A Method for Reducing the Instability of Negawatts Considering Changes in the Behavior of Consumers

Abstract

Negawatt trading is expected to improve energy efficiency via the prediction of peak demands and power-saving requests. However, the amount of power saved by consumers is not stable. The accurate prediction of demands and making appropriate requests with respect to power saving are difficult obstacles that need to be overcome in order to attain useful negawatt trading processes. To increase the accuracy of predictions and requests, earlier research suggests some methods or linear problems. On the other hand, the investigation of factors that affect consumption is important for correcting the current instability. In this research study, weather changes including temperature, time passing, and the consciousness of consumers are considered as important factors against electricity demands. In this paper, we propose a behavioral model of consumers using weather data. By using this behavioral model, the effectiveness of the suggested methods in earlier research for improving negawatt trading and the uncertainty of negawatts caused by weather changes is investigated.

1. Introduction

Electricity utilities in each region monopolized electricity supplies in the conventional Japanese electricity market. However, electricity retailing had been liberalized in stages since March 2000. The liberalization, including low-voltage power services, was completed in April 2016. Thanks to the liberalization of electricity retailing, an increase in electricity market sellers and declines in electricity prices are expected. On the other hand, sustainable electricity supplies are the most important demands for electric systems.

The storage of large amounts of electricity is challenging, so electric utilities have to produce and supply an amount of electricity that is equal to the demand because a disruption of the balance between electricity supply and demand causes periodic turbulence in power systems, which results in the deterioration of electricity [1]. Conversely, electricity consumers always use electricity without attention to the restriction of electricity production. Therefore, demand largely fluctuates throughout the day, and this fluctuation causes a difference between the demand during a peak period and the average demand. A large gap in electricity demand lowers energy utilization efficiency and raises the costs associated with managing electricity utilities. To solve the gap problem in electricity demands, electricity consumers should reduce electricity consumption during peak demand periods. Electricity demand during peak periods and average demand should be as close as possible.

Additionally, the demand response (DR) attracts our attention in terms of limiting electricity demands [2,3]. DR is a system for changing the pattern of electricity consumption, and it can control electricity demands by adjusting electric bills and bestowing rewards. Japanese DR began in the 1990s. After the Great East Japan Earthquake, DR is considered a means toward the achievement of a stable supply of electricity. DR is grouped into two classes according to the approach. When electricity utilities employ price-based DR, they change the electric bill dynamically and control the amount of demand. In terms of incentive-based DR, including negawatt trading, consumers who subscribe to electricity utilities save electricity as much possible, and electricity utilities send incentives according to the amount of power saved [4]. A negawatt denotes electric power that is not consumed due to power saving. In negawatt trading, a monetary reward is sent as an incentive to save a certain amount of demand.

In negawatt trading markets, other than electricity utilities and consumers, aggregators behave as stakeholders. Aggregators manage consumers’ negawatts in an integrated fashion. In this market, electricity utilities predict the peak hours of electricity demand and request aggregators to derive negawatts to ensure a stable supply. Aggregators request consumers to save power in order to fulfill the negawatt supply. Subsequently, aggregators gather the saved electric power and pass them to electricity utilities. Consumers who save electricity are given cash rewards according to negawatts.

The negawatt trading market can be categorized into three types [5]. The first type involves retail electricity companies that source negawatts from their customers. The second type involves retail electricity companies that source negawatts from consumers who are contracted with other electric utilities. The third type involves general electricity utilities that source negawatt to manage the demand and supply. This research study does not depend on the types of negawatt trading because the trading model used in this study abstracts negawatt trading.

Additionally, negawatt trading flow comprises direct trades between electricity utilities or trades made through the Japan Electric Power eXchange(JPEX). Major markets held by JPEX are day-ahead markets and hourly-ahead markets [6]. In day-ahead markets, electric power, which is passed on the next day, is traded. On the other hand, hourly-ahead markets are held to deal with unforeseen disturbances in the supply–demand balance. To simulate the trade to systematically decrease electricity consumption at peak hours, we employ day-ahead markets in this research study.

Although negawatt trading is expected to limit electricity demands by using incentives, the sustention of a stable supply–demand balance is difficult because a reduction in demand depends on the behaviors of consumers. For example, temperatures and the time of day affect the use of electrical appliances and result in unstable demands. Therefore, this research aims to reduce the instability of negawatt trading in electricity markets by considering meteorological phenomena.

