Efficient representation of supply and demand curves on day-ahead electricity markets
EEfficient representation of supply and demand curves on day-aheadelectricity markets
Mariia Soloviova ∗ Tiziano Vargiolu † Abstract
Our paper aims to model supply and demand curves of electricity day-ahead auction in a parsimoniousway. Our main task is to build an appropriate algorithm to present the information about electricity pricesand demands with far less parameters than the original one. We represent each curve using mesh-freeinterpolation techniques based on radial basis function approximation. We describe results of this methodfor the day-ahead IPEX spot price of Italy.
Accurate modeling and forecasting electricity demand and prices are very important issuesfor decision making in deregulated electricity markets. Different techniques were developedto describe and forecast the dynamics of electricity load. Short term forecast proved to bevery challenging task due to these specific features. Figure 1 and 2 demonstrate changingof electricity equilibrium price and quantity during one week. Functional data analysis isextensively used in other fields of science, but it has been little explored in the electricitymarket setting.
Figure 1: Electricity equilibrium prices dur-ing a week Figure 2: Electricity equilibrium quantitiesduring a week ∗ Department of Mathematics “Tullio Levi Civita”, University of Padua; [email protected] † Department of Mathematics “Tullio Levi Civita”, University of Padua and Interdepartmental Centre for Energy Economics andTechnology “Giorgio Levi-Cases”, University of Padua; [email protected] a r X i v : . [ q -f i n . P R ] F e b e consider the Italian electricity market (IPEX). IPEX consists of different markets, includ-ing a day-ahead market. The day-ahead market is managed by Gestore del Mercato Elettricowhere prices and demand are determined the day before the delivery. Supply and demandcurves on day-ahead electricity markets are the results of thousands of bid and ask entries inthe day-ahead auction, this for all the 24 hours. In principle, it would be possible to represent,and forecast, these curves by taking into account each production and each consumption unitas a separate time series, and then joining these together to construct the final curves, andthus the resulting price. However, the huge number of these units makes this naive strategyinfeasible, unless one has extremely high computing capacity with complex machine learningalgorithms available.In this paper, we are going to present a more parsimonious approach. In fact, the idea isto represent each curve using non-parametric mesh-free interpolation techniques, so that wecan obtain an approximation of the original curve with far less parameters than the originalone. The original curve, in fact, in principle depends on about hundreds of parameters and isobtained as follow.The producers submit offers where they specify the quantities and the minimum price atwhich they are willing to sell. The demanders submit bids where they specify the quantitiesand the maximum price at which they are willing to buy. They are then aggregated by anindependent system operator (ISO) in order to construct the supply and demand curves. Oncethe offers and bids are received by the ISO, supply and demand curves are established bysumming up individual supply and demand schedules. In the case of demand, the first step isto replace ”zero prices“ bids by the market maximum price (for Italian electricity market, themarket maximum price is 3000 Euro) without changing the corresponding quantities. Afterthis replacement, the bids are sorted from the highest to the lowest with respect to prices.The corresponding value of the quantities is obtained by cumulating each single demand bid.For supply curve, in contrast, the offers are sorted from the lowest to the highest with respectto prices and the corresponding value of the quantities is obtained by cumulating each singlesupply offer. The market equilibrium is the point where both curves intersect each other andthe price balances supply and demand schedules (see, e.g. Figure 3). This point determinesthe market clearing price and the traded quantity. Accepted offers and bids are those that fallto the left of the intersection of the two curves, and all of them are exchanged at the resultedprice.In the beginning of the 2000s the amount of papers focused on electricity price forecastingstarted to increase dramatically. A great variety of methods and models occurred during lasttwenty years. Weron [17] (2014) made an overview of the existing literature on electricityprice forecasting and divided electricity price models into five different groups: multi-agent,fundamental, reduced-form, statistical and computational intelligence models. A review ofprobabilistic forecasting was done in [10] (2018) by Weron and Nowotarski. Most models have in common that they focus on the price itself or related time series. In such a way these modelsdoes not take