorithms for the same purpose required 2t − 1 tests, none of which could be performed simultaneously.     

Market Power Issues in Bid-Based Hydrothermal Dispatch

The objective of this work is to investigate market power issues in bid-based hydrothermal scheduling. Initially, market power was simulated with a single stage Cournot–Nash equilibrium model. In this static model the equilibrium was calculated analytically. It was shown that the total production of N strategic agents is smaller than the least-cost solution by a factor of (N/(N+1)). Market power analysis for multiple stages was then carried through a stochastic dynamic programming scheme, where the decision in each stage and state is the Cournot–Nash equilibrium of a multi-agent game. Case studies with data taken from the Brazilian system are presented.     

New Cyclic Difference Sets with Singer Parameters Constructed from d-Homogeneous Functions

In this paper, for a prime power q, new cyclic difference sets with Singer para- meters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed by using q-ary sequences (d-homogeneous functions) of period q ⁿ –1 and the generalization of GMW difference sets is proposed by combining the generation methods of d-form sequences and extended sequences. When q is a power of 3, new cyclic difference sets with Singer parameters ((q ⁿ –1/q–1), (q ^n–1–1/q–1), (q ^n–2–1/q–1)) are constructed from the ternary sequences of period q ⁿ –1 with ideal autocorrelation introduced by Helleseth, Kumar, and Martinsen.     

The construction of an iterative fractional scheme: the case of Joe

In his paper on A New Hypothesis Concerning Children’s Fractional Knowledge, Steffe (2002) demonstrated through the case study of Jason and Laura how children might construct their fractional knowledge through reorganization of their number sequences. He described the construction of a new kind of number sequence that we refer to as a connected number sequence (CNS). A CNS can result from the application of a child’s explicitly nested number sequence, ENS (Steffe, L. P. (1992). Learning and Individual Differences, 4(3), 259–309; Steffe, L. P. (1994). Children’s multiplying schemes. In: G. Harel, & J. Confrey (Eds.), (pp. 3–40); Steffe, L. P. (2002). Journal of Mathematical Behavior, 102, 1–41) in the context of continuous quantities. It requires the child to incorporate a notion of unit length into the abstract unit items of their ENS. Connected numbers were instantiated by the children within the context of making number-sticks using the computer tool TIMA: sticks. Steffe conjectured that children who had constructed a CNS might be able to use their multiplying schemes to construct composite unit fractions. (In the context of number-sticks a composite unit fraction could be a 3-stick as 1/8 of a 24-stick.) In the case of Jason and Laura, his conjecture was not confirmed. Steffe attributed the constraints that Jason and Laura experienced as possibly stemming from their lack of a splitting operation for composite units. In this paper we shall demonstrate, using the case study of Joe, how a child might construct the splitting operation for composite units, and how such a child was able to not only confirm Steffe’s conjecture concerning composite unit fractions, but also give support to our reorganization hypothesis by constructing an iterative fractional scheme (and consequently, a fractional connected number sequence (FCNS)) as a reorganization of his ENS.     

Zero divisors and units with small supports in group algebras of torsion-free groups

Kaplansky’s zero divisor conjecture (unit conjecture, respectively) states that for a torsion-free group G and a field 𝔽, the group ring 𝔽[G] has no zero divisors (has no units with supports of size greater than 1). In this paper, we study possible zero divisors and units in 𝔽[G] whose supports have size 3. For any field 𝔽 and all torsion-free groups G, we prove that if αβ = 0 for some non-zero α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥10. If 𝔽 = 𝔽₂ is the field with 2 elements, the latter result can be improved so that |supp(β)|≥20. This improves a result in Schweitzer [J. Group Theory, 16 (2013), no. 5, 667–693]. Concerning the unit conjecture, we prove that if αβ = 1 for some α,β∈𝔽[G] such that |supp(α)| = 3, then |supp(β)|≥9. The latter improves a part of a result in Dykema et al. [Exp. Math., 24 (2015), 326–338] to arbitrary fields.     

