In most countries, on one hand, the growing population of the world and on the other hand, the rise in electricity consumption per capita have contributed to the management of the demand side being more significant. DR is regarded as one of the most impressive methods of demand-side management, where the peak of consumers will be shifted from high-price hours to low-price hours (Abapour et al., 2020). For implementing innovative DR programs, tariffs for electricity purchases must be designed to maximize the economic interest and to alter the pattern of consumption. This can be achieved by introducing incentive mechanisms. The first one is to decrease the customer demand by attempting to offer rewards to the customer during peak hours. The second means is to avoid the occurrence of supply–demand imbalance as far as possible (Ajoulabadi et al., 2020).

The DR programs are classified based on the control mechanism, offered motivations, and decision variable which was clearly described by Rajendhar and Belwin Edward (2019). The programs discussed here like the flat rate(FR), real-time pricing (RTP), day-ahead pricing (DAP), and peak time rebate (PTR) programs fall under time-based DR programs. The earlier studies explored the elasticity of residential loads, that is, the potential elasticity of demand and reduction of demand with regard to various prices (H. A. Aalami et al., 2019). Furthermore, there are some major variables that have not been completely discussed. To be addressed: most of the previous market elasticity studies are focused on price-based DR, such as the possible reduction in demand with a dynamic pricing scheme (Hosseini et al., 2019; Monfared, Ghasemi et al., 2019). There is, however, little work performed to target incentive-based DR′s (IBDR) elasticity, namely, the relationship between the reduction of overall peak demand and financial incentives.

Based on the DR, Jiang et al. (2019) proposed an efficient real-time pricing model. In order to express the relationship between price and energy consumption dynamics, price elasticity is used. RTP can offer many advantages as a measure of demand-side management, such as peak shaving, generation deferral, and network investment, promoting the introduction of renewable energy. The authors introduced a new model in Sharifi et al. (2018) for DR programs for economic models, based on the methodology of time-of-use. The essential advantages of the model were demonstrated based on price-elasticity. In a smart grid system, the real-time electricity pricing mechanism is applied compared to the flat rate pricing mechanism. It results in monetary savings due to the reduction of electricity bills and to involve the consumer directly in the reduction of peak loads by reducing the grid burden and encouraging the interaction of renewable sources with the grid (Dhundia, 2016).

Lu and Hong (2019) proposed a new real-time incentive-based demand response algorithm for reinforcement learning and deep neural network smart grid systems. It intended to help the service provider to buy energy resources from its subscribing customers to manage energy volatility and strengthen the grid. In particular, deep neural networks are used to forecast unpredictable prices and demands for energy in order to solve the potential uncertainties. Reinforcement learning is then implemented to achieve the optimum reward rates for various clients, taking into account the income of both service providers and clients. Chai et al. (2019) suggested an incentive-based demand response model to optimize the benefits of electricity retailers. The breakthrough is that, given their diverse activities during both peak and valley times, the models provide utility and elasticity of different customers. The optimum reduction of clients with a certain reward price can be achieved by solving the customer benefit optimization model at peak times. The variance of customers can be measured with a certain reward price, according to the elasticity during the valley times. Then, the optimum incentive price can be found based on the suggested DR model by evaluating the sensitivity of incentive prices to retailer benefits.

From utility‘s viewpoint, IBDR has significant prospects for power system peak demand control. Shi et al. (2020) incorporated a systematic approach by integrating the technological model and the social–behavioral survey to determine IBDR ability. The outcome validates the strategy suggested and acts as guidance for IBDR initiatives for utilities.

The Pecan Street research network, which was established in 2009, is the world’s only real power, gas, and water test bed. More than 1,000 houses, 250 solar homes, and 65 electric vehicle users have contributed data to it. Every home’s energy production and consumption are tracked continuously, 24/7/365, and can be dissected down to the circuit level. These high-resolution data shed light on the production, usage, and storage of energy (Residential data page of Pecan street data port, 2022).