into account the underlying mechanic which determines the price process – theintersection between the part of the electricity supply and demand.Some of the recent approaches try to to analyse the real offered volumes for selling andpurchasing electricity. This commonly leads to a problem of a large amount of data and,therefore, high complexity. In particular, Eichler, Sollie, Tuerk in 2012 [5] investigated a newapproach that exploits information available in the supply and demand curves for the Germanday-ahead market. They proposed the idea that the form of the supply and demand curvesor, more precisely, the spread between supply and demand, reflects the risk of extreme pricefluctuations. They utilize the curves to model a scaled supply and demand spread using anautoregressive time series model in order to construct a flexible model adapted to changing arket conditions. Furthermore, Aneiros, Vilar, Cao, San Roque in 2013 [2] dealt with theprediction of residual demand curve in elecricity spot market using two functional models.They tested this method as a tool for optimizing bidding strategies for the Spanish day-aheadmarket. Then Ziel and Steinert in 2016 [18] proposed a model for the German EuropeanPower Exchange (EPEX) market, which considers all the supply and demand information ofthe system and discusses the effects of the changes in supply and demand. Their idea was tofill the gap between research done in time-series analysis, where the structure of the market isusually left out, and the research done in structural analysis, where empirical data is utilizedvery rarely and even less thoroughly. They provided deep insight on the bidding behavior ofmarket participants. They also showed that incorporating the sale and purchase data yieldspromising results for forecasting the likelihood of extreme price events. In 2016 Shah [14] alsoconsidered the idea of modeling the daily supply and demand curves, predicting them andfinding the intersection of the predicted curves in order to find the predicted market clearingprice and volume. He used the functional approach, namely, B-spline approximation, to convertthe resulted piece-wise constant curves into smooth functions.As far as we know, non-parametric mesh-free interpolation techniques were never consideredfor the problem of modeling the daily supply and demand curves.We are going to use a relatively new modeling technique based on functional data analysisfor demand and price prediction. The first task for this purpose is to make an appropriatealgorithm to present the information about electricity prices and demands, in particular toapproximate a monotone piecewise constant function.We want to make an appropriate algorithm to present this information, in particular, toapproximate a monotone piecewise constant function. Accuracy of the approximation andrunning time are very important for us. As we already said, the basic novelty of our problemis that we are going to present the information about electricity prices and demands usingfunctional data analysis approach. The main idea behind functional data analysis is, instead ofconsidering a collection of data points, to consider the data as a single structured object. Thisallows to use additional information contained in the functional structure of the data. Oncethe data are converted to functional form, it can be evaluated at all values over some interval.The most promising technique to do so is the use of (integrals of) Radial Basis Functions,which are been used in several other applications (image reconstruction, medical imaging,geology, etc.) and allow a very flexible adaptation of the interpolating curves to real data.The use of radial basis functions have attracted increasing attention in recent years as anelegant scheme for high-dimensional scattered data approximation, an accepted method formachine learning, one of the foundations of meshfree methods and so on. The initial motivationfor RBF methods came from geodesy, mapping, and meteorology. RBF methods were firststudied by Roland Hardy, an Iowa State geodesist, in 1968, when he developed one of thefirst effective methods for the interpolation of scattered data. Later in 1986 Charles Micchelli, n IBM mathematician, developed the theory behind the multiquadric method. Micchellimade the connection between scattered data interpolation and positive definite functions [9].RBF methods are now considered an effective way to solve partial differential equations, torepresent topographical surfaces as well as other intricate three-dimensional shapes, having beensuccessfully applied in such diverse areas as climate modeling, facial recognition, topographicalmap production, auto and aircraft design, ocean floor mapping, and medical imaging (see, forexample, [4], [7], [11]). Now RBF methods are an active area of mathematical research, asmany open questions still remain. We will present different techniques for this interpolation,with their advantages and drawbacks, and with an application to the Italian day-ahead market.The paper is organized as follow. Section 2 describes the theoretical background, namely,mesh-free interpolation techniques based on radial basis function approximation. Section 3presents the database from the Italian electricity market. Section 4 is devoted to a shortdescription of the numerical schemes and to the analysis of the results. Section 5 concludes thepaper. Let us briefly notice some features of supply and demand curves that are relevant for ourmodeling:• By construction, the curves are monotone.