# Data-Driven Consumption Load Monitoring and Adjustment Strategy in Smart Grid

Authors: Bingjie He; Jinxiu Xiao; Qiaorong Dai
Journal: Journal of Mathematics
 (2021)

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## Abstract

The enhancement of the intelligent construction of the power grid and widespread popularity of smart meters enable large amounts of electrical energy consumption data to be collected and analyzed. Based on the data, the energy provider gives a guiding price in the future periods to users. It encourages users to be more economical and smarter in the process of using electricity. By applying the social welfare model to equate demand and supply in every time interval, we gain the optimal prices and generation capacity. Nevertheless, the truth is that there is a great gap between the consumers’ booked electrical energy consumption and the optimal generation capacity, causing the power system overload and even outage. This article puts forward a novel automatic process control strategy in order to monitor the gap between the consumers’ booked electrical energy consumption and optimal generation capacity by using statistical method to predict the future one. When the predicted value exceeds the boundary, the energy provider adopts the changeable electricity price to stimulate consumers to adjust their electrical energy demands so that it can have smoothly actual electrical energy consumption. Our adjustment method is data-driven exponential function-based adjustment. Case study results show that the strategy can obtain small adjustment times, stable actual consumption load, and controllable prediction errors. Different from the linear monitoring and adjustment strategy, our approach obtains almost the same adjustment frequency, less standard deviation of residuals, and higher total social welfare and energy provider profit.


