normality test on the annual electricity consumption and
TRANSCRIPT
Normality test on the annual electricity consumption and PVgeneration in Higashi-Ohmi City
Shiro Yano, Tatsunori Kishi, Tadahiro TaniguchiRitsumeikan University
Abstract
For purpose of effective utilization of energy from decentralized power generation, elec-tricity trading would works effectively. This trading requires big cost for human, so it shouldbe done by the agents with artificial intelligence automatically. This algorithm needs topredict the resident’s load profile. Analysing the characteristics of load profiles (e.g. thenormality of variables) would be help of constructing a more effective learning algorithm.
In this paper, we analyse annual load profiles of Higashi-Ohmi City. At first, we employmacroscopic analysis which aims to understand the seasonality effect on the energy consump-tion and PV generation. Second, we employ microscopic analysis which aims to understandwhether electricity consumption and PV generation shows normality or not. At last, we ex-tend the microscopic analysis through the use of macroscopic analysis result.
Keywords: normality, annual data analysis, PV generation, directional statistics.
1. IntroductionFor the last few years, Japanese Ministry of Internal Affairs and Communications have
promoted the project ”green decentralization reforms” which aims to retain sustainable anddecentralized development throughout the regions of Japan [1]. Higashi-Ohmi City (figure 1) participates in this project and challenges to progress energy independence by developingsmart grid technologies [2]. This project aims to investigate the capacity of solar photo-voltaic(PV) power generation in Higashi-Ohmi City.
(a) Shiga Prefecture, Japan (b) Higashi-Ohmi City
Figure 1: Figure shows the observation site of this study: Higashi-Ohmi city in Shiga pre-fecture, Japan. Green area shows Higashi-Ohmi city, where we exhibit the amount of datacollected specific to each locality.
For purpose of effective utilization of energy from decentralized power generation, re-searchers started exploring the possibility of electricity trading within and between regions[9, 3]. In this concept, we assume that electricity trading should be done automatically byan artificial intelligent agent to save a lot of humans’ costs. For this, developing the algo-rithms, which learn and predict the resident’s load profile, are needed. For the matter, before
developing above mentioned algorithms, it is essential to analyse and understand the patternof load profiles.
In this paper, we analyse the annual and regional load profiles which is from Higashi-Ohmi project. At first, we employ macroscopic analysis which aims to understand the season-ality effect on the energy consumption and PV generation. Second, we employ microscopicanalysis which aims to understand whether electricity consumption and PV generation showsnormality or not. At last, we extend the microscopic analysis through the use of macroscopicanalysis result.
2. Materials and Methods2.1. Materials
2.1.1. Load profile instrument
Electric load profiles of were measured by using an electric power meter ”CK-5” pro-duced by Chugoku Electrical Instruments Co., Ltd. [2, 8]. This can easily record electricload profiles every 15, 30, or 60 minutes. In this paper, load profile data were recorded every30 minutes during November 2010 to October 2011. The number of households is 72.
More specific experimental conditions are obtainable in Higashi-Ohmi’s Report [2].
2.1.2. A look at datas
Fig. 2 shows time course of energy consumption and energy generation from PV through-out year 2010-2011 in Higashi-Ohmi City. With this figure, we can find that the energy con-sumption reaches its peak in January and doesn’t have peaks in summer season. This patternare found commonly in cold climates [7]. The peak consumption is about twice that of thesummer seasons. Energy generation reaches its peak in summer season contrastingly.
Figure 2: Power consumption and generation in Higashi-Ohmi per day are shown. Eachline shows the median value (orange line), 25 - 75 percentile range (blue line), and 10 - 90percentile range (cyan line).
Fig. 3 shows am 6:30 consumption and am 9:30 generation of certain household. Fromthis figure, it seems there exists bimodal distribution. If we model a kind of bimodal data withmonomodal distribution, the case would occur that prediction algorithm underestimates con-sumption profiles and overestimates generation profiles. Then there would occur an electricalpower shortage in a worst case scenario. Thus it is important to understand the characteristicsof consumption and generation profiles.
Figure 3: The histogram of the energy consumption and generation of one household whichis extracted from the same time each day. Histogram exhibits bimodal distribution.
2.2. Macroscopic analysis
In macroscopic analysis, we propose a regression model for the time evolution of energyconsumption and generation (Fig. 2). Our data has a seasonalness, so we have to employsome kind of analytical method which is tailored to cyclic data. Few methods have developedfor satisfying this requirement in the field of time-series analysis such as Seasonal Auto-Regressive Integrated Moving Average (SARIMA)[5]. While these models are expressiveenough to regress seasonal and cyclic data, they are difficult to deal analytically.