• The values attained by the supply curve are roughly clustered around layers , correspond-ing to different production technologies. In Italy they are non-dispatchable renewables,gas, coal, hydro, oil.• The fact that renewables are the first ones make the supply curve intrinsically "meshless".• Demand is much more inelastic than supply.So, we are dealing with a scattered data interpolation problem. We have a large amount ofpoints (each point represents price and amount of electricity) that we want to approximate.We can formalize this problem as follows.Given a set of N distinct data points X N = { x i : i = 1 , , . . . , N } arbitrarily distributed ona domain Ω ⊂ R and a set of data values (or function values) Y N = { y i : i = 1 , , . . . , N } ⊂ R ,the data interpolation problem consists in finding a function s f : Ω → R such that s f ( x i ) = y i , i = 1 , . . . , N. (2.1)Let us recall briefly the most popular methods for the interpolation problem. Polynomialinterpolation is the interpolation of a given data set by the polynomial of lowest possible degreethat passes through the points of the dataset. For given data sites X N and function values Y N here exists exactly one polynomial p ∈ π N − ( R ) that interpolates the data at the data sites.Therefore the space π N − ( R ) depends neither on the data sites nor on the function values butonly on the number of points.Runge’s phenomenon (1901) shows that for high values of N , the interpolation polynomialmay oscillate wildly between the data points. Besides, the polynomial interpolation does notguarantee of monotonicity of the curves (see Figure 4). Figure 4: Approximation of supply curve with polynomials
It is a well-established fact that a large data set is better dealt with splines than with polyno-mials. An aspect to notice in contrast to polynomials is that the accuracy of the interpolationprocess using splines is not based on the polynomial degree but on the spacing of the datasites. In particular, cubic splines are widely used to fit a smooth continuous function throughdiscrete data. However, spline interpolation requires a mesh.Notice that for all methods, the interpolant s f is expressed as a linear combination of somebasis functions B i , i.e. s f ( t ) = d (cid:88) k =1 c k B k ( t ) . The basis functions in e.g. polynomial interpolation does not depend on the data points. nother approach is to use a basis which depends on the data points.One simple way to solve problem (2.1) is to choose a fixed function φ : R → R and to formthe interpolant as s f ( x ) = N (cid:88) i =1 α i φ ( (cid:107) x − x i (cid:107) ) , where the coefficients α i are determined by the interpolation conditions s f ( x i ) = y i . Therefore,the scattered data interpolation problem leads to the solution of a linear system Aα = y, where A i,j = φ ( | x i − x j | ) . The solution of the system requires that the matrix A is non-singular. It is enough to knowin advance that the matrix is positive definite (see [16] for more details). Let us recall thedefinition of strictly positive definite function. Definition 2.1.
A real-valued function
Φ : R −→ R is called positive semi-definite if , for all m ∈ N and for any set of pairwise distinct points x , x , . . . , x m , the m × m matrix A = (Φ( x i − x j )) mi,j =1 is positive semi-definite, i.e. for every column vector z of m real numbers the scalar z T Az (cid:62) .The function Φ : R −→ R is called (strictly) positive definite if the matrix A is positive definite,i.e. for every non-zero column vector z of m real numbers the scalar z T Az > .The most important property of positive semi-definite matrices is that their eigenvalues arepositive and so is its determinant.A radial function is a real-valued positive semi-definite function whose value depends only onthe distance from the center c . One useful characterization for positive semi-definite univariatefunction was given by Schoenberg in 1938 in the terms of completely monotone functions: acontinuous function φ : [0 , ∞ ) → R is positive semi-definite if and only if φ ∈ C ∞ (0 , ∞ ) and ( − k φ ( k ) ( r ) (cid:62) for all r (cid:62) , for k = 0 , , . . . .Some standard radial basis functions are• φ ( r ) = e − ( εr ) (Gaussian),• φ ( r ) = e − εr ( εr + 1) (Mat´ern),• φ ( r ) = (1 − εr ) (4 εr + 1) (Wendland),where ε > denote a shape parameter, r = (cid:107) x (cid:107) .The idea of meshless approximation with radial basis functions is to find an approximant of f in the