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## Body

## 1. Introduction

With the development of urbanization, human beings’ material living quality has improved dramatically. However, some issues such as the environmental pollution also have emerged. In order to decrease the environmental pollution and avoid overconsumption of resources, words like peak carbon dioxide emissions and carbon neutrality have been hotly discussed. Mentioning energy consumption, human beings turn to some clean sustainable energy resources including hydropower and solar energy rather than restricting traditional coal fossil energy.The limitation of traditional power grid’s rigid construction, namely, the lack of flexibility for grid connection with new energy and the delays in the transmission of information due to the backward communication network and so on, may cause problems, for example, the supply-demand imbalance. Due to the defects of the previous generation of grid and the emergence of mobile communication technology, smart grid was put forward by IBM (in America) in 2006, “next generation power grid” [1]. Not only America and EU countries but also China has picked up some cities as pilot ones for SG (smart grid) [2]. Compared with traditional grid, SG has the following advantages: timely reliable two-way communication among data on the network, the supply-demand balance on account of information interaction, simple and convenient storage of the distribution energy for security of the microgrid connection benefiting from the development of high capacity battery technology, and highly efficient calculating ability generated from the creative model and smart algorithm.Smart meter develops rapidly with the gradually mature communication technology, which integrates the metering and data interaction functions of traditional one. So, users and energy providers can exchange data. Meanwhile, it also can analyze, forecast, and manage the consumption load. Equipped with advanced sensor technology and reliable terminal equipment, real-time pricing (RTP) is booming. Different from traditional pricing structures, reasonable RTP can keep supply and demand in balance and keep the consumers’ and suppliers’ comfort maximized, because it has the flexible and intelligent characteristics.The ultimate purpose of RTP research by domestic and foreign experts is to achieve maximized total social welfare [3, 4]. For this aim, a distributed dynamic pricing algorithm was developed to obtain peak-shaving and valley-filling [4]. Lately, this sort of RTP models has been advanced vigorously in terms of model improvement and stronger algorithm convergence. Chiu et al. [5] researched on an energy transaction billing system by using a dynamic pricing mechanism. Zhu et al. [6] got a better rate of convergence and a better operation effect by solving the model with ADMM algorithm.RTP highlights the stable and reliable theoretical pricing policy and optimal generation capacity, but it is out of line with reality. The original intention is to guide users to make rational use of electric energy through changing electricity prices and balance the smart grid. Nevertheless, the truth is that the majority of consumers are unwilling to adjust their electricity demand with the ever-changing electricity price every hour. That causes the practical electrical energy consumption to lose control again and leads to the loss of the smart grid stability and reliability. Even in extreme cases, the fact that the energy provider offers the booked consumption to the users may cause blackouts at peak time. To solve the blackout, the electricity companies and power plants have to face the increasing energy cost, which is far more than the revenues. That is the least thing that the energy provider wants to have. In order to prevent this, we should work out a solution based on the operation of RTP model. It can not only make the power system have a limited changing price through the automatic monitoring but also have a smooth and steady practical electrical energy consumption. In other words, the users’ practical electrical energy consumption is close to the optimal generation capacity from the RTP model. In the existing literature, there are many studies on how to manage the electrical energy consumption in smart grid. However, they rarely consider how to reduce the adjustment frequency of electricity price [7, 8].The automatic process control (APC) strategy can make up for this shortcoming. It can offer effective process monitoring and adjustment. Box [9] applied APC method to product control. We will make an adjustment when the process is beyond the boundary set before. In this way, the adjustment frequency will decrease and the production quality will often be controlled in a certain range. The APC strategy is widely used in product, manufacture, and service fields. Hernández et al. [10] put forward a control tool to monitor variables. Yuan et al. [11] studied an APC chart to identify exceptions. To monitor the gap between the booked and optimal electrical energy consumptions, He et al. [12] researched a line function-based APC strategy. However, the APC strategy has not been widely used in SG [13–15].A new data-driven exponential function-based APC strategy is proposed in this paper. We use exponentially weighted moving average (EWMA) to monitor electrical energy consumption. After obtaining the dynamic pricing from energy providers, the users can book one day or more of electricity in advance through the smart meters. At that time, the energy providers can monitor users’ booked consumption load and calculate the difference between the optimal generation capacity and it. Since then, it is the turn to use data-driven APC scheme to manage electrical energy consumption by changing the dynamic pricing. There is a long research history of the EWMA for scholars from home and abroad. Yang et al. [16] designed a Phase Two EWMA control model to monitor alterable dimension mean vector. In statistical applications, EWMA is often used to predict trends [17–19]. He et al. [20] studied an EWMA prediction model to monitor the process of electrical energy consumption. In this paper, EWMA is applied to predict the next interval gap between the optimal electrical energy generation capacity and the booked electrical energy consumption. When it exceeds the preset boundary, rising or reducing the price in some time interval is supposed to be adopted to stimulate the demand response. In this way, it can get a few adjustments and avoid the side effects on the users caused by the frequent price adjustments. The stable consumption load is finally achieved.The research features and highlights of this article are listed as follows:(1)
This study comes up with an original data-driven exponential function-based automatic process control strategy to manage the gap between the consumers’ booked electrical energy consumption and the optimal generation capacity(2)
A small adjustment number is obtained by the data-driven exponential function adjustment method, which can achieve a practical electrical energy consumption approaching the optimal generation capacity after adjustment(3)
This strategy can make up for the defects of the RTP algorithm and achieve effective peak carbon dioxide emissions effectsThe remaining part of this article is arranged as follows: data-driven APC strategy is offered in Section2. In Section 3, the algorithm is proposed. Case studies and result analysis are included in Section 4. The conclusions are drawn in Section 5.