2. Materials and Methods

2.1. Related Research

Methods for controlling demands using dynamically changing electric bills were researched. Ref. [7] suggested an optimal real-time algorithm for determining electric bills. In the research study, the electrical power system consists of electricity suppliers, some consumers, and regulators. The suggested algorithm is used to optimize electricity consumption, the generation capacity of suppliers, and electric bills. In addition, the study modeled the choices and consumption patterns of consumers based on a utility function in microeconomics and employs optimization problems as the behavioral models of consumers. However, consumers may not always maximize their own utility, so the use of the utility function will not be the best method for achieving accurate predictions.

On the other hand, Ref. [8] suggested a technique for designing electric bills that match electricity supply and demand in each region while ensuring benefits for an entire society. The research presumes an independent system operator (ISO), which matches the consumer, supplier, and the amount of electricity demand and supply and maximizes the benefit for an entire society as market players. Additionally, regret matching is employed in the behavioral models of suppliers and consumers. In regret matching, market players do not always choose an optimal strategy because they stochastically choose their strategies. However, suppliers should always choose an optimal strategy to earn benefits as business operators. Accordingly, methods of using regret matching in the prediction of electricity demand and supplied should be reconsidered. In terms of other research studies on demand responses, research proposing stochastic programming and the alternating direction method of multiplier algorithms (SP-ADMM) exists [9]. This research study considered the use of air conditioners and heaters. As variables affecting behavioral models, outdoor temperature, solar generations, hot water consumptions, and responsive load are employed. Ref. [9] improved the conventional alternating direction method of the multiplier algorithm. On the other hand, it did not consider the consciousness against power saving. Moreover, this research study dealt with both weekdays and weekends in the same manner, although consumers will act differently during the weekends; potential factors are still left in terms of demand response.

Moreover, many research studies in terms of negawatt trading such as [10,11] focus on real-time electricity markets and discuss the management of the prices of incentives and allocation rates. Based these research studies, in our earlier research [12,13], we assumed that aggregators always choose the best strategies and that they adjusted the requirement of power saving for each consumer. The adjustment of aggregators will enable entire markets to control the demand. In particular, aggregators calculate the amount of power saved compared to the requirement of calculating the amount against each consumer based on the records of power saving. Then, aggregators solve optimization problems to minimize the gap between the target amount of power saving and the sum of power saving. In the research study, the behavioral models of consumers are assigned particular probability distributions to show the contribution of power saving and uncertainties. From the simulation result, the suggested technique can acquire negawatts that are closer to the target amount than the technique of requesting that all consumers use the same amount of negawatts. Nevertheless, the research study limits the behaviors of consumers and assumes that consumers consume the same amount of electricity regardless of the time and weather. Therefore, the model in the research study is insufficient for considering real situations.

Hereupon, we suggest applying climate data to the behavioral models of consumers to simulate trading with a consideration of the time course in this paper. We verify the effects of weather in the negawatt trading simulation because such results have not been found. Specifically, we simulate whether the technique for determining the requirement of power saving suggested in our earlier research can reduce the instability of negawatts in the electricity markets where the behavioral model of consumers created in this research is applied.

2.2. Definition

2.2.1. Market Players and Times for Power-Saving Requests

The model of negawatt trading markets in this research is shown in Figure 1. Each market player is expressed as a vertex, and each trading relationship is expressed as an edge. An electric utility contracts with an aggregator and an aggregator contracts with consumers to carry out negawatt trading. In one market, one electric utility and one aggregator exist. A group of consumers is defined as B, and each consumer is expressed as $b_j(j=1,2,\dots ,|B\left|\right)$. In addition, we consider the model in which information related to consumers is updated as often as an electric utility requests an aggregator negawatt. To use the model, the times for power-saving requests are expressed as $t(t=0,1,2,\dots )$. When an electric utility sends an aggregator a t-th power-saving request, an aggregator also sends all consumers a t-th power-saving request.

2.2.2. A Target Amount of Power Saving and Requirement of Power Saving

The amount of power saving requested by an electric utility at t-th time is defined as $T^t$. Additionally, the amount of power saving requested by aggregators toward consumer $b_j$ at t-th time is defined as $x_j^t$, and its limit is $d_j$. If a consumer did not request power saving, this consumer is dealt with as requested $x_j^t=0$ power saving. $d_j$ is determined via a discussion between an aggregator and a consumer when they contract with each other. In addition, $T^t$ and $x_j^t$ are non-negative real numbers, and $d_j$ is defined as a non-negative integer that fulfills the equation below.

$$0\le x_j^t\le d_j,$$

(1)

Furthermore, $T^t$ fulfills the second equation below because an aggregator informs an electric utility of the sum of $d_j$ when contracted with all consumers in advance who did not request a the larger amount of power saving than the sum of the contract.

$$0\le T^t\le \sum _{j=1}^{\left|B\right|}{d}_j,$$

(2)

2.2.3. Behavioral Models of Consumers Considering Weather

Consumers save power when an aggregator requests them, so the amount of consumption changes regardless of whether power saving is requested. Therefore, this paper calculates the electricity consumption of consumer $b_j$ from the difference between electricity consumption when $b_j$ does not save power and the amount of saved power thanks to a power-saving request. The behavior of consumption without saving power is defined as a consumption behavior, and the behavior of power saving is defined as power-saving behavior.