In macroscopic analysis, we employ Null hypothesis that the data is from von Misesdistribution. After the regression estimation with maximum likelihood estimation, we testthis Null hypothesis with kuiper’s test; kuiper’s test is a method to discriminate whethera sample periodical data is from given cyclic distribution. These sequence are based ondirectional statistics which have been developed for the use of manipulating cyclic data, so wecan use a systematic method from directional statistics. Additionally, von Mises distributionis superior in terms of ease in analytical manipulation. Employing a function which allowsanalytical treatment provides the foundation for future development of designing, analysingand simulating the smart grid systems.
2.2.1. Directional Statistics and Regression Model
Equation (1) is known as von Mises distribution which is comparable to Gaussian distri-bution in polar coordinates (Fig. 4)[4]:
v(θ|µ, κ) = 1
2πI0(κ)exp (κ cos (θ − µ)) , (1)
where µ is a measure of most condensed location, and κ is a measure of concentration whichis called concentration parameter. I0(κ) is known as modified bessel function of order zerowhich is a kind of elliptic integrals [4]. This distribution is subject to eq (2)-(4):
v(θ) > 0 (2)
v(θ + 2π) = v(θ) (3)∮θv(θ|µ, κ)dθ = 1. (4)
In this study, we extend von Mises distribution as eq (5):
f(θ|µ, κ, α) = α+ (1− 2πα)v(θ|µ, κ), (5)
where α is a constant parameter. This model means the mixed distribution of von Misesdistribution and uniform distribution(Fig. 4). Due to the bounded domain of the integralinterval in directional statistics, we can mix the uniform distribution to any kind of cyclicdistributions as follows:∮
θf(θ|µ, κ, α)dθ =
∮θαdθ + (1− 2πα)
∮θv(θ|µ, κ)dθ (6)
= 2πα+ (1− 2πα) (7)
= 1. (8)
Figure 4: Typical example of distributions. Blue line is plotted under κ = 4, green is underκ = 1, and red is under κ = 0.25. Right is an extended von Mises distribution (α = 0.1).
2.2.2. Maximum likelihood estimation for extended von Mises distribution
For the standard von Mises distribution, Maximum likelihood estimation is used for pa-rameter estimation[6]. Likelihood function is defined as eq (9):
l(D|µ, κ) =N∏i=1
(1
2πI0(κ)exp(κ cos(θi − µ))
), (9)
for data D = (θ0, θ1, · · · , θN ). Owing to exponential family, we can simplify it with loglikelihood function as
L(D|µ, κ) = −N log(2πI0(κ)) +N∑i=1
(κ cos (θi − µ)) . (10)
Then we use standard method to derive optimized parameters:
dL
dκ=
I1(κ)
I0(κ)−
N∑i=1
(cos (θi − µ)) (11)
dL
dµ= −κ
N∑i=1
sin (θi − µ) , (12)
where I1(κ) is a modified bessel function of order one.Because our proposed extended von Mises distribution is not exponential family, we can-
not adapt above method and we should employ EM-algorithm like method. At first, we setα. Next, we optimize µ and κ. Then we adjust α and repeat optimizing parameters until
likelihood will maximized. In this algorithm, we firstly deform data f(D|µ, κ) according toeq (13) under given α:
F (θ|µ, κ) =f(θ|µ, κ, α)− α
(1− 2πα)
= v(θ|µ, κ). (13)
Then we can employ maximum likelihood method as mentioned above.
2.3. Microscopic analysis
In microscopic analysis, we focus each household’s daily energy consumptions and gen-erations separately. We analyse whether the data which is extracted from the same time eachday has a normality or not. Evaluation for the presence of normality will help us construct-ing an regression and prediction algorithm for the time course of energy consumption andgeneration.
2.3.1. Normality tests
In this analysis, we examine whether energy consumption and generation at the each timestep has normality or not. At first, we sort data by the clock time of observation throughout theyear. For example, we collect and cluster the data which is observed at am 9:30 throughoutthe year as is shown in fig. 3. After the above sorting, we get 72 x 48 x 2 data sets; which iscalculated as follows. The observation cycle is 30 minutes, so there are 48 steps in a day. Thenumber of households we measured is 72. Each house has two time-series data about energyconsumption and energy generation. At last, we examine the normality of each data sets byShapiro-Wilk test and JarqueBera test.
2.3.2. Normality tests with adjustment for seasonality
In this analysis, we examine normality with adjustment for seasonality. Fig. 2 implies thatthe data extracted from the same time throughout the year would be affected by seasonality.For example, bimodality in fig. 3 is suspected to be due to seasonality.