## 2. Data-Driven APC (Automatic Process Control) Strategy

The structure of the SG system discussed in this article is as follows: a power plant, an energy provider, and a few users. The users have installed smart meters. The power plant transmits the power to an energy provider. The energy provider collects the power consumption data from users through smart meters. The energy providers apply the social welfare maximization model to calculate price of next time interval and transmit it to users. After receiving the price asp dollars/kWh, users reserve consumption load from the energy provider (one day or even one week). Set the number of consumers as N, and assume that the time period of electrical energy operation is divided into T intervals. Suppose that set ℵ=1,2,…,N represents consumers and set Γ=1,2,…,T represents time intervals. The energy provider obtains each user i’s i∈ℵ valley and peak electrical energy consumption data in interval t∈Γ according to the past data provided by the smart meter, namely, mit and Mit. Denote xit as user i’s electrical energy consumption in interval t, and its range can be assumed as mit≤xit≤Mit. The detailed social welfare model is available in Appendix.After we solve the optimization problem (C.1)–(C.3) (see Appendix), optimal price pt∗ and theoretical optimal generation capacity Gt∗ in interval t can be gained. The electricity supplier obtains a smooth and steady electrical energy consumption based on Gt∗. But this is just an optimal situation. Most often, consumers’ booked electrical energy consumption observed from smart meters is considerably different from optimal generation capacity Gt∗. Guiding the consumers to use electrical energy appropriately is the most effective way to prevent this kind of phenomenon.Taking users’ demand response mechanism to price into account, we calculate the gap between optimal generation capacity with the social welfare model and the booked consumption loads. Later, when it exceeds the boundary, we use the data-driven APC scheme to change the gap. The energy provider changes prices to make users adjust their actual consumption loads. In the end, the actual electrical energy consumption is near the optimal generation capacity. Moreover, we can obtain higher social welfare and the energy provider can get more profit with data-driven APC strategy than before. We first introduce the definition of the EWMA estimation [12, 20].


### 2.1. EWMA Estimation

We suppose that the users book the electrical energy consumption of the next interval, and the reservation retains an important reference value for accurate adjustment.In order to accurately obtain the extent of gap between booked electrical energy consumptionxt of users in time interval t and optimal generation capacity Gt∗, we set gap dt as(1)dt=xt−Gt∗,and we predict the next gap value dt+1 by the EWMA model from last adjusted gap value. The details of the calculation are as follows.Set the initial gap value asdτ,τ=t,t−1,…, and set the adjusted one as dτ′,τ=t,t−1,…. The EWMA d¯t+1 of gap value dt+1 in time interval t+1 is in the following formula:(2)d¯t+1=θd¯t+μdt′,in which θ=1−μ is the discount factor.Similarly, EWMAp¯t+1 of price pt+1 in interval t+1 is(3)p¯t+1=θp¯t+μpt′,in which pt′ is changed price in interval t.In the process of adjustment, the changed EWMA valuep¯t+1′ in interval t+1 is(4)p¯t+1′=θp¯t+μpt″,where pt″ is readjusted price in interval t.


### 2.2. Data-Driven APC Electrical Energy Monitoring

In this section, we discuss how to develop a data-driven APC electrical energy monitoring strategy in order to minimize the difference from the goal electrical energy gapE. We will change the price when the EWMA value is beyond the boundary as(5)d¯t+1≥B1ord¯t+1≤B2,B1≥0,B2≤0,where B1 is prestipulated upper limit and B2is prestipulated lower limit. In the process of monitoring, when d¯t+1 conforms to (5), the EWMA value d¯t+1 is out of the limits. The action of adjusting it to get nearer to the goal value will be taken. It is obvious in test results that, to achieve a stable subsequent adjustment, it is worthy of discussion to find a way to set parameters E1≥0 and E2≤0 of the target process properly.When monitoring users’ booked consumption load, we obtain a seriesd¯tt=1T of EWMA estimation. If d¯t+1 satisfies (5), the users’ booking electrical energy consumption has been beyond the steady limit. For preventing the consumers’ blind electricity utilization, the energy provider applies the users’ price demand response. It guides the users to use power properly, which achieves smooth and steady electrical energy consumption.