The calculation of electricity consumption refers to a power demand forecasting method suggested for renewable energy use [13]. The average weather forecast error is approximately one percent, and the inputs of the method are relatively easy to access. Additionally, the purpose for predicting electricity consumption is similar, so the forecasting method is applicable to the behavioral models in negawatt trading. The forecasting method is a regression equation, and we calculate the amount of electricity demand at a target period from the prediction formula below. The coefficients from $a_1$ to $a_9$ are regression coefficients that are assigned values based on records described later in order to minimize forecast errors. Each regression coefficients are multiplied by explanatory variables.

$$\begin{array}{ccc}\hfill (\mathrm{amount}\mathrm{of}\mathrm{demand}\mathrm{at}\mathrm{a}\mathrm{certain}\mathrm{period})& =a_1·(\mathrm{an}\mathrm{average}\mathrm{amount}\mathrm{of}\mathrm{demand})\hfill & \\ \hfill & +a_2·(\mathrm{deviation}\mathrm{of}\mathrm{demand}\mathrm{at}\mathrm{a}\mathrm{forecasting}\mathrm{period})\hfill & \\ \hfill & +a_3·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{Saturday})\hfill & \\ \hfill & +a_4·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{Sunday}\mathrm{or}\mathrm{holiday})\hfill & \\ \hfill & +a_5·(\mathrm{a}\mathrm{variable}\mathrm{of}\mathrm{the}\mathrm{air}\mathrm{conditioner}\mathrm{effect})\hfill & \\ \hfill & +a_6·(\mathrm{Global}\mathrm{Horizontal}\mathrm{Irradiance}\left(\mathrm{GHI}\right)\mathrm{at}\mathrm{a}\mathrm{forecasting}\mathrm{period})\hfill & \\ \hfill & +a_7·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{one}\mathrm{year}\mathrm{before})\hfill & \\ \hfill & +a_8·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{two}\mathrm{years}\mathrm{before})\hfill & \\ \hfill & +a_9,\hfill \end{array}$$

(3)

Moreover, if the period for forecasting is before sunrise or after sunset, the equation that removes the effect of GHI from an explanatory variable is used.

$$\begin{array}{ccc}\hfill (\mathrm{amount}\mathrm{of}\mathrm{demand}\mathrm{at}\mathrm{a}\mathrm{certain}\mathrm{period})& =a_1·(\mathrm{an}\mathrm{average}\mathrm{amount}\mathrm{of}\mathrm{demand})\hfill & \\ \hfill & +a_2·(\mathrm{deviation}\mathrm{of}\mathrm{demand}\mathrm{at}\mathrm{a}\mathrm{forecasting}\mathrm{period})\hfill & \\ \hfill & +a_3·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{Saturday})\hfill & \\ \hfill & +a_4·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{Sunday}\mathrm{or}\mathrm{holiday})\hfill & \\ \hfill & +a_5·(\mathrm{a}\mathrm{variable}\mathrm{of}\mathrm{the}\mathrm{air}\mathrm{conditioner}\mathrm{effect})\hfill & \\ \hfill & +a_6·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{one}\mathrm{year}\mathrm{before})\hfill & \\ \hfill & +a_7·(\mathrm{a}\mathrm{dummy}\mathrm{variable}\mathrm{of}\mathrm{two}\mathrm{years}\mathrm{before})\hfill & \\ \hfill & +a_8,\hfill \end{array}$$

(4)

The details of the explanatory variables in the above equations are provided below:

• An average amount of demand: the average amount of electricity consumption in the most recent 49 days within the same time period;
• Deviation of demand at a forecasting period: the difference between the average electricity consumption and the record of demands during a forecasting time period;
• A dummy variable for Saturday: a variable to which 1 is substituted if the forecasting day is a Saturday, which is not a holiday, and to which 0 is substituted if the day is not a Saturday or the day is a holiday;
• A dummy variable for Sunday or a holiday: a variable that is 1 if the forecasting day is a Sunday or a holiday, and it is 0 if the day is not a Sunday nor a holiday;
• A variable of the air conditioner effect: the variable is set to reflect the effect of temperatures toward the behaviors of consumers. The value is calculated using the equation below.

