Enhancing Peak Shaving through Nonlinear Incentive-Based Demand Response: A Consumer-Centric Utility Optimization Approach

. Tis paper presents a comparative study on the implementation of incentive-based demand response programs in power grid management, comparing the results obtained from linear and nonlinear models. Te challenges faced by power grids in balancing supply and demand and managing price spikes during peak periods are addressed, and demand response programs are proposed as an efective strategy. Te focus is on implementing incentive-based programs where load serving entities determine the optimal incentive price and demand reduction. A novel approach is presented for simulating consumer behavior based on the price elasticity of demand and consumers’ utility function, incorporating both linear and nonlinear economic models. Te calculation of demand reduction aims to maximize consumers’ welfare, while the determination of the optimal incentive price maximizes the proft of the load serving entities. Real data are utilized, and the proposed models are implemented using the mixed-integer nonlinear programming (MINLP) method. Te results demonstrate the efectiveness of providing incentives to consumers during peak hours compared to other approaches, with a comparison between the linear and nonlinear models.


Introduction
1.1.Motivation and Incitement.Te smart grid has become an important part of modern society and the economy by increasing the interconnection of electricity with other sectors, including transportation, communication, and distributed energy resources (DERs) [1].Te integration of DERs has revolutionized the energy infrastructure, ofering a sustainable and cost-efective solution to replace fossil fuel-based generation and enabling decentralized networks [2].One critical distributed resource is demand response programs (DRPs), which balance supply and demand without changing power generation.DRPs incentivize changes in electricity usage patterns, reducing operating costs, greenhouse gas emissions, improving grid efciency, and defending against cyberattacks such as false data injection (FDIA) [3][4][5][6].
With two-way communication, smart meters, and home energy management systems (HEMS), optimal demand-side planning and management are considered suitable solutions for implementing DRPs [3].By exchanging data with consumers, power companies can implement DRPs, reducing the need for additional generation and transmission capacity expansion [8].Such programs can also help facilitate retail choice and promote competition, ultimately leading to lower energy costs and prices that more closely refect those of a free market [9][10][11].
However, implementing DRPs requires investment from various stakeholders.Consumers bear costs such as installing smart meters, telecommunications infrastructure, peak load control, HEMS, and investing in renewable energy.Tey are also responsible for operation and maintenance costs [6].Meanwhile, system operators or service providers should pay for training consumers and telecommunications infrastructure on the control side, among others [12].
In summary, the shift towards DERs and DRPs ofers limitless potential for reshaping the energy landscape and creating a sustainable energy system that is environmentally friendly and economically viable.However, it requires investment from various stakeholders, and careful planning and management are necessary to ensure efcient implementation.
Generally, DRPs are divided into two basic categories: price-based demand response programs (PBDRPs) and incentive-based demand response programs (IBDRPs).PBDRP ofers the same price rate over a time period for all consumers with diferent consumption levels, which is one of its disadvantages [13,14].
As a fundamental part, IBDRPs play a signifcant role in improving network performance, system reliability, and cost management.Tese programs aim to reduce consumer demand in response to price spikes, emergencies that threaten grid reliability, or critical peak hours [13,14].Unlike the PBPRP, these programs provide more fexibility for electric utilities to access responsive resources in addition to paying incentives to consumers instead of paying electricity bills at diferent prices during the day [14].Moreover, these programs are more attractive to consumers than PBDRPs because people are motivated to accept an incentive program to receive more encouragement [15].However, IBDRPs account for about 93% of peak demand reduction for available resources in the United States [16].
Today, in a competitive electricity market, the load serving entity (LSE) plays a critical role in establishing a link between wholesale markets and retail consumers to connect to an optimal operating framework.Teir purpose in establishing this connection is to maximize their proft based on the uncertainties such as the electricity price and the consumer's demand [17].Te LSE participates in wholesale markets such as the real-time market and buys electricity at variable wholesale prices for retail consumers such as residential and small business consumers.Ten, retail price signals are sent to consumers' smart meters through the advanced metering infrastructure (AMI) [18].After receiving the price and establishing two-way communication between HEMS and the smart meter, HEMS controls the demand consumption of controllable appliances during peak hours by moving, disconnecting, or reducing the demand for these devices.Tis causes consumers' electric utilities cost reduction [19].
Tese programs must be optimally implemented to take full advantage of the potential of IBDRPs.Terefore, the problems in this feld can be designed as an optimization of IBDRP implementation by the LSE.