### 2.3. Data-Driven APC Adjustment

If the automatically calculated estimated valued¯t+1 exceeds the upper boundary B1, it means the booked electrical energy consumption is beyond expectation. Meanwhile, the real-time price will be increased to induce consumers to reasonably reduce the booked electrical energy consumption. By the same token, if d¯t+1 is lower than the lower boundary B2, it means scheduled electrical energy consumption is lower than expectation and the remaining power is sufficient. It is necessary to reduce the real-time price to encourage consumers to add more booked electrical energy consumption at that moment. Energy provider can even encourage users to store electricity in their own batteries to get through the period of rising prices. Through the above adjustments, users can be guided to reasonable electrical energy consumption. Therefore, a smooth and steady supply of electricity can be ensured from the energy supplier.The strategy needs to be discussed in terms of the quantitative relation between price changes and the gap between users’ booked electrical energy consumption and optimal generation capacity. The relationship can be tested by relevant data. In order to explain the adjustment strategy, we provide the following theorem.Theorem 1.
Set the demand function as an exponential function. The gap EWMA estimationd¯t+1 is exponential to electricity price EWMA value p¯t+1, and the form is d¯t+1=k1ek2p¯t+1ord¯t+1=−k1e−k2p¯t+1; k1>0 and k2<0 are constants. When the load gap satisfies d¯t+1≥B1>0, d¯t+1 is adjusted to E1∈0,B1, and then the price variation is(6)δt+1=1μk2lnE1−lnd¯t+1.Whend¯t+1≤B2<0, d¯t+1 is adjusted to E2∈B2,0, and then the price variation is(7)δt+1=1−μk2lnE2−lnd¯t+1.Proof.
In intervalt+1, when d¯t+1≥B1>0, d¯t+1 has to adjust to E1∈0,B1. Meanwhile, the EWMA price is shifted from p¯t+1 to p¯t+1′. Under the assumed condition, we have(8)d¯t+1=k1ek2p¯t+1,E1=k1ek2p¯t+1′,which can be written as(9)lnd¯t+1=lnk1+k2p¯t+1,lnE1=lnk1+k2p¯t+1′,lnE1−lnd¯t+1=k2p¯t+1′−p¯t+1.
According to (3), (4), (8), and (9), we have(10)lnE1−lnd¯t+1=k2μpt″−pt′.
Settingδt+1:=pt″−pt′ as change in price, we have(11)δt+1=1μk2lnE1−lnd¯t+1.
Similarly, whend¯t+1≤B2<0, we adjust d¯t+1 to E2∈B2,0; under the assumed condition, we have(12)d¯t+1=−k1e−k2p¯t+1,E2=−k1e−k2p¯t+1′,so we obtain(13)lnd¯t+1=ln−k1−k2p¯t+1,lnE2=ln−k1−k2p¯t+1′,lnE2−lnd¯t+1=−k2p¯t+1′−p¯t+1=−μk2δt+1.
Hence, from above, formula (7) is established.


## 2.1. EWMA Estimation

We suppose that the users book the electrical energy consumption of the next interval, and the reservation retains an important reference value for accurate adjustment.In order to accurately obtain the extent of gap between booked electrical energy consumptionxt of users in time interval t and optimal generation capacity Gt∗, we set gap dt as(1)dt=xt−Gt∗,and we predict the next gap value dt+1 by the EWMA model from last adjusted gap value. The details of the calculation are as follows.Set the initial gap value asdτ,τ=t,t−1,…, and set the adjusted one as dτ′,τ=t,t−1,…. The EWMA d¯t+1 of gap value dt+1 in time interval t+1 is in the following formula:(2)d¯t+1=θd¯t+μdt′,in which θ=1−μ is the discount factor.Similarly, EWMAp¯t+1 of price pt+1 in interval t+1 is(3)p¯t+1=θp¯t+μpt′,in which pt′ is changed price in interval t.In the process of adjustment, the changed EWMA valuep¯t+1′ in interval t+1 is(4)p¯t+1′=θp¯t+μpt″,where pt″ is readjusted price in interval t.


