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To decarbonize the heating sector, residential consumers may install heat pumps. Coupled with heating loads with high thermal inertia, these thermostatically controlled loads may provide a significant source of demand side flexibility. Since the capacity of residential consumers is typically insufficient to take part in the day- ahead electricity market (DAM), aggregators act as mediators that monetize the flexibility of these loads through demand response (DR). In this paper, we study the strategic interactions between an aggregator, its consumers and the DAM using a bilevel optimization framework. The aggregator-consumer interaction is captured either as a Stackelberg or a Nash Bargaining Game, leveraging chance-constrained programming to model limited controllability of residential DR loads. The aggregator takes strategic positions in the DAM, considering the uncertainty on the market outcome, represented as a stochastic Stackelberg Game. Results show that the DR provider-aggregator cooperation may yield significant monetary benefits. The aggregator cost-effectively manages the uncertainty on the DAM outcome and the limited controllability of its consumers. The presented methodology may be used to assess the value of DR in a deregulated power system or may be directly integrated in the daily routine of DR aggregators.

A transition from generation on demand to consumption on demand is one of the solutions to overcome the many limitations associated with the higher penetration of renewable energy sources. Such a transition, however, requires a considerable amount of load flexibility in the demand side. Demand response (DR) programs can reveal and utilize this demand flexibility by enabling the participation of a large number of grid-interactive efficient buildings (GEB). Existing approaches on DR require significant modelling or training efforts, are computationally expensive, and do not guarantee the satisfaction of end users. To address these limitations, this paper proposes a scalable hierarchical model-free transactional control approach that incorporates elements of virtual battery, game theory, and model-free control (MFC) mechanisms. The proposed approach separates the control mechanism into upper and lower levels. The MFC modulates the flexible GEB in the lower level with guaranteed thermal comfort of end users, in response to the optimal pricing and power signals determined in the upper level using a Stackelberg game integrated with aggregate virtual battery constraints. Additionally, the usage of MFC necessitates less burdensome computational and communication requirements, thus, it is easily deployable even on small embedded devices. The effectiveness of this approach is demonstrated through a large-scale case study with 10,000 heterogenous GEB. The results show that the proposed approach can achieve peak load reduction and profit maximization for the distribution system operator, as well as cost reduction for end users while maintaining their comfort.

Due to their heat/cool storage characteristics, thermostatically controlled loads (TCLs) play an important role in demand response programmers. However, modeling of the heat/cool storage characteristic of large numbers of TCLs is not simple. In this paper, the heat exchange power is adopted to calculate the power instead of the average power, and the relationship between the heat exchange power and energy storage is considered to develop an equivalent storage model, based on which the time-varying power constraints and the energy storage constraints are developed, to establish the overall day-ahead scheduling model. Finally, the proposed scheduling method is verified using the simulation results of a six-bus system.

Ancillary services rely on operating reserves to support an uninterrupted electricity supply that meets demand. One of the hidden reserves of the grid is in thermostatically controlled loads. To efficiently exploit these reserves, a new realization of control of voltage in the allowable range to follow the set power reference is proposed. The proposed approach is based on the deep reinforcement learning (RL) algorithm. Double DQN is utilized because of the proven state-of-the-art level of performance in complex control tasks, native handling of continuous environment state variables, and model-free application of the trained DDQN to the real grid. To evaluate the deep RL control performance, the proposed method was compared with a classic proportional control of the voltage change according to the power reference setup. The solution was validated in setups with a different number of thermostatically controlled loads (TCLs) in a feeder to show its generalization capabilities. In this article, the particularities of deep reinforcement learning application in the power system domain are discussed along with the results achieved by such an RL-powered demand response solution. The tuning of hyperparameters for the RL algorithm was performed to achieve the best performance of the double deep Q-network (DDQN) algorithm. In particular, the influence of a learning rate, a target network update step, network hidden layer size, batch size, and replay buffer size were assessed. The achieved performance is roughly two times better than the competing approach of optimal control selection within the considered time interval of the simulation. The decrease in deviation of the actual power consumption from the reference power profile is demonstrated. The benefit in costs is estimated for the presented voltage control-based ancillary service to show the potential impact.