QR factorization of Toeplitz matrices

Summary This paper presents a new algorithm for computing theQR factorization of anm×n Toeplitz matrix inO(mn) operations. The algorithm exploits the procedure for the rank-1 modification and the fact that both principal (m–1)×(n–1) submatrices of the Toeplitz matrix are identical. An efficient parallel implementation of the algorithm is possible.     

Average-case results on heapsort

A new algorithm for rearranging a heap is presented and analysed in the average case. The average case upper bound for deleting the maximum element of a random heap is improved, and is shown to be less than [logn]+0.299+M(n) comparisons, *) whereM(n) is between 0 and 1. It is also shown that a heap can be constructed using 1.650n+O(logn) comparisons with this algorithm, the best result for any algorithm which does not use any extra space. The expected time to sortn elements is argued to be less thann logn+0.670n+O(logn), while simulation result points at an average case ofn log n+0.4n which will make it the fastest in-place sorting algorithm. The same technique is used to show that the average number of comparisons when deleting the maximum element of a heap using Williams' algorithm for rearrangement is 2([logn]–1.299+L(n)) whereL(n) also is between 0 and 1, and the average cost for Floyd-Williams Heapsort is at least 2nlogn–3.27n, counting only comparisons. An analysis of the number of interchanges when deleting the maximum element of a random heap, which is the same for both algorithms, is also presented.     

Maximizing Queueing Network Utility Subject to Stability: Greedy Primal-Dual Algorithm

We study a model of controlled queueing network, which operates and makes control decisions in discrete time. An underlying random network mode determines the set of available controls in each time slot. Each control decision “produces” a certain vector of “commodities”; it also has associated “traditional” queueing control effect, i.e., it determines traffic (customer) arrival rates, service rates at the nodes, and random routing of processed customers among the nodes. The problem is to find a dynamic control strategy which maximizes a concave utility function H(X), where X is the average value of commodity vector, subject to the constraint that network queues remain stable.We introduce a dynamic control algorithm, which we call Greedy Primal-Dual (GPD) algorithm, and prove its asymptotic optimality. We show that our network model and GPD algorithm accommodate a wide range of applications. As one example, we consider the problem of congestion control of networks where both traffic sources and network processing nodes may be randomly time-varying and interdependent. We also discuss a variety of resource allocation problems in wireless networks, which in particular involve average power consumption constraints and/or optimization, as well as traffic rate constraints.     

A probabilistic analysis of the switching algorithm for the euclidean TSP

The well-known switching algorithm proposed by Lin and Kernighan for the Euclidean Travelling Salesman Problem has proved to be a simple efficient algorithm for medium size problems (though it often gets trapped in local optima). Although its complexity status is still open, it has been observed to be polynomially bounded in practice, when applied to uniformly distributed points in the unit square. In this paper this polynomial behaviour is derived theoretically. (However, we will come up with a bound of O(n ¹⁸) with probability 1 –c/n, whereas in practice the algorithm works slightly better.)     

Joint planning of combined heat and power and electric power systems: An efficient model formulation

The growing importance of combined heat and power (CHP) around the world has increased the need to consider its role within electric power systems. In this paper, we show how the problem of joint planning of CHP and electric power systems may be formulated more efficiently than has previously been done, by exploiting the special structure of the optimal solution. Numerical tests indicate that this reformulation typically allows a reduction in the time required for problem solution by a factor of two to five.     

On optimal simultaneous rational approximation to with being some kind of cubic algebraic function

It is shown that each rational approximant to (ω,ω²)^τ given by the Jacobi–Perron algorithm (JPA) or modified Jacobi–Perron algorithm (MJPA) is optimal, where ω is an algebraic function (a formal Laurent series over a finite field) satisfying ω³+kω-1=0 or ω³+kdω-d=0. A result similar to the main result of Ito et al. [On simultaneous approximation to (α,α²) with α³+kα-1=0, J. Number Theory 99 (2003) 255–283] is obtained.