## 2.2. Data-Driven APC Electrical Energy Monitoring

In this section, we discuss how to develop a data-driven APC electrical energy monitoring strategy in order to minimize the difference from the goal electrical energy gapE. We will change the price when the EWMA value is beyond the boundary as(5)d¯t+1≥B1ord¯t+1≤B2,B1≥0,B2≤0,where B1 is prestipulated upper limit and B2is prestipulated lower limit. In the process of monitoring, when d¯t+1 conforms to (5), the EWMA value d¯t+1 is out of the limits. The action of adjusting it to get nearer to the goal value will be taken. It is obvious in test results that, to achieve a stable subsequent adjustment, it is worthy of discussion to find a way to set parameters E1≥0 and E2≤0 of the target process properly.When monitoring users’ booked consumption load, we obtain a seriesd¯tt=1T of EWMA estimation. If d¯t+1 satisfies (5), the users’ booking electrical energy consumption has been beyond the steady limit. For preventing the consumers’ blind electricity utilization, the energy provider applies the users’ price demand response. It guides the users to use power properly, which achieves smooth and steady electrical energy consumption.


## 2.3. Data-Driven APC Adjustment

If the automatically calculated estimated valued¯t+1 exceeds the upper boundary B1, it means the booked electrical energy consumption is beyond expectation. Meanwhile, the real-time price will be increased to induce consumers to reasonably reduce the booked electrical energy consumption. By the same token, if d¯t+1 is lower than the lower boundary B2, it means scheduled electrical energy consumption is lower than expectation and the remaining power is sufficient. It is necessary to reduce the real-time price to encourage consumers to add more booked electrical energy consumption at that moment. Energy provider can even encourage users to store electricity in their own batteries to get through the period of rising prices. Through the above adjustments, users can be guided to reasonable electrical energy consumption. Therefore, a smooth and steady supply of electricity can be ensured from the energy supplier.The strategy needs to be discussed in terms of the quantitative relation between price changes and the gap between users’ booked electrical energy consumption and optimal generation capacity. The relationship can be tested by relevant data. In order to explain the adjustment strategy, we provide the following theorem.Theorem 1.
Set the demand function as an exponential function. The gap EWMA estimationd¯t+1 is exponential to electricity price EWMA value p¯t+1, and the form is d¯t+1=k1ek2p¯t+1ord¯t+1=−k1e−k2p¯t+1; k1>0 and k2<0 are constants. When the load gap satisfies d¯t+1≥B1>0, d¯t+1 is adjusted to E1∈0,B1, and then the price variation is(6)δt+1=1μk2lnE1−lnd¯t+1.Whend¯t+1≤B2<0, d¯t+1 is adjusted to E2∈B2,0, and then the price variation is(7)δt+1=1−μk2lnE2−lnd¯t+1.Proof.
In intervalt+1, when d¯t+1≥B1>0, d¯t+1 has to adjust to E1∈0,B1. Meanwhile, the EWMA price is shifted from p¯t+1 to p¯t+1′. Under the assumed condition, we have(8)d¯t+1=k1ek2p¯t+1,E1=k1ek2p¯t+1′,which can be written as(9)lnd¯t+1=lnk1+k2p¯t+1,lnE1=lnk1+k2p¯t+1′,lnE1−lnd¯t+1=k2p¯t+1′−p¯t+1.
According to (3), (4), (8), and (9), we have(10)lnE1−lnd¯t+1=k2μpt″−pt′.
Settingδt+1:=pt″−pt′ as change in price, we have(11)δt+1=1μk2lnE1−lnd¯t+1.
Similarly, whend¯t+1≤B2<0, we adjust d¯t+1 to E2∈B2,0; under the assumed condition, we have(12)d¯t+1=−k1e−k2p¯t+1,E2=−k1e−k2p¯t+1′,so we obtain(13)lnd¯t+1=ln−k1−k2p¯t+1,lnE2=ln−k1−k2p¯t+1′,lnE2−lnd¯t+1=−k2p¯t+1′−p¯t+1=−μk2δt+1.
Hence, from above, formula (7) is established.