Literature Review.
It is evident that IBDRPs play a fundamental role in balancing electricity supply and demand.Te implementation of these programs has been shown to ofer benefts such as reducing greenhouse gas emissions, preventing or postponing costs associated with investment in the generation, transmission, and distribution sectors, and reducing consumer bill payments.However, there are challenges in optimizing IBDR, such as minimizing peak demand during critical periods while maximizing the proft of the LSE.Various researchers have proposed different IBDR models with diferent features and goals, including three-level hierarchical models, consumption management contracts, and interruptible load program models.Although some of these models have been criticized due to their assumptions or potential negative consequences, they hold promising potential in mitigating the challenges faced by the electricity market today, such as renewable energy uncertainty and price spikes.
Yu and Hong propose a three-level hierarchical model for IBDR that includes a system operator, aggregator, and consumer level.Te model incentivizes the aggregator to implement IBDR on consumers by ofering them a payment.Once motivated, the aggregator acts as an intermediary and reduces consumers' demand to maximize their own proft from exchanging DR with the system operator while minimizing consumers' incentive payments [20].Pratik and Debapriya discuss new approach to managing energy in microgrids.Te approach involves using both an IBDRP and a reconfguration method to minimize costs and maximize profts for the microgrid operator.However, the reduction of energy demand resulting from DR programs can cause discomfort for consumers, which is modeled in terms of cost [21].Additionally, Yu et al. introduce an IBDRP that allows consumers to reduce their demand in exchange for incentives that can be used as capacity resources in the wholesale market [22].Consumers are incentivized to reduce their demand only if the aggregator calls upon them; otherwise, they face heavy penalties.Asadinejad et al. designed the IBDR model to address the negative impact of high price fuctuations on consumer satisfaction, LSE proft, and overall stress on the power system.Te model calculates the optimal incentive price to encourage consumers to change their consumption patterns to prevent market price fuctuations.Tis model assumes a fxed incentive price of $10/MWh higher than the fxed retail price and a linear relationship between the reduction in load and the incentive price [23].Asadinejad and Tomsovic aimed to maximize LSE proft and designed an optimal DR plan using TOU and IBDR based on price elasticity to reduce the optimal demand during times where wholesale price increases above retail price.In the IBDR section, they consider a linear relationship between load reduction and incentive price [16].Chai et al. present an IBDR model that maximizes retailers' profts during price spikes and peak consumption time by considering the consumer utility function and demand price elasticity.Te model determines the optimal demand reduction by solving the consumer proft optimization model and fxing a certain incentive price.Te optimal incentive price is then calculated by using the result of optimal demand reduction in the retailer's proft function and examining the efect of the incentive price on the retailer's proft [11].Wang et al. proposed a two-level hierarchical model that uses controllable thermal loads, such as heating and air conditioning systems, as fexible sources for IBDR.

