1.2. Literature Review
Hybrid energy system composed of wind turbines, solar photovoltaic panels, batteries and auxiliary fuel generator have been mathematically modelled by Gupta et al. [
1] and Billionet et al. [
2]. Similar problems are also studied with considering different kinds of generators as mentioned by Gupta et al. [
1] such as: micro-hydro generator, biomass generator, and biogas generator. The aim of their model is to identify the most economic and appropriate power supply and operating the diesel engine generator under constant output with high efficiency to reduce pollution. The authors considered remote rural areas, and formulated the problem as a mixed integer linear programming with the objective of minimizing the energy costs.
Uncertainties in energy generation and demand had been handled by Prabhu et al. [
3], Zhou et al. [
4], and Billionet et al. [
2]. A two-stage robust approach with constraint generation algorithm is used by Billionet et al. [
2], where each sub-problem is reformulated by a mixed-integer, linear program and hence solved by a standard solver. The authors use a polynomial time dynamic programming algorithm for the recourse problem and showed that, in some cases, this algorithm is much more efficient than mixed-integer linear programming approach. Stochastic/probabilistic models for the same problem were considered by Prabhu et al. [
3]. To solve this problem Zhou et al. [
4] used a two-stage decomposition based solution strategy with genetic algorithm performing the search on the first stage variables and a Monte Carlo method dealing with uncertainties in the second stage.
The problem of energy efficiency versus effectiveness was solved by Prabhu et al. [
3]. They optimized energy efficiency considering manufacturing system effectiveness as a constraint, or they optimize manufacturing system effectiveness for a given profile. They used the total available power as a constraint, or they defined a simultaneous optimization of both to get a balanced solution.
The economic efficiency and energy intensity consumption had been analyzed by Bojnec and Papler [
5] as determinants of sustainable economic development for 33 selected European countries. The methods used are multivariate factor analyses, regression, and correlation.
The trade off between the costs of renewable energy technologies and the reliability of a multi-sources generation system had been studied by Bilal et al. [
6]. They proposed a fast and elitist multi-objective non-dominated sorting genetic algorithm NSGA-II to optimize the use of renewable energy technologies taking into account the annualized cost and the power system reliability in terms of load supplied and including renewable sources in power generation.
Buildings with multisources of energy need optimization to supply its load with less cost and more efficiency. A multisource system technology’s proper sizing and energy demands are optimized by Barbieri et al. [
7], they used genetic algorithm for minimum energy consumption and net present value, the method is applied on a thirteen-floor tower. To reduce the energy demand, cost and emissions Galvão et al. [
8] developed an energy model based on a mixed system of renewable energy. This energy model is a new sustainable standard about energy consumption efficiency of a small hotel building and a relevant contribution to certify the building. For minimum life-cycle costs of meeting the energy demand (power, heating, cooling) of a commercial building, Safaei et al. [
9] integrated cogeneration, solar and conventional energy sources. They optimized the investment planning and operating strategies of the energy systems using general algebraic modeling system. The analysis showed varied operating strategies and output levels among cogeneration technologies and the energy systems coupled with them.
Various problems face multisources of energies in microgrids. Achieving better economic and environmental benefits of microgrids under multiple uncertainties in renewable energy resources and loads is problem solved by Wang et al. [
10]. The energy production scheduling method is based on robust multi-objective optimization with minimax criterion, they used a hierarchical meta-heuristic solution strategy including multi-objective cross entropy algorithm. Distributed intelligent multi-agent technology is applied by Logenthiran et al. [
11] to make the power system more reliable, efficient and capable of exploiting and integrating alternative sources of energy. The simulation results of a power system with distributed resources comprising three microgrids and five lumped loads show that the proposed multiagent system allows efficient management of micro-sources with minimum operational cost. To manage the problematics for different levels of information sharing in a smart grid, Caron and Kesidis [
12] proposed a dynamic pricing scheme incentivizing consumers to achieve an aggregate load profile suitable for utilities, and study how close they can get to an ideal flat profile depending on how much information they share. They provide distributed stochastic strategies that successfully exploit this information to improve the overall load profile. For future smart grid, Mohsenian-Rad et al. [
13] considered autonomous and distributed demand-side energy management system among users that take advantage of a two-way digital communication infrastructure envisioned. The procedure utilized is game theory and formulation of energy consumption scheduling game. To achieve a stable operation after islanding with minimum fuel cost during the grid-connected opteration, Nam et al. [
14] developped economic dispatch problem and the constraints considering reservation for variation in load demand, power outputs of non-dispatchable distributed generators, and stable islanded operation, with flow limits between two different areas. For optimally designing distributed energy resources within cooling and heating power based microgrids, Zhang et al. [
15] coupled environmental and economic sustainability in a multi-objective optimisation model using genetic algorithm which integrates the results of a life cycle assessment.