## 3. Algorithm

According to Theorem1, the adjusted electrical energy consumption in interval t+1 is(14)xt+1′=xt+1−k1ek2δt+1,xt+1,δt+1>0,δt+1=0,xt+1+k1ek2δt+1,δt+1<0.Then we have(15)dt+1′=dt+1+xt+1′−xt+1=dt+1−k1ek2δt+1,dt+1,δt+1>0,δt+1=0,dt+1+k1ek2δt+1,δt+1<0.We get optimal solutionpt∗t=1T and Gt∗t=1T by applying Lagrange dual method to solve the social welfare maximization problem (C.1)–(C.3) (see Appendix). Smart meters feed users’ booked electrical energy consumption series xtt=1T back to the energy supplier. According to (1), we calculate the series dtt=1T of gap between booked electrical energy consumption xt of users in interval t and optimal generation capacity Gt∗. Let the initial adjusted consumption load gap d1′=d1, p1′=p1∗, and d¯1=0, so that the initial predicted error is e1=d1′−d¯1=d1′. Set the initial price adjustment as δ1=0. Suppose that the parameters k1>0 and k2<0, 0≤E1<B1, B2<E2≤0, and μ∈0,1,ω∈1,4. In interval t∈Γ, applying data-driven APC strategy, the monitoring and adjustment algorithm is summarized as Algorithm 1.Algorithm 1: Data-driven APC algorithm.
Step 1: calculated¯t+1 according to (2). If (5) holds, turn to Step 4. Otherwise, δt+1=0, turn to Step 2.Step 2: calculatext+1′ according to (14), dt+1′ according to (15), et+1=dt+1′−d¯t+1.Step 3: repeat Step 1.Step 4: whend¯t+1≥B1>0, let δt+1=1/μk2lnE1−lnd¯t+1, when d¯t+1≤B2<0, let δt+1=1/−μk2lnE2−lnd¯t+1, then turn to Step 2.


## 4. Case Analysis

The operation effect of data-driven exponential function-based APC monitoring and adjustment strategy is analyzed through Singapore’s power market data [21] in this part.


### 4.1. Power Load

We select RTP data from Mar 5, 2017, to Mar 6, 2017, and electrical energy consumption data from Mar 3, 2017, to Mar 6, 2017, for simulation. In Algorithm1, we set the RTP data as the initial booked sequences pt∗t=1T. In equation (1), we set the electrical energy consumption data from Mar 5, 2017, to Mar 6, 2017, as booked electrical energy consumption xtt=1T. Past electrical energy data at the corresponding time from Mar 3, 2017, to Mar 4, 2017, is regarded as optimal generation capacity Lt∗t=1T in equation (1). The original power loads are shown in Figure 1.Figure 1
Comparison between optimal generation capacity and booked electrical energy consumption.As illustrated in Figure1, the users’ booked consumption power load runs far away from the optimal generation capacity. In order to encourage users to reasonably consume power, the data-driven APC strategy needs to be adopted. This means that the adjustment of electricity prices is set by suppliers. Then it will guide consumers to adjust real electrical energy consumption.


### 4.2. Numerical Analysis for APC Adjustment

Letp¯1=75, and see Section 3 for the other initial arguments. Set the parameters in Algorithm 1 as follows: k1=20,k2=−1, μ=0.3, B1=800,B2=−800, E1=B1/2, and E2=B2/2. Assume that the arguments a,b,c in equation (B.3) are 0.01,0,0. Figures 2–5 show the APC strategy simulation results.Figure 2
APC strategy process.Figure 3
Electrical energies comparison.Figure 4
Price adjustment effects.Figure 5
Comparison of total social welfare and energy provider’s profit, with original and adjusted pricing strategies.Figure2 depicts that electrical energy gap series are steadier than the ones without adjustment after 11 adjustments. By experience, the average adjustment interval is 47/11 = 4.3, and the standard deviation of residuals is σ=∑t=1Tet2/T−1=∑t=148et2/47=734. No points outside the range 3σ indicate that there is no sign of the abnormality.As can be seen in Figure3, adjusted electricity consumption is nearer optimal generation capacity than the one without adjustment, and expected effects can be achieved. Figure 4 shows that the electricity price has changed 11 times. The biggest change in price is −3.376×10−3 units. During these periods, we encourage the consumers to buy and use more electrical energy consumption. We apply equation (C.1) to calculate the total social welfare to get 6.34×108, and we apply equation (B.4) to calculate the profit to get 3.82×107. As can be seen from Figure 5 by running our strategy, we can obtain higher social welfare and profits than those without adjustment.Besides improving energy provider’s profit and total social welfare, the data-driven APC adjustment strategy helps to balance power supply and prevent SG outages.