2
International Transactions on Electrical Energy Systems Te system operator determines the incentive received by the aggregator for reducing the load.Te aggregator then maximizes its proft from the revenue received from the system operator and the consumer's payment [24].incentive-based demand response, enabling companies to determine optimal bidding strategies, including prices and capacities, to achieve maximum proft and mitigate energy price volatility [29].Gul and Suchitra presented a novel incentive-based demand response scheme.Te scheme involves a service provider acting as a price maker and selling capacity resources obtained from consumers through demand reductions to the wholesale market.Consumers provide demand reductions in exchange for incentives.Te interaction between the service provider and consumers is formulated as optimization problems solved using GAMS with nonlinear programming.Te approach focuses on reducing demand during peak periods when energy prices are high [30].Rana et al. focused on the efcient management of electricity grids through demand response (DR) programs, specifcally in the context of a community microgrid (CMG).Te study considers an incentive-based DR model where an aggregator provides fexibility to the CMG.Te objective is to minimize the cost of fexibility management, which includes incentives for shifting demands and penalty payments for contractual violations.To achieve this, a two-stage optimization approach is adopted, utilizing a biobjective formulation followed by a single objective formulation [31].Fotouhi Ghazvini et al. aimed to maximize the daily proft of the LSE while minimizing peak demand to avoid electricity purchases during market risk times [32].To achieve this, the LSE calculated the optimal hourly incentive price for residential consumers based on generation capacity.However, the efect of distribution system operators on designing the optimal incentive price was not considered in their study.Currently, the electricity market faces two signifcant challenges: the entry of expensive power plants into the grid during peak hours and renewable energy uncertainty, which can cause wholesale prices to soar above retail prices.Tis price spikes poses fnancial risks to both LSEs and consumers and threatens the power grid's reliability.To mitigate these issues, Vu et al.
proposed an IBDR model to minimize LSEs' fnancial losses during peak hours [33].Te IBDRP incentivizes consumers to reduce their energy consumption during peak hours below their baseline demand response level by ofering incentives.Te model utilizes a linear approach to optimize the problem and accounts for limited consumer response related to the IBDR implementation.
Numerous studies have been performed on diferent models of IBDR to determine the optimal incentive price and load reduction, as well as the impact of these programs on consumer behavior and LSE proft.Te novelty of this paper lies in proposing nonlinear economic voluntary incentive models that consider the efect of the incentive price paid on electricity consumption to maximize the welfare of consumers participating in IBDR.In most cited papers such as [34], the relationship between incentive price and optimal load reduction is assumed linear.However, residential electricity consumption is variable and dependent upon consumer behavior, which plays a signifcant role in energy consumption [35].Terefore, understanding consumer behavior and determining the appropriate incentive price to encourage consumers to participate in the program is International Transactions on Electrical Energy Systems particularly essential [36].By utilizing these models, it is possible to fnd the best balance between incentivizing consumers to participate and maximizing the LSE's profts.Overall, this paper presents a novel approach by considering consumer behavior to develop an efective incentive model for IBDR programs that can both maximize consumer welfare and LSE profts.

Contribution and Paper
Organization.Te goal of this study is to propose an efective incentive pricing scheme for managing residential and small business sector electricity consumption in response to price spikes, emergencies, or peak demand, while ensuring the welfare of consumers.
Te contributions of this paper are outlined below in a bulleted format.
(i) A nonlinear IBDR model is extracted from consumers' utility functions based on the efect of consumer behavior on electricity consumption (ii) An optimization algorithm is developed to minimize consumer costs and maximize utility profts, ensuring a mutually benefcial outcome (iii) Variable optimal incentives are determined based on factors such as the wholesale price, system operator price, and the level of IBDR implementation (iv) Emphasis is placed on the coordination between the utility and system operator to ensure the reliability of the system (v) Finally, this scheme provides a structured framework for retailers to select the most appropriate regions for IBDR implementation, resulting in better decision-making and optimized resource allocation Tese contributions put forth an efective and holistic approach to managing residential and small business sector electricity consumption, which can beneft both the utility company and its consumers.
Tis paper is divided into fve main sections as follows.Section 2 presents nonlinear economic load models based on utility functions.In Section 3, the LSE proft function, considering nonlinear models, is demonstrated.Simulation results are presented and discussed in Section 4. Finally, conclusion points are drawn in Section 5.

Demand Response in the IBDR Model
Te problem and studied system are presented in this section.Tis paper aims for reducing consumer demand by developing the optimal incentive price in critical times.In the power system, each consumer has independent behavior, and their energy demand depends on various parameters, including their priorities, weather conditions, and electricity prices.For this reason, a proper understanding of consumer behavior and how consumers respond can help to develop DR programs [32,33].
Consumer utility functions are used to model the impact of consumers' behavior on the implementation of the program.According to this function, linear and nonlinear economic voluntary incentive models were extracted as a function of consumer behavior to determine the efect of the incentive price on electricity consumption.Based on these derived economic models, voluntary incentive-based programs are proposed to maximize consumer welfare.Ten, these models are used to fnd the optimal incentive price and load reduction by maximizing the LSE proft.
Four demand functions are illustrated in Table 1.Te table provides the formula of some demand functions in terms of D and P (price and demand) for certain coefcients such as a and b.
A simple and low-accuracy method of analyzing consumer behavior and utility functions is the IBDR linear model.However, optimization problems in the power system and related constraints tend to be nonlinear.It is necessary to extract nonlinear models and compare their results to the linear model for the accuracy and efciency of the IBDR model improvement.Due to this, exponential, logarithmic, and power nonlinear models are used.As a result, these functions are used more often because their formulas are simple to understand [37].