At first, we calculate averaged consumption and averaged generation per 30 minutes eachday. Second, we subtract them from every time step on the same day. Then we make the dataset in a similar way we explained in previous paragraph: we get 72 x 48 x 2 data sets. At last,we examine the normality of each data sets by Shapiro-Wilk test and JarqueBera test.
3. Results3.1. Results of Macroscopic analysis
Fig. 5 shows von Mises Regression.After this regression, we adapt kuiper’s test for the real data and regression curve. Result
shows p>0.05, so Null hypothesis wasn’t dismissed. This result doesn’t necessary mean thatreal data is from von Mises distribution, but means von Mises distribution as a some kind ofreasonable model.
3.2. Results of Microscopic analysis
3.2.1. Normality tests
We show results of normality tests. Fig. 6 shows the result of Shapiro-Wilk test and JBtest for the energy consumption, Fig. 7 for the energy generation.
Figure 5: Regression for energy consumption (left) and generation(right). Orange line showsthe real data and green shows the regression curve. Orange line is from the power consump-tion and generation per week.
Figure 6: Normality test for the energy consumption. Shapiro-Wilk test(left) and JBtest(right).
3.2.2. Normality tests with adjustment for seasonality
We present two types of adjustment for seasonality. First case, we subtracted the medianof the region, which is shown in orange line in fig. 2, from each hosehold’s load profiles.Second case, we subtracted the mean value of the each day and of each household load profilefrom each household’s load profiles. Fig. 8 and fig. 9 are from the first case. Fig. 10 and fig.11 are from the second case.
Figure 7: Normality test for the energy generation. Shapiro-Wilk test(left) and JB test(right).
Figure 8: Seasonality subtraction case 1: Normality test for the energy consumption withadjustment for seasonality. Shapiro-Wilk test(left) and JB test(right).
Figure 9: Seasonality subtraction case 1: Normality test for the energy generation with ad-justment for seasonality. Shapiro-Wilk test(left) and JB test(right).
Figure 10: Seasonality subtraction case 2:Normality test for the energy consumption withadjustment for seasonality. Shapiro-Wilk test(left) and JB test(right).
Figure 11: Seasonality subtraction case 2: Normality test for the energy generation withadjustment for seasonality. Shapiro-Wilk test(left) and JB test(right).
4. Conclusion and DiscussionIn this study, we presented one macroscopic analysis and one microscopic analysis. In
macroscopic analysis, we proposed the regression method for the median values of energyconsumption and generation in Higashi-Ohmi City. In microscopic analysis, we examinedwhether energy consumption and generation at the each time step has normality or not. Atfirst, there were few households which have normality in consumption data and generationdata. However, subtraction of seasonality increased the number of households which havenormality.
Our study suggests that subtracting seasonality would contribute to construct the learningand predicting algorithm: it permits us to use the normality-based algorithms. Our study alsoshows that von Mises distribution is usable to represent median time course of consumptionand generation in Higashi-Ohmi City. There remains other questions that whether time courseof consumption and generation in other cities can be expressed by the von Mises distribution.It is expected that there exist some regions whose time course of consumption are expressedby mixed-bimodal von Mises distribution: there would be very cold at winter and very hot atsummer.
As a future work, more research is required to analyse the profile of each households.
5. ACKNOWLEDGEMENTSThis work was supported by Grant in Aid for Regional Innovation Strategy Support Pro-
gram 2011 for Promotion of Environmental Industries in Biwako Region by The Ministry ofEducation, Culture, Sports, Science and Technology (MEXT).
References[1] New Growth Strategy Blueprint for Revitalizing Japan, Jun. (2010).
[2] Report for green decentralization reforms in Higashi-Ohmi City, Feb. (2011).
[3] i-rene web site, [Online] Available: http://www.i-rene.org/
[4] N.I. Fisher, Statistical analysis of circular data, Cambridge Univ Pr. (1996)
[5] J.D. Hamilton, Time series analysis, Cambridge Univ Pr. (1994)
[6] S.R. Jammalamadaka, and A. Sengupta, Topics in circular statistics,World ScientificPub. Co. Inc. (2001)
[7] Portal Site of Official Statistics of Japan (Electrical Power Demand Research), NationalStatistics Center, (2012)
[8] “Chugoku electrical instruments co., ltd.” http://www.chukeiko.co.jp/01product/energy/ck-5/
[9] P. Vytelingum and T.D. Voice and S.D. Ramchurn and A. Rogers and N.R. Jennings,“Agent-based micro-storage management for the smart grid”, Proceedings of the 9thInternational Conference on Autonomous Agents and Multiagent Systems: volume 1,39-46 (2010)