### 4.3. Comparison between Two Different Demand Function Adjustments

Reference [12] points out that there is a linear relationship between the EWMA predicted value d¯t+1 of consumption load gap and the EWMA predicted value p¯t+1 of price. This paper proposes that d¯t+1 and p¯t+1 are presented as an exponential function. The arguments are k1=120,k2=−1,k=500, μ=0.5, B1=1000,B2=−1000, E1=B1/2, and E2=B2/2. The comparison of the electrical energy consumption results adjusted by these two methods is shown in Table 1 and Figures 6–8.Table 1
Comparison results.
Exponential adjustmentLinear adjustmentAdjustment frequency109Standard deviation585625Social welfare6.35 × 1085.24 × 108Profit3.82 × 1073.76 × 107Figure 6
Comparison of adjustments in different APC strategies.Figure 7
Comparison of total social welfare and energy provider’s profit, with linear and exponential adjustment pricing strategies.Figure 8
Comparison of price adjustment effects in different adjustment strategies.From Table1 and Figure 6, we can learn that the adjustment frequency of the exponential adjustment is slightly higher than that of the linear one, but the standard deviation of the exponential adjustment is smaller than that of the linear adjustment.Table1 and Figure 7 illustrate that total social welfare and energy provider’s profit of the exponential demand function are higher than those of the linear one. Figure 8 presents that price adjustment effects of exponential demand function are better than those of the linear one. In particular, even the adjustment frequency with the exponential adjustment is slightly more than that in the linear one, and the standard deviation of residuals, total social welfare, and energy provider’s profit with exponential demand function are better than those of the linear function adjustment.From the observation results, we can conclude that, in general, the effect of exponential function adjustment is better than that of linear function adjustment.


## 4.1. Power Load

We select RTP data from Mar 5, 2017, to Mar 6, 2017, and electrical energy consumption data from Mar 3, 2017, to Mar 6, 2017, for simulation. In Algorithm1, we set the RTP data as the initial booked sequences pt∗t=1T. In equation (1), we set the electrical energy consumption data from Mar 5, 2017, to Mar 6, 2017, as booked electrical energy consumption xtt=1T. Past electrical energy data at the corresponding time from Mar 3, 2017, to Mar 4, 2017, is regarded as optimal generation capacity Lt∗t=1T in equation (1). The original power loads are shown in Figure 1.Figure 1
Comparison between optimal generation capacity and booked electrical energy consumption.As illustrated in Figure1, the users’ booked consumption power load runs far away from the optimal generation capacity. In order to encourage users to reasonably consume power, the data-driven APC strategy needs to be adopted. This means that the adjustment of electricity prices is set by suppliers. Then it will guide consumers to adjust real electrical energy consumption.