  $$max(d-20,18-d,0),$$

  (5)
  d denotes the record of temperature during a forecasting period. A variable of the air conditioner effect is assigned a value using a particular function. When d is higher than 293.15 Kelvin, the variable increases in proportion to the temperature to reflect the effect of coolers. In this weather, consumers may use coolers in their homes. When d is lower than 291.15 Kelvin, the variable exhibits an inverse relation relative to the temperature to reflect the effect of heaters. In this weather, consumers use heaters to retain comfort. To simplify the formula, the variables related to temperatures are dealt with as degree Celsius.
• A variable of solar irradiance: the value is calculated from the equation below.

  $$\omega ·(\mathrm{Global}\mathrm{Horizontal}\mathrm{Irradiance}\mathrm{at}\mathrm{the}\mathrm{forecasting}\mathrm{period}),$$

  (6)
  Here, Global horizontal irradiance is the amount of energy obtained from the sun. $\omega$ is a coefficient for correcting the difference in solar insolation effects due to the difference in the amount photovoltaic systems installed each year. When a large number of photovoltaic systems are installed by many consumers, the effects of solar irradiance will be high.
• A dummy variable for one year before: a variable that takes 1 if the record for using the coefficient in a regression equation comprises data from one year before a forecasting year, and it takes 0 if not.
• A dummy variable for two years before: A variable that takes 1 if the record for using the coefficient in a regression equation comprises data from two years before a forecasting year, and it takes 0 if not.

The coefficients of an equation are determined based on records of the past two years. In particular, coefficients from $a_1$ to $a_9$ are determined in order to minimize forecasting errors against records during four recent weeks in the forecasting year, records in the past four weeks in the year before the forecasting day, and records in the past four weeks two years before the forecasting day. When we calculate these coefficients, we prepare simultaneous equations for electricity consumption. In one-negawatt trades, electricity consumption within thirty minutes is referred to, so 35,040 equations can be used to calculate the coefficients, although approximations are employed as the calculation result. Listing A2 and Figure A1 show how consumption behaviors are calculated.

In the inputs of the forecasting method, climate data published in [14] and demand data published in [15] are used. The published data in [15] comprise the data of entire regions relative to the supplied electricity, so we multiply $1.0\times {10}^{(-5)}$ to match the demand of consumers who contracted high-voltage power services (contracted power between 50 kW and 2000 kW). The reason why we take these consumers as the standard is [16,17] because we are considering the minimum negawatt trading unit as 100 kW. Moreover, consumers who use high-voltage power services switch contracts with the power producer and supplier (PPS), and their dominance in electricity demands is high. We define the electric power calculated from the forecasting method as the tentative electricity consumption, $f_j^t$—this is the electricity that consumers will consume.

Particular probability distributions that express the awareness of saving power and uncertainties are used to calculate an amount of power saved. The reason for defining particular probability distributions is that we can express power-saving actions by expressing each characteristic. The motivation for using uncertainties is that the same consumer will not always save power in the same manner.

Firstly, the probability distribution for showing the awareness of power saving $C{U}_m$ is defined. This paper assumes that consumers can be grouped based on awareness and defines the continuous uniform distribution as comprising of shortage, negative, standard, accurate, positive, and excess. To simplify models, a continuous uniform distribution is employed in this research. The consumer, $b_j$, tries to save a certain amount power, $a_j^t{x}_j^t$, which is equal to the requested saving power, $x_j^t$, multiplied by a random number, $a_j^t$, according to $C{U}_m$. The range of $C{U}_m$ is shown in

Table 1. Range of $C{U}_m$.

Tendency	m	Range of ${\mathit{CU}}_{\mathit{m}}$
shortage	1	$0.1\le C{U}_1\le 0.3$
negative	2	$0.6\le C{U}_2\le 0.8$
standard	3	$0.9\le C{U}_3\le 1.1$
accurate	4	$0.95\le C{U}_4\le 1.05$
positive	5	$1.2\le C{U}_5\le 1.4$
excess	6	$1.7\le C{U}_6\le 1.9$

. A higher $C{U}_m$ implies higher consciousness toward power saving. This paper defines the situation as standard when $a_j^t$ fulfills $0.9\le a_j^t\le 1.1$. To avoid resulting a negative power-saving value from calculations with uncertainties, the lower limit of $a_j^t$ was set to 0.1. Additionally, to balance the amount of power saving between consumers so that the value is higher than a positive value and lower than a negative value, the upper limit of $a_j^t$ was set to 1.9.