Researchers and practitioners had different attempts for managing energy pricing. A quantitative analysis on real time pricing and the energy market had been studied by Poletti and Wright (2017) [
16] for New Zealand introducing different assumptions on the retail market and the shape of the demand function. For scheduling electrical appliances for an individual household, taking into account a grid connected system with a battery and an in-house renewable energy generator, Mitra and Dutta [
17] offered dynamic prices as function of the planned consumption and forecasted grid load in the optimization model for minimum electricity bill. To reduce the consumption of electricity in peak periods for industries, Safdarian et al. [
18] considered time varying prices by establishing a stochastic model for the medium-term decision making problem faced by a distribution company. The model is formulated as a mixed integer linear programming and using a price elasticity matrix the demand response to time of use prices is captured. To bridge the speed of response gap between suppliers and consumers yet adhere to the principle of marginal cost pricing of electricity, Çelebi and Fuller [
19] examined the complementarity programming models of equilibrium. They developped a computable equilibrium model to estimate ex ante time of use prices for retail electricity market. To study the biomass supply contract pricing and policy making in the biofuel industry, Huang and Hu [
20] proposed an agent-based simulation model formulated as a mixed integer optimization model. In this model, the agents include a biofuel producer and farmers. Farmers’ decision-making is assumed to be profit driven. Due to the dissimilarity between the peak and off-peak hours prices, Sulaima et al. [
21] introduced Demand Side Management through Demand Response technique for the modification of the demand profile by implementing different strategies of measures. The objective of this study is to optimize the energy profile for commercial sector, as well as analyze the significance of electricity cost reduction by using Evolutionary Algorithm Meta-heuristic optimization technique in Malaysia. In electricity markets under uncertain price Luo et al. [
22] set up a new self-scheduling model based on robust optimization methodology. By using optimal dual theory, the proposed model is reformulated to an ordinary quadratic and quadratic cone programming problems in the cases of box and ellipsoidal uncertainty respectively. The model is tested on IEEE 30-bus system is used to test the new model.
There are attempts for optimizing the total energy costs for machines. For example, two new mathematical models to reduce total energy consumption cost of a single machine manufacturing system are presented by Aghelinejad et al. [
23]. The first model improved the formulation of Shrouf et al. [
24] problem considering a predetermined jobs sequence. Meanwhile, the second model studies production scheduling on job levels and machine, proposing an optimal sequence for them by minimizing the occurrence number of each machines state, the optimal allocation of these states during the periods minimizes energy costs and jobs within the processing state. The optimal solution of the problem is provided by calculating the shortest path between the first node and last node representing respectively the first and the last periods. Machines consume different amount of energy in function of its state: processing, idle or off state. So for single machine scheduling problems, Aghelinejad et al. [
25] proposed a dynamic programming approach to solve these problems by using a finite graph. A non-preemptive single-machine manufacturing environment had been investigated by Aghelinejad et al. [
26] to reduce the total costs. They Improved the mathematical formulation of scheduling problem in a predetermined order at machine level to process the jobs. Second, the authors generalized the model to deal simultaneously with the production scheduling at machine level as well as job level. A heuristic algorithm and a genetic algorithm were applied on the second model because the problem is NP hard to provide good solutions in reasonable computational time in an accurate and efficient manner. To minimize the total energy consumption costs over the planning horizon of several energy-oriented single-machine, Aghelinejad et al. [
27] studied two cases: constant energy price and increasing energy prices during all the time-slots. Moreover, two versions are investigated: with and without the fixed sequence for the jobs. A multi-item capacitated lot-sizing and scheduling problem had been discussed by Masmoudi et al. [
28] in a flow-shop system with energy consideration. They formulated a mixed integer linear programming to solve this issue, and because the capacitated lot-sizing problems are NP-hard, a fix-and relax heuristic is put in place. A single-item capacitated lot-sizing problem in a flow-shop system with energy consideration is also addressed by Masmoudi et al. [
29]. In addition to fix-and-relax heuristic method, a genetic algorithm is formed for better quality solutions and to deal with the complexity of the NP-hard problem. In a two stage production system, where a product is manufactured on a machine and delivered to the subsequent production stage in batch shipments, Zanoni et al. [
30] proposed an analytical model of this system to minimize the total cost of the production, storing, and energy. To optimally plan saving for energy aware scheduling of manufacturing processes, Bruzzone et al. [
31] proposed a mixed integer programming model where the reference schedule is modified to account for energy consumption without modifying the jobs’ sequencing and assignment provided by the reference schedule. For sustainable consideration in manufacturing scheduling, Mansouri et al. [
32] incorporated energy consumption as an explicit criterion in shop floor scheduling. They explore the potential for energy saving in manufacturing. They analyzed the trade-off between total energy consumption, minimizing makespan, and a measure of service level. To find the Pareto frontier comprised of makespan and total energy consumption, they developed a mixed integer linear multi-objective optimization model. For Sustainable scheduling Gahm et al. [
33] developed a research framework for energy-efficient scheduling.