## 4.2. Numerical Analysis for APC Adjustment

Letp¯1=75, and see Section 3 for the other initial arguments. Set the parameters in Algorithm 1 as follows: k1=20,k2=−1, μ=0.3, B1=800,B2=−800, E1=B1/2, and E2=B2/2. Assume that the arguments a,b,c in equation (B.3) are 0.01,0,0. Figures 2–5 show the APC strategy simulation results.Figure 2
APC strategy process.Figure 3
Electrical energies comparison.Figure 4
Price adjustment effects.Figure 5
Comparison of total social welfare and energy provider’s profit, with original and adjusted pricing strategies.Figure2 depicts that electrical energy gap series are steadier than the ones without adjustment after 11 adjustments. By experience, the average adjustment interval is 47/11 = 4.3, and the standard deviation of residuals is σ=∑t=1Tet2/T−1=∑t=148et2/47=734. No points outside the range 3σ indicate that there is no sign of the abnormality.As can be seen in Figure3, adjusted electricity consumption is nearer optimal generation capacity than the one without adjustment, and expected effects can be achieved. Figure 4 shows that the electricity price has changed 11 times. The biggest change in price is −3.376×10−3 units. During these periods, we encourage the consumers to buy and use more electrical energy consumption. We apply equation (C.1) to calculate the total social welfare to get 6.34×108, and we apply equation (B.4) to calculate the profit to get 3.82×107. As can be seen from Figure 5 by running our strategy, we can obtain higher social welfare and profits than those without adjustment.Besides improving energy provider’s profit and total social welfare, the data-driven APC adjustment strategy helps to balance power supply and prevent SG outages.


## 4.3. Comparison between Two Different Demand Function Adjustments

Reference [12] points out that there is a linear relationship between the EWMA predicted value d¯t+1 of consumption load gap and the EWMA predicted value p¯t+1 of price. This paper proposes that d¯t+1 and p¯t+1 are presented as an exponential function. The arguments are k1=120,k2=−1,k=500, μ=0.5, B1=1000,B2=−1000, E1=B1/2, and E2=B2/2. The comparison of the electrical energy consumption results adjusted by these two methods is shown in Table 1 and Figures 6–8.Table 1
Comparison results.
Exponential adjustmentLinear adjustmentAdjustment frequency109Standard deviation585625Social welfare6.35 × 1085.24 × 108Profit3.82 × 1073.76 × 107Figure 6
Comparison of adjustments in different APC strategies.Figure 7
Comparison of total social welfare and energy provider’s profit, with linear and exponential adjustment pricing strategies.Figure 8
Comparison of price adjustment effects in different adjustment strategies.From Table1 and Figure 6, we can learn that the adjustment frequency of the exponential adjustment is slightly higher than that of the linear one, but the standard deviation of the exponential adjustment is smaller than that of the linear adjustment.Table1 and Figure 7 illustrate that total social welfare and energy provider’s profit of the exponential demand function are higher than those of the linear one. Figure 8 presents that price adjustment effects of exponential demand function are better than those of the linear one. In particular, even the adjustment frequency with the exponential adjustment is slightly more than that in the linear one, and the standard deviation of residuals, total social welfare, and energy provider’s profit with exponential demand function are better than those of the linear function adjustment.From the observation results, we can conclude that, in general, the effect of exponential function adjustment is better than that of linear function adjustment.


## 5. Conclusions

In our smart grid system, users can book a day or more of electrical energy consumption according to dynamic pricing provided by the energy provider. This energy provider monitors the real-time booked consumption loads and obtains the stable consumption load through the price demand response mechanism. The automatic process control strategy put forward in the article is as follows. Manage power consumption process. That is to say, the energy supplier monitors the gap between the optimal generation capacity given by the social welfare maximization problem and consumers’ booked electrical energy consumption. Then predict next time interval electrical energy consumption gap with statistical average model. It is only when predicted average number is beyond the presupposed boundary that price rises and cuts are used to change the price and to stimulate demand response. In this way, the adjustment frequency is not too great, and the users will change their initial consumption plan (i.e., reservation consumption) during the actual power consumption process. So the electrical energy consumption can become stable and the grid can run reliably and safely. The case analysis show that the network system of the energy provider automatically monitors and adjusts the price so as to get a small adjustment frequency, a stable actual electrical energy consumption, and a controllable residual standard deviation. After comparison, the exponential function adjustment method proposed in this paper is also shown to be more suitable than the linear one.



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*Source: 9373204-2021-09-23.xml*