In addition, we define uncertainty $r_j^t$, which is the random number according to the continuous uniform distribution $U(-0.1,0.1)$. By multiplying this random number with $x_j^t$, uncertainty is expressed. This uncertainty shows a gap from $a_j^t{x}_j^t$. When an aggregator requests a larger $x_j^t$, the larger gap in uncertainty will result because consumers have to stop electric facilities that consume large amounts of electric power to achieve a large value of $x_j^t$, and such large amounts of power saving occurring all at once impedes the precise control of power saving. For example, consumers cannot control one-watt power saving by stopping refrigerators or televisions. The actual amount of saved power when the consumer, $b_j$, receives the t-th request amounting to $x_j^t$ is defined as $w_j^t$. $w_j^t$ is shown in Equation (7) below.

$$w_j^t=(a_j^t+r_j^t)x_j^t(w_j^t\ge 0),$$

(7)

Furthermore, we define the sum of $w_j^t$ as $W^t$ and express Equation (8) below.

$$W^t=\sum _{j=1}^{\left|B\right|}{w}_j^t,$$

(8)

From the definition, the actual amount of electric power consumed by a consumer $b_j$ who receives a power-saving request at time t can be calculated from the difference between the amount of electricity consumption calculated from the consumption behavioral model and the actual saved power calculated from the power-saving behavioral model. We define this value as $c_j^t$ and express Equation (9) below.

$$c_j^t=f_j^t-w_j^t,$$

(9)

In addition, the total actual amount of electric power consumed by all consumers is defined as $C^t$, and it is shown in Equation (10) below.

$$C^t=\sum _{j=1}^{\left|B\right|}{c}_j^t,$$

(10)

A clear measurement of the amount of saved electric power (thanks to consumers) is difficult for an electric utility supplier and an aggregator. To solve this difficulty, they estimate electricity consumption based on the records of consumption. The estimated amount of electric power is defined as a baseline, and an amount that is lower than the baseline is considered as a negawatt. Ref. [18] described the method for setting a baseline for the day of DR based on the latest consumption records. Based on this method, the baseline of consumer $b_j$ at the $t-$th power-saving request is expressed as $l_j^t$, and negawatt is shown as $n_j^t$. They fulfill Equation (11) below.

$$n_j^t=l_j^t-w_j^t,$$

(11)

Moreover, the sum of the baselines of consumers is shown as $L^t$, and the sum of the negawatt is shown as $N^t$. They fulfill Equation (12) below.

$$N^t=L^t-W^t,$$

(12)

2.2.4. Estimated Achievement Rate

The aggregator is not informed of $a_j^t$. On the contrary, an aggregator needs to know $a_j^t$ to accurately control the amount of saved power. Accordingly, an aggregator estimates the expectation of $a_j^t$ based on the records of power saving. The expectation is defined as the estimated achievement rate $E_j^t$. The idea is based on the assumption that $w_j^t$ can be predicted from records of the first to $(t-1)$th power-saving instance. Moreover, this paper assumes that power-saving behavior does not substantially change. We define $E_j^t$ as shown in Equation (13) below. We define $x_j^0=1$ and $w_j^t=1$.

$$E_j^t=\frac{{\sum }_{k=1}^{t-1}{w}_j^t}{{\sum }_{k=1}^{t-1}{x}_j^k},$$

(13)

2.2.5. The Estimated Amount of Saved Power and the Maximum Estimated Amount of Saved Power

Based on $E_j^t$, an aggregator estimates the actual amount of saved power. At this time, the estimated actual amount of electric power saved by $b_j$ is defined as the estimated amount of saved power $E_j^t{x}_j^t$. The sum of the estimated amount of saved power $P^t$ is expressed in Equation (14) below.

$$P^t=\sum _{j=1}^{\mid B\mid }{E}_j^t{x}_j^t,$$

(14)

Furthermore, when an aggregator requests that all consumers save the amount of power equal to $d_j$, the sum of the estimated amount of saved power is the maximum estimated amount of saved power $M^t$. $M^t$ is expressed in Equation (15) below.

$$M^t=\sum _{j=1}^{\mid B\mid }{E}_j^t{d}_j,$$

(15)

In addition, from the definitions of the equations above, the relationship between $P^t$ and $M^t$ is expressed in Equation (16).

$$0\le P^t\le M^t,$$

(16)

2.2.6. The Error in the Amount of Saved Power

We define the error in the amount of saved power as $y_j^t$ as an index to show the accuracy of the amount of saved power and express Equation (17) below.

$$y_j^t=\sqrt{\frac{1}{t-1}\sum _{k=1}^{t-1}{(x_j^k-w_j^k)}^2},$$

(17)

$y_j^t$ shows the root mean square error (RMSE) to evaluate the size of errors in $x_j^t$ and $W_j^t$ from the first to $(t-1)$th power-saving instance. When the value of RMSE is small, the accuracy of the prediction and request is considered to be of high quality.