Deriving Exponential IBDR Model.
In normal circumstances, consumers purchase their electricity at retail prices P(t) from the LSE.According to the utility function U(D(t), w) in Figure 1, the following equation is used to calculate consumers' proft S(D(t), w) [38]:

S(D(t), w) � U(D(t), w) − P(t) × D(t).
( By implementing IBDR during peak periods and price spikes, consumers can reduce their consumption ∆D(t) and receive rewards P inc (t).Terefore, the beneft function can be written as follows [33]: To efectively model consumer behavior towards IBDR, maximizing consumer beneft is the key.Tis involves analyzing the multivariable function S(D(t), w), where D (t) represents electricity demand and w denotes consumer's willingness to their consumption.Te ideal value of D (t) that maximizes consumer beneft is determined by calculating the sensitivity of function S to changes in D (t), while keeping other variables constant.Tis involves taking partial derivatives of the function with respect to D (t) to assess their impact on consumer beneft.Te analysis also considers the partial derivatives of the utility function and electricity demand, evaluating their collective infuence on the overall change.By deriving the consumers' beneft function from D (t) and considering these factors, we gain a comprehensive understanding of consumer behavior in relation to IBDR [38].
4 International Transactions on Electrical Energy Systems Ten, equation ( 4) is written as equation ( 5) after derivation and sorting in accordance with equation (2) [38].
Various nonlinear functions are used to model consumer behavior, such as logarithmic, exponential, or quadratic functions.Te Taylor series expansion is used for consumers' proft maximization [39].Moreover, before the implementation of IBDR, the proft function is as follows [38]: Te price elasticity of demand E(t) shows the sensitivity of demand to price changes as shown following relationship [38]: Based on the exponential demand function of Table 1 and the second-order Taylor series expansion, the exponential model is written in the following form [37]: Deriving from equation ( 8), we have the following equation: In order to extract the exponential model of the incentive-based program, we put equation (5) in equation ( 9) and sort the formula, assuming the price remains constant before and after IBDR implementation (P 0 (t) and P(t)) and equals to P retail (t), as indicated by equation (11); the exponential IBDR model is obtained as follows: 2.2.Deriving Logarithmic IBDR Model.Based on the assumptions related to the consumers' proft function and the demand function of Table 1, the logarithmic model can be represented as follows [37]: Derived from equations ( 13) and ( 14), the following equation is obtained [37]: According to the same process as in the previous model, the following equation is obtained: Utilizing the exponential function of the demand function and the work process as described in the previous two models, we achieve the following relations [37]: zU(D(t), w) zD(t) � P 0 (t) For small demand elasticities and using assumption equation (10), the fnal power IBDR model is given by the following equation: Figure 2 compares linear, exponential, logarithmic, and power models based on the changes in elasticity from 0 to −0.3.Te power model experiences fewer load reductions than the others.Load reduction rates in the four models become similar at low elasticity values but diverge as elasticity increases.Also, the diference between the four curves is small since the Taylor expansion omits the second term due to the low elasticity in the four models.