Exelon Corporation (NASDAQ: EXC), a Fortune 100 energy corporation with around 10 million customers for electricity and natural gas, is based in Chicago and owns ComEd. More than 4 million customers, or 70% of the state’s population, in northern Illinois rely on ComEd for power. The service area of ComEd is essentially bordered to the south by Iroquois County (about Interstate 80), to the north by Wisconsin, to the west by Iowa, and to the east by Indiana (ComEd’s Hourly Pricing, 2011).

The objectives of the article are as follows:

1. To develop a smart grid simulation setup to implement a dynamic DR program with real-time data on load, generation, pricing signals, and seasons;

The novelties of the article are as follows:

1. To propose a dynamic DR program based on price elasticity that integrates with the loss aversion characteristics;

Risk aversion is the behavior of human beings (in particular, investors and consumers) in finance and economics, who strive to minimize uncertainty, if they are subjected to uncertainty. It is the reluctance of a person to participate in an unpredictable payoff scenario rather than a more predictable payoff but maybe a lower expected payoff scenario. (For example, a risk-averse investor may prefer to invest their money in a lower but guaranteed bank rate instead of a stock that may have a high return rate, which also involves an opportunity to lose value) (Mohajeryami, Schwarz, & Baboli, 2015).

The fact that losses and disadvantages have a greater effect on perceptions than benefits and advantages is a well-established behavioral fact. This results in a feature of utility that is steeper for losses than for gains. A traditional presumption that assumes a symmetry between the gain and loss value simplifies the study of an individual decision. However, it is not practical. It can, therefore, lead to over- or under-estimated assumptions. In Figure 5, a clear illustration of a value function, that can describe a broad range of outcomes, is shown. In riskless and risky situations, loss aversion is distinct. In risky situations, the value function of loss aversion is clearly steeper. The riskless scenarios include both real-time pricing and peak time discounts.

During peak hours, the utilities charge more in real-time pricing (RTP), so any load that can be shifted to peak hours appears to the consumer as a loss relative to the price of off-peak times. On the other hand, based on their load reduction, the peak time rebate (PTR) relies on rewarding the clients at the peak time. Any load that can be shifted to off-peak times is also equivalent to a gain. Their perceived value, therefore, belongs to the two opposite sides of the value function, although they are the same method, namely, shifting the flexible loads from peak to off-peak times.

The price-elasticity curve is shown in Figure 6. When the price decreases, the demand for almost all goods and services increases. This demand shift is not linear. The non-linear demand curve can be linearized around the particular time to quantify this effect. It is regarded as the demand elasticity of the price. In other words, market elasticity is a standardized indicator of the shift in demand because price increases. E = P 0 d 0 . ∂ d ∂ p (12)

Here, E = demand elasticity; d = the value of demand; P = the price of electricity; and P 0 and d 0 are the initial price of electricity and the demand value, respectively. If electricity prices differ over different periods, the following reactions to demand are

(i) Some of the loads (e.g., illuminating loads) are not capable of moving from one time to the next and can only be on/off. Thus, these loads have just a single period of sensitivity and it is called “self-elasticity,” and it often has a negative value: E x x = ∆ d x ∆ p x ≤ 0 . (13)

(ii) Such periods of consumption can be changed from high to off-peak or low periods. Such behavior is referred to as sensitivity for several periods, and “cross-elasticity” is measured. This value is positive at all times.

In this article, we will model the DR program depending on price elasticity and articulate how RTP, DAP, and PTR programs affect the prices and demands of electricity and how these programs can achieve the maximum customer benefit.

Assume that

d(i) is the consumer’s demand; p(i) is the electricity spot price; I(i) is the incentive ($/kWh); and R (d(i)) is the revenue of the consumer. Also suppose that, depending on the price chosen for the incentive I(i)), the consumer changes his/her own demand from d₀(i) (original value) to d(i). Thus, due to running PTR, P ($) would be as P (∆d(i)) = I(i).∆d(i) (H. Aalami et al., 2008).