2.3. Formulation of Problems

2.3.1. The Minimization Problem of the Gap between the Sum of the Estimated Amount of Saved Power and the Target Amount of Saved Power

The sum of the actual amount of saved power, $M^t$, should be close to the target amount of saved power, $T^t$, in order to achieve a stable balance between electricity demand and supply [11]. However, the complete control of the actual amount of saved power is difficult, so this research aims to approximate values from $P^t$ to $T^t$ as much as possible. Therefore, we express $P^t$ in Equation (18) below using vectors and define linear programming problems.

$$P^t=(E_1^t{E}_2^t\dots E_{\left|B\right|}^t),\left(\begin{array}{c}{x}_1^t\\ x_2^t\\ ⋮\\ x_{\left|B\right|}^t\end{array}\right)$$

(18)

Problem 1. $M^t<T^t$

$$\begin{array}{ccc}\hfill & \mathrm{maximize}{P}^t\hfill & \hfill \end{array}$$

(19a)

$$\begin{array}{ccc}\hfill & \mathrm{subject}\mathrm{to}0\le x_j^t\le d_j\hfill & \hfill \end{array}$$

(19b)

Problem 2. $T^t\ge M^t$

$$\begin{array}{ccc}\hfill & \mathrm{minimize}{P}^t\hfill & \hfill \end{array}$$

(20a)

$$\begin{array}{ccc}\hfill & \mathrm{subject}\mathrm{to}0\le x_j^t\le d_j\hfill & \hfill \end{array}$$

(20b)

$$\begin{array}{ccc}\hfill & T^t\le P^t\hfill & \hfill \end{array}$$

(20c)

In terms of problem 1, the gap between $P^t$ and $T^t$ can be minimized by maximizing $P^t$ because $P^t<M^t$. In terms of problem 2, the gap between $P^t$ and $T^t$ can be minimized by minimizing $P^t$ in restriction $T^t\le P^t$. This restriction exists due to our assumption that the target amount of saved power is fulfilled.

2.3.2. The Minimization Problem of the Error in the Sum of the Amount of Power Saved by All Consumers

An algorithm for preferentially requesting consumers with a high amount of saved power after calculating $y_j^t$ for each consumer exists [12]. In this algorithm, the appropriate requirement for power saving for each consumer is calculated. Thanks to this algorithm, consumers who accurately save power and who do not can be distinguished, and inequality is solved. Based on the algorithm, we prepare the problem. The sum of errors in the amount of power saved by all consumers, $Y^t$, is defined in Equation (21) below, and some linear programming problems are defined below. When $M^t$ fulfills $M^t<T^t$, all consumers should have requested the maximum requirement $d_j$; thus, Equation (23) is fulfilled.

$$Y^t=\sum _{j=1}^{\left|B\right|}{x}_j^t{y}_j^t,$$

(21)

Problem: If $T^t\le M^t$, then

$$\begin{array}{ccc}\hfill & \mathrm{minimize}{Y}^t\hfill & \hfill \end{array}$$

(22a)

$$\begin{array}{ccc}\hfill & \mathrm{subject}\mathrm{to}0\le x_j^t\le d_j\hfill & \hfill \end{array}$$

(22b)

$$\begin{array}{ccc}\hfill & T^t\le P^t\hfill & \hfill \end{array}$$

(22c)

$$x_j^t=d_j(M^t<T^t)$$

(23)

If $T^t\le M^t$, the requirement for the consumers with high accuracy becomes higher by minimizing $Y^t$ under the restriction of $T^t\le M^t$.

2.3.3. The Algorithm to Request Power Saving

The algorithm for solving the formulated problems in Section 2.3.2 is shown in Algorithm 1.

In this algorithm, the target amount of saved power is assigned to variable $rest$. Whenever the amount of 1 with respect to saved power relative to the request is distributed, 1 is subtracted from $rest$. This computation is repeated until $rest$ becomes lower than 0, and the requirement with respect to the amount of saved power is determined. The inputs of the algorithm include the following: the target amount of saved power $T^t$; the number of consumers $\left|B\right|$; list $l^{t-1}$ that contends with the indexes, j, of consumers $b_j$; the contracted amount to save $d_j$; and the estimated achievement rate of each consumer $E_j^t$. The i-th element in list $l^t$ is expressed as $l^t\left[i\right]$, and all elements in list $l^0$ are defined as 1. The output of the algorithm is the requirement of the amount of saved power $x_j^t$. $sort\left(\right)$ in Algorithm 1 is the function for sorting j in the list $l^{t-1}\left[i\right]$ in the ascending order of $y_j^t$.