Load Serving Entities' Profit
Usually, the wholesale price P wholesale (t) is lower than the retail price.Terefore, the LSE income is derived from the diference between the two prices.However, there are times when unexpected factors such as high demand (demand D 0 is more than supply D s ) or price spikes prevent the LSE from providing electricity to its consumers, resulting in outage costs.In this regard, the LSE cooperates with the system operator to implement IBDR and receives incentive price In the case of D 0 � D S , the following proft function is obtained after IBDR implementation [11]: max profit �  24 t�1 D 0 (t) − ∆D(t)  × P retail (t) − P wholesale (t)  + ∆D(t) × P dso (t) − P inc (t) Te fowchart of the proposed IBDR scheme is shown in Figure 3.In Figure 3, steps 1 and 2 are designed to derive a nonlinear IBDR scheme based on customer proft maximization, while steps 3 and 4 are presented to determine the optimum incentive price and load reduction by using MINLP method aimed at maximizing the LSE benefts during times of high demand or price spikes.Te MINLP method helps identify the optimal combination of incentive price and load reduction to achieve this goal.

. Simulation and Results
Tis paper focuses on the DPL area of the PJM market in the mid-eastern Atlantic region of the United States, where residential customers account for 46% of electricity consumption.Te selected bus for analysis is BETHANY with identifcation number 49865, which is a load type with a voltage level of 69 kV and located in Bethany Beach (zip code 19930).Tis paper presents price and demand data International Transactions on Electrical Energy Systems from a peak summer day in July 2020, with a particular emphasis on residential energy consumption (Figure 4) [40,41].Furthermore, it is essential to note that the considered demand is an aggregate of consumers' loads.Several LSEs in this area ofer fxed-rate electricity plans for both the short and long term.Terefore, the estimated retail price is 69.9 $/MWh [42].In addition, the day is divided into three periods: midpeak, of-peak, and peak period.Terefore, the price elasticities of diferent types of demand are calculated for each of these periods (see Table 2) [39].
Some studies such as [16,39] have examined the potential impact of implementing IBDR on reducing overall energy load.Tese studies suggest that such implementation could result in load reductions ranging from 10% to 40% compared to the baseline load.
To determine which devices to prioritize during power outages, consumer devices are usually categorized into four main groups: critical, interruptible, noninterruptible, and shiftable to of-peak and midpeak times.Further analysis can be conducted based on the operating time of these devices: (1) Long operation period (3 hours or more): Tis category includes appliances such as refrigerators, TVs, computers, sound systems, lighting, and water heaters (2) Medium operation period (between 1 and 3 hours): Appliances falling into this group typically include ovens, irons, vacuum cleaners, and dishwashers (3) Short operating period (maximum 1 hour): Washing machines and electric kettles are examples of devices belonging to this group By strategically disconnecting and transferring loads from devices such as washing machines, dryers, dishwashers, irons, vacuum cleaners, and pool pumps, and by reducing loads from devices such as water heaters and air conditioners, it becomes possible to achieve a signifcant load reduction of up to 30%.
In summary, the studies indicate that implementing IBDR can lead to substantial load reductions by categorizing devices based on their operating time and selectively managing their power consumption during periods of high demand or power outages.
Furthermore, GAMS software has been used to determine the optimal incentive price which is lower than the wholesale price and the amount of load reduction based on the LSE's objective function.
Eslaminia and Mashhadi [34] assumed a linear relationship between incentive price and load reduction.While linear modeling is easy to analyze, it lacks accuracy especially when dealing with nonlinear functions such as optimization in power systems and related constraints.
To improve the accuracy and efciency of the model and its implementation on a real system, it is necessary to use nonlinear models.Terefore, this section presents the results of nonlinear modeling and compares them with those obtained from the linear model presented in paper [34].By doing so, we aim to enhance the accuracy of the model and make it more suitable for practical applications.
Overall, this research highlights the importance of accurately modeling incentive-based programs for load reduction in power systems.Te use of nonlinear models can signifcantly improve the accuracy of these programs and help ensure their efectiveness in practice.In order to validate the superiority of our proposed model, we have compared our calculations with another state-of-the-art existing scheme.Specifcally, we compared our nonlinear modeling results with the linear model presented in paper [34].Our results show that the nonlinear model outperforms the linear model in terms of accuracy and efciency.Terefore, our proposed nonlinear model can be considered as a better option for practical applications.