Assume that the consumer’s benefit for the ith period is as shown in B d i = R d i − d i . p i + P ∆ d i $ , (15) P ∆ d i = λ . I B i . ∆ d i , (16) where IB(i) is an incentive bonus, and in this scenario, this is a reward paid on every peak decrease in kWh. λ, It is a coefficient that represents the actual value of the incentive or reward’s nominal payment. It is believed that each person is trying to maximize their benefit. So, ∂ B d i ∂ d i = ∂ R d i ∂ d i − p i + ∂ P ∆ d i ∂ d i = 0 . (17)

Consequently, ∂ R d i ∂ d i = p i + λ . I B i . (18) Marginal utility is equivalent to the energy price at the optimum value. According to H. Aalami et al. (2008), for quadratic customer revenue function, the Taylor series expansion is used. So, R d i = R d 0 i + ∂ R d 0 i ∂ d i × ∆ d i + 1 2 × ∂ 2 R d 0 i ∂ d 2 i × ∆ d i 2 , (19) where d(i) represents a change from d₀(i) (optimal point) in client demand. The following relation must be maintained if the initial demand is optimum prior to the introduction of the DR program. ∂ B 0 ∂ d i = ∂ R d 0 i ∂ d i − p 0 = 0 , (20) ∂ R d 0 i ∂ d i = p 0 . (21)

Using (18) and the concept of demand price elasticity (12), ∂ 2 R d i ∂ d 2 i = ∂ p ∂ d = 1 E × p 0 d 0 . (22) Plugging (21) and (22) into the extension to the Taylor series gives R d i = R d 0 i + p 0 . ∆ d i + 1 2 . 1 E i . p 0 d 0 . ∆ d i 2 . (23)

It is easy to rewrite Eq. 23 as follows: R d i = R d 0 i + p 0 . ∆ d i 1 + ∆ d i 2 × E i × d 0 (24)

Expanding d(i) = d(i)-d₀(i) and afterward relating (24) to (18) offer p i + λ . I B i = p 0 i × 1 + d i − d 0 i E i × d 0 i , (25) p i + λ . I B i = p 0 i + p 0 i × d i − d 0 i E i × d 0 i . (26)

So, the consumption of the consumer can also be evaluated as follows: d i = d 0 i × 1 + E i × p i − p 0 i + λ . I B i p 0 i . (27) When I(i) is zero (i.e., no incentive award) in the aforementioned equation, d(i) would be equal to d₀(i). Consequently, the price of energy would not change and the elasticity of the price will be equal to 0.

The cross-elasticity between the periods i and j is described as follows: E i , j = P 0 j d 0 i × ∂ d i ∂ p j ,   i ≠ j , (28) E i , j ≤ 0 , i f   i = j , E i , j ≥ 0 , i f   i ≠ j .

In (28), we conclude that ∂ d i ∂ ρ j is constant. The demand response to price variance could, therefore, be characterized as a linear function. Here, 15 min is known as a one-time slot. The demand response model for 24 (96-time slots) hours a day can be obtained by incorporating self- and cross-elasticity of demand as follows: d i = d 0 i + E i × d 0 i p 0 i × p i − p 0 i + λ . I B i + ∑ j = 1 j ≠ i 96 E i , j × d 0 i p 0 j × p j − p 0 j + λ . I B j ,   i = 1,2 … 96 . (29)

The aforementioned equation shows how high the customer’s consumption should be in order to reach maximum profits in 24 h. The change in demand in Eq. 29 comes from two sources. The source is self-elasticity, expressed in the first term, and cross-elasticity, expressed in the second term, is the other source. Both price- and reward-sensitive demands are expressed in the model. In order to achieve the maximum profit in 24 h, the aforementioned equation indicates how high the consumption of the consumer should be. In the following section, we will illustrate how incentives could change the demand curve through executing FR, RTP, DAP, and PTR programs in the numerical results segment. Load profiles of the consumers are now pre-classified as low consumption, off peak and peak categories based on the different bands of power consumption and the time duration where it was obtained. The categorization of load profile is shown in Table 1. The self and cross elasticity values are fixed based on the literature and by examining with various set of values to obtain the desired outcomes. The pre-fixed self and cross elasticity values are shown in Table 2.