Algorithm 1 The algorithm for requesting power saving

Require:$T^t$$\left|B\right|$$I^{t}t-1$$d_j$$E_j^t$

Ensure:$x_j^t$

$I^t$ ← $sort\left(I^{t-1}\right)$

$rest\leftarrow T^t$

for $i\leftarrow 1$$\left|B\right|$do

$request\leftarrow 0$

if $rest>0$ then

for $n\leftarrow 1$ to $d_{I^t\left[i\right]}$ do

if $rest\ge n*$$E_{I^t\left[i\right]}^t$ then

$request\leftarrow request+1$

else

break

end if

end for

$x_{I^t\left[i\right]}^t$$\leftarrow request$

$rest\leftarrow rest-request\ast$$E_{I^t\left[i\right]}^t$

end if

end for

return $x_j^t$

2.4. Simulation

2.4.1. Simulation Overview

We conduct simulations using the technique for determining the requirement of the amount of saved power in earlier research and the simple determination technique for comparison purposes. In particular, the method for distributing $T^t$ to all consumers equally (Method I), the method for solving the problems suggested in Section 2.3.1 using the solver (Method II), and the method for solving the problem in Section 2.3.2 using Algorithm 1 (Method III) are compared in terms of the effect relative to negawatt uncertainties from weather changes using the RMSE of $T^t$ and $N^t$. The simulation term takes place for one year from January to December. The target terms of DR are from 11:00 to 13:00 and from 17:00 to 19:00 in January, November, and December. Additionally, the time period from 13:00 to 17:00 in August, September, and October is also the target term of DR. In the simulation, 30 min is dealt with as a period. The declaration of DR is conducted in a 6-day cycle. The parameters used in the simulation are shown in

Table 2. Parameters in the simulation.

Parameter	Value
Target amount $T^t$	${\sum }_{j=1}^{\left|B\right|}0.06c_j^t$
Number of consumers $\left|B\right|$	300
Contracted amount to save $d_j$	100

. These parameters are set in Listing A3 for the purposes of programming. Ref. [18] aims to acquire a 6% negawatt of the maximum demand that is equal to a negawatt in the United States. Therefore, $T^t$ is assigned 6% of the sum of $L^t$. Moreover, temperature and GHI as the inputs of the behavioral models of consumers were assigned data from 2016 to 2019 [14]. The demand amount is based on data from 2016 to 2018 [15]. The data comprise records of one hour, so the calculated average of around 3 hours is used to convert data.

The process to import these data is shown in Listing A1. The percentage of 6 types of consumers defined in 3.4 exhibited 3 patterns in this simulation: the equable market, the market with a dominance of positive consumers, and the market with a dominance of negative consumers. In the equable market, the consciousness of consumers is equally distributed. In the market with a dominance of positive consumers, only consumers whose $C{U}_m$ is higher than 0.9 occupy this market. In the market with a dominance of negative consumers, the $C{U}_m$ of all consumers is lower than 1.05. The specific values of consumers in markets are shown in

Table 3. The types and numbers of consumers in each market.

Type	Number
	Equable Market	Majority Positive	Majority Negative
Shortage	50	0	75
Negative	50	0	75
Standard	50	75	75
Accurate	50	75	75
Positive	50	75	0
Excess	50	75	0

.

The simulation process’s programming in shown in Listing A4.

2.4.2. Evaluation Function

RMSE is employed as the evaluation function for $T^t$ and $N^t$. The applied RMSE is shown in Equation (24) below as an evaluation function A.

$$\sqrt{\frac{1}{t}\sum _{k=1}^t{(T^k-N^k)}^2},$$

(24)

An index of prediction errors is used to evaluate the difference between the predicted values and actual values; thus, the RMSE applies to the difference between $T^t$ and $N^t$ when $T^t$ is used as a prediction value and $N^t$ is used as the actual value. Evaluation function A shows low instability and low effects from weather changes when the value of A is low. Moreover, the evaluation of the error in the difference between $T^t$ and $W^t$ is expressed as an evaluation function B in Equation (25) below. These evaluation functions are calculated in the last part of Figure A2 and Listing A4.

$$\sqrt{\frac{1}{t}\sum _{k=1}^t{(T^k-W^k)}^2},$$

(25)

In this evaluation function, a lower RMSE means that an aggregator gains an expected amount of saved power in the entire society and ensures the accuracy of estimations.

3. Results

Listing A5, Listing A6 and Listing A7 and

Table 4. RMSE of the three methods.