Case 1: Not Receiving Incentives from the System Operator.
In the frst case, the analysis compares the results of the base case and of the linear and nonlinear models without considering the efect of the incentive price paid by the system operator.Te focus is on the sensitivity of consumers to IBDRPs.Te fndings indicate that without incentives from the operator, there is a low load reduction percentage observed, suggesting low consumer sensitivity to IBDRPs.In this case, the assumption is made that the initial demand (D 0 ) is equal to the supply generated by power plants (D s ).Tis implies that at the beginning of the observed period or scenario, the total electricity demand matches the total supply available from power generation sources.Te optimal incentive price and the demand reduction amount were then calculated assuming that the maximum demand reduction was equal to 15% of the initial demand.
During peak times, supplying consumers becomes a signifcant challenge.Terefore, demand reduction is of major importance for the system operator during these times.As a result, the highest increase in proft and decrease in cost were observed in the linear model (Table 3).Additionally, the maximum load reduction occurred at 18:00 for all models, with the linear model achieving the greatest load reduction (the most optimistic state) while the power model experienced the least reduction (the most pessimistic or conservative model).

Case 2:
Receiving Incentives from the System Operator.Tis case examines the impact of receiving incentives from the system operator on the proftability of the LSE and the costs borne by the company.High electricity prices during peak hours result in increased costs for the LSE, including outage costs.Te LSE must purchase electricity at a higher price from the grid and sell it to consumers at a retail price.Additionally, the LSE incurs outage costs due to supply-demand imbalances, resulting in power cuts and an average payment of $1,000/MW for outage power.Consequently, the LSE's proft signifcantly declines, leading to substantial losses for the company.However, implementing IBDRPs with operator incentives can help address these challenges by reducing demand, preventing outages, and improving the LSE's proftability.Te supply is assumed to be 15% lower than the initial demand from 12 to 8 pm.Hence, the LSE must cut the power and pay an average of $1,000/MW for outage power.After paying the outage cost, Table 4 shows that the LSE's proft has plummeted signifcantly, and the company burdens a huge loss.
In this case, it is assumed that the initial demand, denoted as D 0 (t), is equal to the supply generated by power plants, represented by D s .Terefore, implementing IBDRPs can meet consumers' demand and prevent power outages and outage costs.It is assumed that the incentive price paid by the system operator is 1.5 times the wholesale market price, and the maximum load reduction is equal to 15% of the initial load.According to Figures 5 and 6, the incentives given by the operator increase the incentive price and reduce demand, proving the accuracy of the proposed models.Figures 5 and 6 validate the proposed models by demonstrating the impact of operator incentives on incentive price and demand.Te fgures show that ofering incentives increases the incentive price and decreases demand.During peak hours, load reduction remains constant across all models due to maximum constraints, limiting further demand reductions.Additionally, the introduction of incentives from the operator signifcantly afects the LSE's proft and consumers' costs, with the LSE's proft increasing.Overall, the optimization models successfully maximize proft (Table 5).

Case 3: Increasing the Maximum Load Reduction Limit.
In Case 3, increasing the maximum allowable load reduction limit has been performed to investigate the potential for additional demand reduction and its efects.Te analysis shows that raising the maximum load reduction results in higher incentive prices and lower demand.It also emphasizes the connection between incentive prices, consumer satisfaction, and demand reduction.Terefore, the maximum load reduction is set at a fxed percentage of 30% of the initial load, with D s being equal to D 0 .Additionally, the system operator pays an incentive price assumed to be 1.5 times the wholesale price.