There are different electricity tariffs for residential, commercial and industrial customers. The service types applicable to industrial customers are further classified into low-tension and high-tension. The rate schedules available for high-tension service are based on time of use and maximum demand. This paper focuses on the electricity contract decisions of energy consumers such as high-tension industrial customers.
Most industrial and commercial customers are required to pay for their peak demand, besides the energy they consume. Electric utilities charge them for the highest average demand measured in any 15 or 30 min during their billing period. Billing demand is based on consumers’ measured maximal demand and their contract demand with utility companies as per the supply agreement [
34].
In mainland China, utilities allow large customers to adjust their contract demand monthly [
35], if the peak demand does not exceed the contract demand, a fixed demand charge is levied; on the other hand, if the peak demand exceeds the contract demand, a penalty charge twice as the basic rate is levied. In Southern Africa, the contract demand is determined by the notified maximum demand [
35]. Customers can temporarily or permanently increase/decrease their notified maximum demand, and their demand charge is based on the maximum of the measured demand and notified maximum demand. Jemena Electricity Networks Ltd., of Victoria, Australia, also has a similar contract demand reset policy, which allows their customers to permanently/temporarily increase and permanently decrease their contract demand through request. Based on Shanghai, the timely adjustment and notification of the contract demand is a valuable information source for utilities’ load forecast and maintenance planning [
36]. Along with the development of advanced metering infrastructures and data analysis platform, it is expected that a wider range of customers, to provide more accurate and complete information for smart grid operation and smart city functionalities, can adjust the contract demand more frequently and conveniently.
When it comes to the industrial customer side, matching system requirements with the offers in the energy market is one of the important decisions that must be made [
37]. Many industrial customers opt to sign a maximum contracted demand. Such an electricity bill consists of an energy charge and a capacity charge. The energy charge is based on kilowatt-hours, while the capacity charge is based on maximum demand consumed during each time of use period. If the peak demand does not exceed the contract capacity, a fixed capacity charge is levied. On the other hand, if the peak demand exceeds the contract capacity, a penalty charge from two to three times the basic rate is levied [
35]. Hence, choosing an excessively low contract capacity will impose high capacity charges, while choosing an excessively high contract capacity may result in an unnecessary basic capacity charge. Therefore, optimal contract capacity decisions have received significant attention from customers with high electricity usage.
In some countries, electricity bills are composed of five primary components: capacity cost, demand of energy cost, power factor adjustment, penalty cost, and expanding construction cost. The energy costs include the active and reactive energy charges but we left the demand of energy out of the context, since it does not affect the optimization problem. Improper contract capacities capacity scheduling would also cause high expanding construction cost when users modify their contract capacities. If customers modify their contract capacities, electricity suppliers would adopt expanding line construction cost schedules. Capacity cost is based on kilowatt hours, with the unit price varying by peak, medium and off-peak and capacity charge is determined by kilowatts per month based on maximum demand (in 15 minutes average) during the time of use period [
38]. In our case, we consider only peak contract capacity for simplicity.
There are different contributions to find the optimal contract capacity for energy consumers. To determine the electricity contract capacity for industrial customers in Taiwan Chen et al. [
35] used a linear programming approach. To calculate the optimal capacity for peak, semi-peak, and off-peak capacities respectively, Tsay et al. [
39] used meta-heuristic evolutionary programming method. The method is applied on Drow-Ing refinery contract, Ho-Jin region contract, Ling-Jan refinery contract, and Da-Liau station contract. To solve the same problem for selection of optimal capacities, for peak, semi-peak, and off-peak capacities Lee et al. [
38] used another meta heuristic method called iteration particle swarm optimization. A new index, called iteration best is incorporated into particle swarm optimization to improve solution quality and computation efficiency. Demand contract decision for the taiwanese industries are optimized by Hwang et al. [
40]. The techniques employed are cat swarm optimization and particle swarm optimization. This research aims at exploring the benefit on load management options and to provide decision-makers and leaders with useful operation and management strategies as reference. For the assignment of the optimal production planning and energy contract Rodoplu et al. [
37] used linear programming in CPLEX software. The optimization model minimize energy and production costs with respect to constraints of energy supplier contract conditions and production systems. As an application, they chose different contract capacities of multisource of energy in three machines for minimum costs and optimal production planning. Optimization of contract capacity setting for industrial consumer with self-owned generating units is a highly discrete and a nonlinear model. To resolve this issue Yang and Peng [
41] considered the peak, semi-peak, and off-peak contract capacities in their model. To clear up this muddle the authors used an improved Taguchi method by combining it with particle swarm optimization algorithm. Their paper uses data derived from the SCADA system of a large optoelectronics factory with self-owned generating units in September 2004 to March 2005. When several rates are available in the market Ferdavani et al. [
42] proposed new procedure to unfold the contract capacity optimization problem. They include peak contract capacity in their optimization model. The proposed technique gave a better concept for optimizing contract capacity with errors in the forecasted prices or forecasted maximum demand and is faster than the linear programming. In case of variable contract capacity price and uncontracted demand price, using Newton-Raphson method the solution is found within maximum two iterations. To highlight the effectiveness of this method, the proposed approach is performed on the data of various scenarios of a large real electrical user in Singapore. Optimal demand contracting strategy under uncertainty is a complex problem unraveled by Feng et al. [
43]. They proved the convexity of the objective function under uncertainty and consequently that there exists one global optimal solution. To calculate the optimal contract value they adopted Newton–Rapson-based numerical method. In the absence of the wide adoption of the energy performance contracting, due to the reasons of its investment-assessing deficiency, high uncertainty, and the complexity profit allocation, Guo et al. [
44] applied real option analysis for finding the new metrics of the optimal scheme and incorporated contractual flexibility. The method real option analysis utilizing the binomial tree pricing model evaluates the investment value. It single out the feasible and attractive optimal scheme for the service company and the owner.