	Market	RMSE for A	RMSE for B
Method I	equable market	1,076,482.02	419.18
	negative market	1,529,869.49	453,806.65
	positive market	622,671.29	453,391.55
Method II	equable market	1,076,842.45	779.60
	negative market	1,076,842.45	779.60
	positive market	1,075,252.26	810.59
Method III	equable market	1,074,627.82	1435.02
	negative market	1,077,474.046	1411.20
	positive market	1,073,341.63	2721.21

show the numerical result of negawatt trading, including the RMSE. Figure 2 shows the result of the RMSE in terms of evaluation function A. In terms of Method I, the value of the RMSE between the target amount and the total negawatt largely differs. The target amount of power saving did not change in all markets. Therefore, the difference in the RMSE is caused by the difference in the total negawatt. In contrast, the value of RMSE is quite close to each other in Method II and Method III. When all three methods are compared, almost the same amount of negawatts seemed to be acquired in the equable market. On the other hand, the RMSE in the market dominated by negative or positive consumers show differences. In the market dominated by negative consumers, a larger RMSE is outputted in Method I than that of Method II and III. On the contrary, a smaller RMSE is outputted in Method I than that of Method II and III.

Figure 3 shows the result of RMSE in terms of evaluation function B. In this method, Method I shows a large difference in the RMSE between the equable market and other markets. The other two methods also output different RMSEs, but the differences are quite small compared to the difference in Method. Not only is the difference between the values of RMSEs small, but the size of the RMSE is also small in Method II and III. Moreover, in Method I, an RMSE that is quite small is outputted into the equable market. Comparing three the methods, all methods show quite small RMSEs in the equable market. Although the RMSE in the equable market in Method I is smaller than that of the other two methods, the RMSE in the other markets in Methods II and III are much smaller than that of Method I.

4. Discussion

From the RMSE between the target amount of power saving and the amount of saved power, Method II and III enable an aggregator to gather an amount of saved power that is close to the target in any market situation. Although Method I succeeds in gathering sufficient saved power in the equable market, this method is affected by the market situation. The difference between Method I and other methods in terms of function B suggested that the methods used in earlier research are excellent for sufficiently saving power. On the other hand, Figure 2 shows a large gap in negawatts compared to the target. In the hypothesis before the simulation, negawatt instability can be reduced in all market patterns by introducing weather data because electricity consumption is affected by the weather effect.

Despite the hypothesis, in the simulation result, the suggested methods and the compared method do not show a large difference, and the market dominated by positive consumers shows that Method I is superior to Method II and III. The cause of this result is the consumed electric power on the day DR is disconnected from the baseline. The aggregator determines baselines to estimate electricity consumption on the day of DR and determines the required amount. On the contrary, electricity consumption on the day of DR is affected by the weather. Consumers try to save power under the effect of the weather. This flow results in a large gap between negawatts and the target amount of saved power, although the difference between the actual amount of saved power and the target amount of saved power is small. In addition, earlier research [12] indicates that Method II and III are better for reducing negawatt uncertainties than Method I in the positive market. Ironically, these two methods in the simulation stably maintain low negawatts. The values of the RMSE between the target amount of power saving and total negawatt should be smaller.

The described cause introduced lower electricity demands than the baselines, so an excessive reduction in electricity consumption occurred and resulted in the large RMSE in function A and the small RMSE in function B. In negawatt trading, excessive power saving causes disadvantages for the aggregator. The amount of saved power that is larger than the target amount of negawatt in 3% is called an imbalanced penalty, and excessive negawatts are retrieved by electric utilities for free. Therefore, from the perspective of the aggregator, a lower RMSE between the target amount of power saving and total negawatt is much more desired than a larger power-saving amounts than the target. From these results, to achieve low values in both function A and B, the methods for setting the baselines should be revised. Additionally, how weather data can be used to correct consumption forecast should be considered.

5. Conclusions

In this research study, weather changes and time passing are assumed to affect the behaviors of electricity consumers, and we described the result of the research study about the effect of time passing against negawatt trading. This research sets the behavioral models of electricity consumers based on how weather data are created, and the instability of negawatts gathered from consumers is investigated. Past research studies suggest using more efficient or accurate methods to request power saving, so these methods are compared in negawatt trading with a consideration of the weather. As a result, power-saving requests based on methods suggested in earlier research successfully gather the actual amount of saved power that is close to the target amount. However, in terms of negawatt, a large error compared to the target is caused. This error may have come from the difference between the baseline set in advance and the consumption on the day of DR. In future research studies, we will investigate the cause of the difference between baselines and electricity consumption. Moreover, after finding s the correct cause, an appropriate method for setting baselines in negawatt trading will be researched.