The impact of the pricing scheme on the total cost, total demand, and peak demand for all seasons has been clearly analyzed here. Initially, the analysis has been done for 5th of February. Corresponding results are tabulated in Table 3. In contrast to flat-rate pricing, the RTP scheme does not show any effect on parameters such as overall cost, total demand, and peak demand here. However, with a loss of utility income of –30% without any impact on the total demand and peak demand, the DAP scheme will have a negative impact. Compared to other pricing schemes, peak demand was reduced in the case of PTR_1,2,3 (–13.06 percent to –26.13 percent). A fair amount of peak demand is reduced in PTR₁, that is., -26 percent with a loss of -15 percent in utility profits. The potential to minimize the peak demand to half (-26.13 percent to -13.06 percent) and half of the utility income (15.26 percent to -7.63 percent) can be saved relative to PTR₁ by considering the loss aversion (λ = 0.5) in PTR₂. In conclusion, PTR₂ has a moderate impact as opposed to PTR₁, both in terms of peak reduction and total cost. The doubling of the cross-elasticity values in PTR₃ as opposed to PTR₂ indicates no effect in terms of peak reduction. However, it shows a good impact in maintaining utility in profits (-7.63% to -1.2%) and also by improving the overall carrying capacity of the demand (2.53 percent to 8.96 percent).

In the case of PTR₄, the peak demand was drastically reduced, that is, -50 percent with a huge compromise of -64.6 percent in the total utility income, which is an adverse effect and not advisable. Of all PTR conditions, PTR₃ is the best solution for peak reduction (-13 percent) without compromising on utility revenue (-1.2 percent) with load capacity improvement (9 percent) as a result of the shifting load to non-peak hours and low-price hours. Figure 7 shows the demand curve of all pricing schemes for the month of February. Figures 8, 9 shows the bar chart of all parameters and all pricing schemes for the month of February and May respectively. Figure 8 shows the bar chart of all parameters and all pricing schemes for the month of February.

Then, the analysis has been done for 5th of May, August and November. Corresponding results are tabulated in Tables 4–6 respectively. If utility income is only the criterion, DAP is considered the best in May. According to the Pecan Street report (Rate Structure Information for Time-of-Use Pricing, 2011; McCracken and George, 2014), the explanation for low income in May is due to the very low night-time rate that applies during the five windiest months (March–May and November–December). When the key concern is peak demand reduction, then PTR₁ is the best option. However, because of the compromise in utility revenue (36.2 percent), it is not advisable. As far as all parameters are concerned, then PTR₃ is the best choice for May with all the parameters in a moderate range. In PTR_3, with a compromise of 9.5% utility revenue, 13.09% peak demand can be reduced with increased loading capability by 9.7%. In August and November, the trend of all the parameters remains the same. PTR₃ is the optimal solution by keeping all parameters in concern.

The percentage change in total cost, total demand, and peak demand is shown in bar charts for all pricing schemes in Figures 10–12, respectively. Percentage change in total cost: utility revenue has dropped in the range of 3% to 6 percent for RTP pricing in all seasons, and PTR₁ for utility revenue or total cost of all seasons has dropped in the range of 15% to 40 percent. Due to λ ′s variation from 1 to 0, the total cost of PTR₂ is half of PTR₁. A range of 1 percent decreased to 12 percent for PTR₃. Percentage change in total demand: the total demand reduction is in the range of 1 to 4 percent for RTP pricing in all seasons. The percent increase in the total demand range is 0 to +5 percent in PTR₁. In PTR₂, the percentage increase in the total demand range is 0 to +2%, that is, the exact half of PTR₁. Percentage change in peak demand: the percentage peak decrease range for RTP is 0 to –8 percent. The percentage peak reduction range for PTR₁ is 26 percent. The percentage peak reduction range for PTR₂ is exactly half of PTR₁ at –13 percent. The percentage peak reduction range for PTR₃ is exactly the same as PTR₂ at 13 percent.