Thermostatically controlled loads such as refrigerators are exceptionally suitable as a flexible demand resource. This paper derives a decentralised load control algorithm for refrigerators. It is adapted from an existing continuous time control approach, with the aim to achieve low computational complexity and an ability to handle discrete time steps of variable length -- desirable features for embedding in appliances and high-throughput simulations. Simulation results of large populations of heterogeneous appliances illustrate the accurate aggregate control of power consumption and high computational efficiency. Tracking accuracy is quantified as a function of population size and time step size, and correlations in the tracking error are investigated. The controller is shown to be robust to errors in model specification and to sudden perturbations in the form of random refrigerator door openings., Comment: arXiv admin note: substantial text overlap with arXiv:1904.12401

This paper proposes a framework for controlling grid frequency by engaging the generation-side and demand-side resources simultaneously, via a fast transactive control approach. First, we use a proportional frequency-price relation to build and analyze a transactive frequency droop controller for a single-area power grid. Then, we develop a transactive demand response system by incorporating a large population of thermostatically controlled air conditioning loads. A proportional-integral controller is used to adjust the setpoint temperature of the air conditioners based on price variations. A battery storage system is then developed and augmented to the system to capture the energy arbitrage effects. A nonlinear price-responsive battery management system is developed to enable effective charging and discharging operations within the battery’s state-of-charge and power constraints. Simulation results indicate that the proposed transactive control system improves the steady-state and transient response of the grid to sudden perturbations in the supply and demand equilibrium. To decouple frequency from price during daily operation and maintain frequency near the nominal value, we propose adding a feedforward price broadcast signal to the control loop based on the net demand measurement. Through various simulations, we conclude that a combination of feedback transactive controller with feedforward price broadcast scheme provides an effective solution for the simultaneous generation-side and demand-side energy management and frequency control in smart power grids.

Due to the increased use of variable renewable energy sources, more capacity for reserves is required. Non-generating resources such as thermostatically controlled loads (TCLs) can arbitrage energy prices and provide reserves due to their thermal energy storage capacity. The quantity of reserves depends not only on the aggregate power capacity, but also on information and communication technology, exogenous parameters, and system operator requirements. Specifically, the practical limitations origin from (i) communication constraints, (ii) ambient temperature, and (iii) the dispatch time of the activation signal. This paper explores the impact of these parameters on the amount of reserves that an aggregator of TCLs can provide to the system operator based on centralized control of a TCL population. We propose a decision support tool that can be used by aggregators to decide on maximum dispatchable reserve bids. The method can accommodate the specific control algorithm and TCL population of an aggregator and is based on offline computation. It constitutes a powerful reserve bid library to be used when optimization tools become computationally intractable due to the increased number of decentralized flexible loads. QC 20201216

This paper proposes a hierarchical control of Thermostatically Controlled Loads (TCLs) to provide primary frequency control support. The control architecture is comprised of three levels. At the high level, an aggregator coordinates multiple distribution substations and dispatches the primary reserve references. At the middle level, distribution substations estimate the available power of TCLs based on the aggregated bin model, and dispatch control signals to individual TCLs. At the local level, a supplementary frequency control loop is implemented at the local controller, which makes TCLs respond to the frequency event autonomously. Case studies show that the proposed controller can efficiently respond to frequency events and fulfill the requirement specified by the system operator. The users’ comforts are not compromised and the short cycling of TCLs is largely reduced. Due to the autonomous control, the communication requirement is minimized.

A fundamental requirement of the electric power system is to maintain a continuous and instantaneous balance between generation and load. The intermittency and uncertainty introduced by renewable energy generation requires the expansion of ancillary power system services to maintain such a balance. In this paper, we examine the potential of thermostatically controlled loads (TCLs), such as refrigerators and electric water heaters, to provide generation following services in real-time energy markets (1 to 5 minutes). To simulate the non-linear dynamics of hysteretic dead-band systems in a manner suitable for convex optimization, we introduce an alternative control trajectory representation of the TCLs and their discrete input signals. To perform distributed optimization across large populations of TCLs, we propose a variation of the alternating direction method of multipliers (ADMM) algorithm. Based on our simulation results, we have demonstrated the potential for controlling a population of TCLs within an error tolerance of 10 kW.