International Transactions on Electrical Energy Systems
It can be seen from Figures 7 and 8 that reducing the maximum allowable load to 30% of the initial load resulted in a higher incentive price and a lower demand.In Table 6, the percentage of acceptable load reduction per model varies at 6 pm, depending on the demand reduction.Moreover, demand can be reduced by up to 30% to ensure consumer satisfaction.Based on the relationship between Figures 1 and  9, it can be inferred that as consumers reduce their load, their level of satisfaction and utility function decreases.Tis function refects their satisfaction based on their electricity consumption.It is essential to observe Figure 9 to understand that higher incentive prices received by consumers during peak times, such as 6 A.M., result in greater demand reduction through the implementation of IBDRP.In other words, the higher the incentive price, the more willing consumers are to reduce their demand.Furthermore, curves for all the models overlap in the frst 5% of the load reduction, but after that, they diverge, and their trends difer.In the long run, as consumption is reduced, people will show more resistance and require higher incentive prices.It is essential to note that the higher LSE proft observed in the linear model does not indicate its superiority over nonlinear models.Te linear model assumes a linear relationship between incentives and consumption reduction, which is not realistic.In contrast, nonlinear models capture complexities and individual preferences better.Logarithmic and power models provide a better ft than linear models.Te focus of this paper is on the practical applicability of nonlinear models in understanding incentivized energy consumption dynamics, rather than comparing LSE profts based on Table 6 alone.Using these models enhances decision-making processes in this feld.

Case 4: Te Proper Incentive Price Paid by the System
Operator (P dso ) for Implementation.Te analysis focuses on determining the optimal P dso and its impact on demand reduction.Diferent values of P dso were considered, including 1.5 times, 1.75 times, and 2.0 times the wholesale price.Te fndings show that increasing P dso leads to a greater reduction in demand.However, it is important to strike a balance as excessively high incentive prices may result in unnecessary demand reduction.Based on the analysis, an appropriate incentivizing price of 1.75 times the wholesale price is suggested for achieving the desired maximum allowable load reduction of 30%.Te graph illustrates the load reduction resulting from diferent values of P dso in both linear and power models, comparing them to the 30% threshold in relation to the initial demand.
During of-peak hours, such as 9 AM, 11 AM, and 9 PM, both the linear and power models experienced a decrease in demand when the P dso was increased by 2.0 times.However, this reduction may not always be necessary and may result in reduced demand that is not needed.Considering this, the appropriate incentivizing price may be 1.75 times the wholesale price, as it reduces the demand more appropriately than the other two prices.
In summary, Figure 10 shows how diferent values of P dso afect demand reduction and provides insights into the optimal incentive price for achieving a maximum allowable load reduction of 30%.

Conclusions
In conclusion, this paper provides a comprehensive analysis of incentive-based demand response programs in power grid management, comparing the results obtained from linear and nonlinear models.Te study utilizes real data from the hottest day of the year and considers both linear and nonlinear models to determine optimal incentives and load reductions.Te fndings demonstrate that providing incentives to consumers during peak hours is a more efective approach compared to alternative strategies such as implementing blackouts and paying outage costs.In the frst case, where no incentives were ofered, the maximum load reduction resulted in a decrease of approximately 6% in the power model's load.However, in the second case, where incentives were provided, it became evident that incentivizing consumers during peak hours aligns with the interests of both the LSE and consumers.Furthermore, the nonlinear model employed in this study yielded additional insights compared to the linear model, highlighting its value in understanding and optimizing demand response programs.Te results indicate that both P dso and the maximum load reduction have positive efects on consumers' load reductions and incremental increases in the incentives provided (P inc ).Beyond the fnancial benefts of increased company profts and reduced consumer billing costs, the proposed method also addresses the issue of consumption during peak hours and price spikes.By encouraging load reduction during these periods, the proposed approach contributes to a more sustainable and efcient energy system.Overall, this research enhances our understanding of demand response programs in power grid management and provides valuable insights for system operators and policymakers.By considering the efectiveness of incentives and comparing the results between linear and nonlinear models, this study ofers practical guidance for implementing successful demand response initiatives that beneft both energy providers and consumers alike.

Figure 2 :
Figure 2: Comparison of 4 models based on elasticity change.

Figure 4 :
Figure 4: Total demand and P wholesale .

Figure 9 :
Figure 9: Te relationship between the incentive price and the load reduction.

Figure 10 :
Figure 10: Load reduction changes in the linear and power model due to the P dso changes.

Table 1 :
Linear or nonlinear demand functions.

Table 3 :
Te LSE proft and consumers' cost in case 1.

Table 4 :
Comparison of LSE proft before and after charging outage costs.

Table 5 :
Te LSE proft and consumers' cost in case 2.

Table 6 :
Te LSE proft and consumers' cost in case 3.