PTR₁ has a positive effect on the reduction of peak demand (-26%) and the rise in total demand (0–5%). However, the reduction in utility income also had a major effect (15 to 40 percent). The credit for PTR_2,3 is half the credit for PTR₁ in all seasons. The cause is that λ = 0.5 is taken into account in PTR_2,3, and in the demand response equation, reward 0 is replaced by 0.5*reward0. The value of λ is also expressed in the value of the credit. At the λ = 1 condition, peak demand is drastically reduced in all the seasons. Peak demand is decreased and halved at λ = 0.5 compared to λ = 1; peak demand comparison at different conditions of λ = 0(FR), 0.5(PTR₂), and 1(PTR₁) is shown in Figure 13.

The total demand for February and May is between 440 and 660 units. Also, the total cost is between $11 and $24 with average flat rate values of 0.0564 and 0.03. The total demand surged by almost 80% in August and November, that is, in the range of 1,050 to 1180 kwh, and cost ranges from 33 $to 50$.With the same range of the total demand, the total cost of November is almost half of that of August because the RTP price signal variation is based on the season and nature of the day. For the total demand range of ≈440units in February and ≈1,097 units in November, it charges almost in the same range of ≈24 to 30 $.

The FEB is reviewing this situation. The corresponding changes in parameters due to doubling of rebate are tabulated in Table 7. The flat rate is 0.0564 for the FEB. The incentive is now boosted to 0.2820 (i.e., 5.2 times the flat rate) during peak hours. Peak demand is decreased by 49 percent relative to the flat rate. Also, the total demand is raised by 9%, and the utility’s total revenue is decreased by 64%. Compared to PTR₁, just by changing the rebate value, much change has been observed in the percentage change in peak demand and total cost. However, it is not at all suggested as it is happening at the loss of utility revenue.

The equal loss scenario criterion is the one that evaluates how much peak reduction can be accomplished with the same amount of economic loss. In the peak reduction scenario, utilities are interested in peak reduction, regardless of the cost. The month of May is considered for analyzing these scenarios. For equal loss scenario, total demand and peak demand responses are shown in Figure 14. The implications of equal loss scenario and peak reduction scenario on the parameters are listed out in Tables 8, 9 respectively.

To retain equivalent financial loss, i.e., –6.34 percent, the PTR₁ rebate is reduced to 0.0343 from $0.07845 per kWh. PTR₁ can achieve a –11.4 percent peak reduction in PTR₁ without taking into account the loss aversion scenario (i.e., λ = 1), compared to a –8.1 percent peak reduction in RTP. In the same way, a –7.91 percent reduction compared to RTP is accomplished by considering loss aversion in PTR₂. In terms of peak reduction, PTR₃ shows the same effect as PTR₁, i.e., –11.51 percent. The added advantage with PTR₃ is the improvement in total demand serving capability of 8.5% and peak reduction of –11.4% is attained by considering loss aversion and doubling the cross elasticity constants.

Compared to RTP, the corresponding peak reduction of 8.11 percent in PTR is achieved at λ = 0.31. RTP is responsible for utility loss of 6.34 percent for the same peak load reduction, while 11.24 percent in PTR₁. Compared to RTP (4.26 percent), the total demand serving capacity is better in PTR₁ (+0.7 percent). The selection of either RTP or PTR₁ is solely dependent on the priority of the cost, else load demand serving capacity for the same peak reduction.

For the same value of rebate, change in loss aversion value effects the parameters that are shown in Table 10. For the same reward value, a decrease in peak demand is caused by λ approaching from 0 to 1. The value range is from 313.4 to 251.4 for May and 562.5 to 440.5 for August. At the same time, this allows the total demand to rise. For May, its value ranges from 579.4 to 619.6 and for August, it ranges from 1,068.3 to 1,071.3. In conclusion, the pattern of a significant reduction in peak demand and a rise in total demand is observed as λ approaches 0 to 1.

For the same value of reward and λ values, but elasticity constants are doubled. Then the corresponding implications on the parameters are shown in Table 11. The total demand is significantly increased through the doubling of cross elasticity values. Their values change from 612 to 664 for May. The range is 1,073–1,149 for August. The peak demand remains unchanged for May (296–296) and August (518–518). Therefore, the peak demand has been shown to remain unaltered, and overall demand has risen by 7–8.4 percent.