For thermostatically controlled loads (TCLs) to perform demand response services in real-time markets, online methods for parameter estimation are needed. As the physical characteristics of a TCL change (e.g. the contents of a refrigerator or the occupancy of a conditioned room), it is necessary to update the parameters of the TCL model. Otherwise, the TCL will be incapable of accurately predicting its potential energy demand, thereby decreasing the reliability of a TCL aggregation to perform demand response. In this paper, we investigate the potential of various unscented Kalman filter (UKF) algorithm variations to recursively identify a TCL model that is non-linear in the parameters. Experimental results demonstrate the parameter estimation of two residential refrigerators. Finally, simulation results demonstrate the incorporation of the recursive parameter estimation methods into a model predictive controller for demand response.

In this paper, a fuzzy neural network controller for regulating demand-side thermostatically controlled loads (TCLs) is designed with the aim of stabilizing the frequency of the smart grid. Specifically, the balance between power supply and demand is achieved by tracking the automatic generation control (AGC) signal in an electric power system. The particle swarm optimization (PSO) and error back propagation (BP) algorithms are used to optimize the control parameters and consequently reduce the tracking errors. The fuzzy neural network can be applied to solve load control problems in power systems, since its self-learning and associative storage functions can deal with the highly nonlinear relationship between input and output. Simulation results show the advantage of the fuzzy neural network control scheme in terms of frequency regulation error and consumer comfort.

Thermostatically controlled loads (TCLs) are promising to offer demand-side regulation with proper control. In this paper, the aggregate power of TCLs is used to track the automatic generation control (AGC) signal by changing the temperature setpoint. The dynamics of the indoor temperature are described by a Monte Carlo model, and population dissatisfaction is described by the predicted percentage of dissatisfied (PPD). The objective is optimization from two aspects, minimizing both population dissatisfaction and tracking error. We propose an improved active target particle swarm optimization (APSO) algorithm to optimize the model, making it possible to ensure that the user&rsquo, s dissatisfaction is as small as possible while the aggregate power tracks the AGC signal. The novelty of this paper is to introduce PPD into the model and at the same time establish three models using PPD as the objective function and constraints. The simulation results are shown to verify the efficiency of the designed model.

This paper studies the wind power regulation problem by controlling a population of thermostatically controlled loads (TCLs) in smart grids. A decentralised bi-level control approach is proposed, where the first-level controller achieves load following based on wind power forecast, while the second-level controller deals with the forecast error effectively using the frequency deviation information. Specifically, to realise the fair dispatch of wind power in a users’ comfort sense, the wind power following problem is formulated as a quadratic optimisation problem, which is solved by the first-level controller. The second-level adopts a local proportional controller to handle the frequency deviation caused by the forecast errors of wind power. The proposed algorithm converges fast and is decentralised in the sense that the TCLs conduct local computation and keep the parameters’ privacy from the aggregator. Simulations are given to show the performance of the proposed approach.

International audience; This paper presents an approach of demand response (DR) scheduling with thermostatically controlled loads (TCL) in a distribution grid with high penetration of distributed generations (DG). In this approach, household TCL are employed as flexibility resources to support the distribution network for mitigating voltage or congestion constraints. A two-stage rolling optimization based control scheme is proposed to determine the optimal operating status of flexible loads using the forecast of generation and demand in the distribution system. The proposed methodology is conducted in a distribution test feeder with realistic scenarios. The simulation results have shown the usefulness and efficiency of the proposed method in improving the network operation and increasing the hosting capacity of DG.

The improvement of intelligent appliances provides the basis for the demand response (DR) of residential loads. Thermostatically controlled loads (TCLs) are one of the most important DR resources and are characterized by a large load and a high degree of control. Due to its distribution characteristic, the aggregation of TCLs and their control are key issues in implementing the load control for the DR. In this study, we focus on air conditioning loads as an example of TCLs and propose a simple and transferable aggregate model by establishing a virtual house model, which accurately captures the aggregate flexibility. The deviation of the aggregate model is analyzed for the model evaluation. An air conditioning DR control scheme is proposed based on the aggregate model, it has the advantage of simple implementation and convenient control for the individual units. Simulations are performed in Gridlab-D to evaluate the accuracy and effectiveness of the proposed model and control method.

Thermostatically controlled loads such as refrigerators are exceptionally suitable as a flexible demand resource. This paper derives a decentralised load control algorithm for refrigerators. It is adapted from an existing continuous time control approach, with the aim to achieve low computational complexity and an ability to handle discrete time steps of variable length -- desirable features for embedding in appliances and high-throughput simulations. Simulation results of large populations of heterogeneous appliances illustrate the accurate aggregate control of power consumption and high computational efficiency.