1.3. Contribution
This paper focuses on the electricity contract capacity considering multisources of energy for electric consumers such as industries, home residence and regions. Industries pay for different electric services, like connection, energy, demand, and reactive power consumption depending on how they use electricity as mentioned by Wang et al. [
34]. Most industrial and commercial customers, besides the energy that they consume, are required to pay for their peak demand, and the consumers’ contract capacity (CC) and the penalty of the excess peak demand over the contract capacity.
Energy suppliers need to know the capacity demand to plan the generation and the transmission of the energy for better service to the customers. Choosing an excessively high contract capacity may result in an unnecessary basic capacity charge, while choosing an excessively low contract capacity will impose high penalty charges.
For example, when the demand exceeds the upper or lower limits of the contract option, the double power price is charged on the excess quantity as introduced by Rodoplu et al. [
37]. In Taiwan, an excess within 10% of the contract demand is penalized at twice the rate of the contract demand, whereas the excess over 10% of the contract demand is charged at thrice the rate, and the contract demand can be changed by high-tension industrial customers each month as modeled by Chen and Liao [
35].
Multisources of energy contract capacity optimization adding an ecofriendly encouragement factor is introduced by Hamze et al. [
45]. In this work, discrete contract capacity of different types are chosen optimally taking into consideration the penalty price for the excess demand over the total combination of contract capacities. The model is solved using linear programming and the effect of the encouragement factor is examined on the total costs (TC) and the total renewable energy contract capacities used. This paper is an extension of our previous work, stochastic features are introduced to the demand of energy considering continuous contract capacities of multisources of energy.
Different types of energy contracts can be proposed by energy suppliers: traditional and renewable energy contracts (solar, wind, biomass …) as done by Direct Energy electric company in France [
46]. The government encourages the use of renewable energy sources by adding bonuses for the renewable energy contracts because of the pollution and for social reasons, it also discourages the use of traditional energy through taxing policy. It is necessary for industries to find the optimal combination of contracts to increase the percentage of green energy used for marketing purposes at the first hand, and to satisfy their peak demand with minimum costs. Moreover, industries them-self support the use of renewable energy contracts. So in general, this support of renewable energy contracts and discouragement of the use of traditional energy contracts is described in this paper by
. The objective is to form an optimal mix between the renewable energy energy contract capacity and the traditional non-renewable contract capacity. The more is the percentage of the renewable energy sources used, the more the contracts are considered ecofriendly.
The demand of energy changes from time to time depending on the user and many other factors, so the demand of energy is subjected to uncertainty. This paper addresses a nonlinear model to optimize the energy capacity contracting strategy under demand uncertainty while considering traditional and renewable energy sources as a new concept. This model identifies optimum combination of different types of contract capacities taking into account different factors such as the maximum demand, penalties of over consumption and the ecofriendly price for encouraging the use of renewable energies. The influence of the different values of the ecofriendly price, the penalty prices, and uncertainty on the optimal solution are studied in the presence of the different types of contract capacities. Since the penalty price is previously defined by the producer and the uncertainty is uncontrollable, this approach gives an idea about how much the ecofriendly support should be in the presence of different penalty prices and uncertainties.
The remainder of the paper is organized as follows. In
Section 2, the problem statement and the mathematical formulation are described.
Section 3 explains the interior point algorithm used in the optimization approach. In
Section 4, the model is tested several numerical examples inspired from based on real data.
Section 5, summarizes the contribution of this work and introduces some perspectives.