Thermostatically controlled loads (TCLs) can effectively support network operation through their intrinsic flexibility and play a pivotal role in delivering cost effective decarbonization. This paper proposes a scalable distributed solution for the operation of large populations of TCLs providing frequency response and performing energy arbitrage. Each TCL is described as a price-responsive rational agent that participates in an integrated energy/frequency response market and schedules its operation in order to minimize its energy costs and maximize the revenues from frequency response provision. A mean field game formulation is used to implement a compact description of the interactions between typical power system characteristics and TCLs flexibility properties. In order to accommodate the heterogeneity of the thermostatic loads into the mean field equations, the whole population of TCLs is clustered into smaller subsets of devices with similar properties, using k-means clustering techniques. This framework is applied to a multi-area power system to study the impact of network congestions and of spatial variation of flexible resources in grids with large penetration of renewable generation sources. Numerical simulations on relevant case studies allow to explicitly quantify the effect of these factors on the value of TCLs flexibility and on the overall efficiency of the power system.

In this paper, we present a flexibility estimation mechanism for buildings&rsquo, thermostatically controlled loads (TCLs) to enable the distribution level consumption of the majority of solar photovoltaic (PV) generation by local building TCLs. The local consumption of PV generation provides several advantages to the grid operation as well as the consumers, such as reducing the stress on the distribution network, minimizing voltage fluctuations and two-way power flows in the distribution network, and reducing the required battery storage capacity for PV integration. This would result in increasing the solar PV generation penetration levels. The aims of this study are twofold. First, spectral (frequency) analyses of solar PV power generation together with the power consumption of multiple building TCLs (such as heating, ventilation, and air conditioning (HVAC) systems, water heaters, and refrigerators) are performed. These analyses define the bandwidth over which these TCLs can operate and also describe the PV generation frequency bandwidth. Such spectral analyses, in frequency domain, can help identify the flexible components of PV generation that can be consumed by the various TCLs through optimal building load utilization. Second, a quadratic optimization problem based on model predictive control is formulated to allow consuming most of the low and medium frequency content of the PV power locally by building TCLs, while maintaining occupants&rsquo, comfort and TCLs&rsquo, physical constraints. The solution to the proposed optimization problem is achieved using optimal control strategies. Numerical results show that most of the low and medium frequency content of the PV generation can be consumed locally by building TCLs. The remaining high-frequency content of the PV generation can then be stored/offset using energy storage systems.

In this paper, we study the demand response of the thermostatically controlled loads (TCLs) to control their set-point temperatures by considering the tradeoff between the electricity payment and TCL user&rsquo, s comfort preference. Based upon the dynamics of the TCLs, we set up the relationship between the set-point temperature and the energy demand. Then, we define a discomfort function with respect to the associated energy demand which represents the discomfort level of the set-point temperature. More specifically, the system is equipped with a coordinator named electric energy control center (EECC) which can buy energy resources from the electricity market and sell them to TCL users. Due to the interaction between EECC and TCL users, we formulate the specific energy trading process as a one-leader multiple-follower Stackelberg game. As the main contributions of this work, we show the existence and uniqueness of the equilibrium for the underlying Stackelberg games, and develop a DR algorithm based on the so-called Backward Induction to achieve the equilibrium. Several numerical simulations are presented to verify the developed results in this work.

Demand Response (DR) is a Smart Grid technology aiming to provide demand regulation for dynamic pricing and ancillary services to the grid. Thermostatically controlled loads (TCLs) are among those with the highest potential for DR. Some of the challenges in modelling TCLs is the various factors that affect their duty cycle, mainly human behaviour and external conditions, as well as heterogeneity of TCLs (load parameters). These add an element of stochasticity, with detrimental impact on the aggregated level. Most models developed so far use Wiener processes to represent this behaviour, which in aggregated models, such as those based on Coupled Fokker-Planck Equations (CFPE), have a negligible effect as “white noise”. One of the main challenges is modelling the effect of external factors on the state of TCLs' aggregated population and their impact in heterogeneity during operation. Here we show the importance of those factors as well as their detrimental effect in heterogeneity using cold loads as a case study. A bottom up detailed model has been developed starting from thermal modelling to include these factors, real world data was used as input for realistic results. Based on those we found that the duty cycle of some TCLs in the population can change significantly and thus the state of the TCLs' population as a whole. Subsequently, the accuracy of aggregation models assuming relative homogeneity and based on small stochasticity (i.e. Wiener process with typical variance 0.01) is questionable. We anticipate similar realistic models to be used for real world applications and aggregation methods based on them, especially for cold loads and similar TCLs, where external factors and heterogeneity in time are significant. DR control frameworks for TCLs should also be designed with that behaviour in mind and the developed bottom up model can be used to evaluate their accuracy.

Thermal loads are recognized as a valuable source of flexibility in face of the increasing variability caused by the large shares of renewable production. Lockout constraints can significantly reduce the flexibility of thermostatically controlled loads (TCLs). We propose a novel way of modifying the loads' lockout durations to achieve non-intrusive centralized control without relying on local computations and estimations. We derive analytical expressions for the flexibility reduction and validate them via simulations, which show that the proposed method describes the TCLs flexibility accurately. We further show that a simple stochastic centralized controller, which does not rely on local temperature measurements, outperforms the commonly used priority-stack controller in terms of system robustness against infeasible trajectories.

Due to increased use of variable renewable energy sources, more capacity for balancing and ancillary services (AS) is required. Non-generating resources such as thermostatically controlled loads (TCLs) can arbitrage energy prices and provide AS due to their thermal energy storage capacity. This paper explores the impact of energy/AS price notice time, i.e. the time between when the price is announced and when it takes effect, on the TCL energy consumption and AS capacity bids, and quantifies trade-offs between notice time and flexibility. We first optimize the energy consumption and AS capacity offers at a given notice time, varied from 24 hours ahead to real-time. We then introduce uncertainty in TCL availability, formulate the stochastic optimization problem, and evaluate how the trade-offs change. We find that price notice time impacts TCL profits, but does not significantly affect the total AS capacity offered over the day. However, AS capacity offers are impacted by uncertainty, which is likely to increase with notice time. QC 20180717. QC 20191203

This paper focuses on the optimal intraday scheduling of a distribution system that includes renewable energy (RE) generation, energy storage systems (ESSs), and thermostatically controlled loads (TCLs). This system also provides time-of-use pricing to customers. Unlike previous studies, this study attempts to examine how to optimize the allocation of electric energy and to improve the equilibrium of the load curve. Accordingly, we propose a concept of load equilibrium entropy to quantify the overall equilibrium of the load curve and reflect the allocation optimization of electric energy. Based on this entropy, we built a novel multi-objective optimal dispatching model to minimize the operational cost and maximize the load curve equilibrium. To aggregate TCLs into the optimization objective, we introduced the concept of a virtual power plant (VPP) and proposed a calculation method for VPP operating characteristics based on the equivalent thermal parameter model and the state-queue control method. The Particle Swarm Optimization algorithm was employed to solve the optimization problems. The simulation results illustrated that the proposed dispatching model can achieve cost reductions of system operations, peak load curtailment, and efficiency improvements, and also verified that the load equilibrium entropy can be used as a novel index of load characteristics.

This paper proposes a model of the thermal dynamics and of the end-use of domestic refrigerators (fridges) and water heaters (boilers) in the foretasted scenario of the Sardinian electric network in 2020. This model is employed to evaluate the potential variations of power demand of the aggregates of fridges and boilers during one year in the considered scenario. The resulting quantities can be considered as a form of power reserves to be used for contributing to the frequency regulation through a proper demand side response strategy. The particular case of the system for frequency control proposed by the European Network of Transmission System Operators for Electricity (ENTSO-E) is then analysed by simulations in order to show advantages and drawbacks.

There is an increasing interest in exploiting the flexibility of loads to provide ancillary services to the grid. In this paper we study how response delays and lockout constraints affect the controllability of an aggregation of refrigerators offering primary frequency control (PFC). First we examine the effect of delays in PFC provision from an aggregation of refrigerators, using a two-area power system. We propose a framework to systematically address frequency measurement and response delays and we determine safe values for the total delays via simulations. We introduce a controllability index to evaluate PFC provision under lockout constraints of refrigerators compressors. We conduct extensive simulations to study the effects of measurement delay, ramping times, lockout durations and rotational inertia on the controllability of the aggregation and system stability. Finally, we discuss solutions for offering reliable PFC provision from thermostatically controlled loads under lockout constraints and we propose a supervisory control to enhance the robustness of their controllers., Journal of Modern Power Systems and Clean Energy, 5 (1), ISSN:2196-5420

With the proportion of air conditioners increasing gradually, they can provide a certain amount of frequency-controlled reserves for a micro-grid. Optimizing utilization of air conditioners and considering load response characteristics and customer comfort, the frequency adjustment model is a quadratic function model between the trigger temperature of the air conditioner compressor, and frequency variation is provided, which can be used to regulate the trigger temperature of the air conditioner when the micro-grid frequency rises and falls. This frequency adjustment model combines a primary frequency modulation method and a secondary frequency modulation method of the energy storage system, in order to optimize the frequency of a micro-grid. The simulation results show that the frequency modulation strategy for air conditioners can effectively improve the frequency modulation ability of air conditioners and frequency modulation effects of a micro-grid in coordination with an energy storage system.

Constructing microgrids with renewable energy systems could be one feasible solution to increase the penetration of renewable energy. With proper control of the battery energy storage system (BESS) and thermostatically controlled loads (TCLs) in such microgrids, the variable and intermittent energy can be smoothed and utilized without the interference of the main power grid. In this paper, a decentralized control strategy for a microgrid consisting of a distributed generator (DG), a battery energy storage system, a solar photovoltaic (PV) system and thermostatically controlled loads is proposed. The control objective is to maintain the desired temperature in local buildings at a minimum cost. Decentralized control algorithm involving variable structure controller and dynamic programming is used to determine suitable control inputs of the distributed generator and the battery energy storage system. The model predictive control approach is utilized for long-term operation with predicted data on solar power and outdoor temperature updated at each control step.

This paper discusses the possibility of providing voltage control-based ancillary services using thermostatically controlled loads (TCLs). The idea is to change the voltage at the point of common coupling in order to control power consumption of the TCLs through time. This allows to utilize the most common type of loads in the grid to provide ancillary services. These services differ from usual demand response strategies that operate on short to medium time scales. In this paper the authors propose a controller that regulate the power consumption by minimizing the power consumption error signal and by taking physical restrictions into account. Voltage restrictions arise due to the small window of voltage variability about the nominal value of the controlled voltage. In addition, capacitor bank switching interactions with motor and thermostatically controlled loads is discussed. QC 20161213 EU funded FP7 iTesla project

Due to increasing shares of renewable energy sources, more frequency reserves are required to maintain power system stability. In this paper, we present a decentralized control scheme that allows a large aggregation of refrigerators to provide Primary Frequency Control (PFC) reserves to the grid based on local frequency measurements and without communication. The control is based on stochastic switching of refrigerators depending on the frequency deviation. We develop methods to account for typical lockout constraints of compressors and increased power consumption during the startup phase. In addition, we propose a procedure to dynamically reset the thermostat temperature limits in order to provide reliable PFC reserves, as well as a corrective temperature feedback loop to build robustness to biased frequency deviations. Furthermore, we introduce an additional randomization layer in the controller to account for thermostat resolution limitations, and finally, we modify the control design to account for refrigerator door openings. Extensive simulations with actual frequency signal data and with different aggregation sizes, load characteristics, and control parameters, demonstrate that the proposed controller outperforms a relevant state-of-the-art controller., Comment: 44 pages, 17 figures, 9 Tables, submitted to IEEE Transactions on Power Systems

In this paper, we study the demand response of the thermostatically controlled loads (TCLs) to control their set-point temperatures by considering the tradeoff between the electricity payment and TCL user’s comfort preference. Based upon the dynamics of the TCLs, we set up the relationship between the set-point temperature and the energy demand. Then, we define a discomfort function with respect to the associated energy demand which represents the discomfort level of the set-point temperature. More specifically, the system is equipped with a coordinator named electric energy control center (EECC) which can buy energy resources from the electricity market and sell them to TCL users. Due to the interaction between EECC and TCL users, we formulate the specific energy trading process as a one-leader multiple-follower Stackelberg game. As the main contributions of this work, we show the existence and uniqueness of the equilibrium for the underlying Stackelberg games, and develop a DR algorithm based on the so-called Backward Induction to achieve the equilibrium. Several numerical simulations are presented to verify the developed results in this work., Applied Sciences, 8 (8), ISSN:2076-3417