following form: s f ( x ) := N (cid:88) i =1 α i φ ( (cid:107) x − x i (cid:107) ) where: the coefficients α i and the centers x i are to be chosen so that the interpolant is as nearas possible as the original function f ;• φ : R → R is a radial basis function (RBF).Notice that the radial basis function φ (cid:62) , with α i (cid:62) , so M (cid:88) i =1 α i φ ( (cid:107) x − x i (cid:107) ) (cid:62) . As we need to approximate piecewise constant monotone function from [0 , M ] to R + , we decidedto use the integrals of RBF. Namely, we want to find an approximant of the form s f ( t ) = (cid:90) t M (cid:88) i =1 α i φ ( λ i (cid:107) x − x i (cid:107) ) dx = M (cid:88) i =1 α i (cid:90) t φ ( λ i (cid:107) x − x i (cid:107) ) dx where λ i is a shape parameter for every center x i . As radial basis functions, we choose Gaussianfunctions for analytical tractability.Evidently, any supply curve and any demand curve can be approximated by a combinationof error functions, which is the integral of a normalized Gaussian function. The standard errorfunction is defined as: erf( x ) = 1 √ π (cid:90) x − x e − t dt = 2 √ π (cid:90) x e − t dt. In order to find unknown coefficients α i , λ i , x i we need to solve global minimization problem: min p (cid:107) s f ( x i , p ) − y i (cid:107) , where p = ( α i , λ i , x i ) i =1 ,...,N and s f ( t, p ) := M (cid:88) i =1 α i (cid:90) t φ ( λ i (cid:107) x − x i (cid:107) ) dx and φ ( t ) = (erf( t ) + 1) / is the primitive of a Gaussian kernel. However, this optimizationproblem is very heavy, as it is a nonlinear and nonconvex minimization over p ∈ R M .For this reason, we divide our global problem in simpler subproblems, with lower dimen-sionality, so that the final result is faster. We describe two realization of this approach inSection 4. ith the same price, are aggregated in the price layer. Even in this form, we are dealing with amassive amount of data. For instance, offer and
558 926 bid layers were observedduring this period.
Table 1: DataDate Hour Volume (MW) Price (Euro)01-01-2017 1 13392.7 001-01-2017 1 25 0.101-01-2017 1 113.8 101-01-2017 1 11 3.501-01-2017 1 270.3 501-01-2017 1 0.5 6.................. ...... ...................... ....................31-12-2017 24 370 554.231-12-2017 24 352 554.331-12-2017 24 365 554.531-12-2017 24 97 70031-12-2017 24 60000 3000
This means, that on average there are 324 offer and 65 bid layers for each hour of the year,which corresponds to one supply curve and one demand curve respectively.It is a known fact that the dynamics of electricity trade displays a set of characteristics:external weather conditions, dependence of the consumption on the hour of the day, the dayof the week, and time of the year. Variation in prices are all dependent on the principles ofdemand and supply. First of all, on the day-ahead market the energy is typically traded on anhourly basis and this means that the prices can and will vary per hour. For example, at 9:00a.m. there could be a price peak, while at 4:00 a.m. prices could be only half of the peak price.Second, the weekly seasonal behaviour matters. Usually, it is necessary to differentiate betweenthe two weekend days (Saturday and Sunday), the first business day of the week (Monday), thelast business day of the week (Friday) and the remaining business days (see e.g. [1]). Thirdly,electricity spot prices display a strong yearly seasonal pattern: for instance, demand increasesin summer, as consumers turn their air conditioners on, and also in winter because of electricheating in housing.As far as the number of offers (or bids) affects directly the complexity of approximation,we decided to explore the relationship between the number of bids and offers and such acharacteristics as the hour of the day, the day of the week, and the month of the year. Basedon the dependence between this three factors and electricity prices we could expect that somehours, days have much less offers and bids than another one. This analysis is presented onFigures 5 – 7.The main conclusion that we have made is that there is no direct relationship between the umber of offer and bid layers and the hour of the day, the day of the week, and the time ofthe year. In particular, during 24 hour of the day the number of offer layers varies between 299and 332, and the number of bid layers varies between 61 and 66. With regard to dependence ofthe day of the week the number of offer layers varies between 310 and 320, and the number ofbid layers varies between 55 and 68. Based on this observation we decided to choose the samenumber of basis functions independently of the hour of the day, the day of the week, and thetime of the year. Hour Number Hour Numberof offers of bids of offers of bids
300 64
329 64
299 64
329 64
300 64
330 64
300 64
332 64
301 63
332 63
303 63
332 63
307 62
331 64
318 63
329 65
325 65
329 66
326 64
323 64
329 64
321 63
329 65
314 61Figure 5: Hour dependence of the number of offer and bid layersMonth Numberof offers of bids
Sunday
310 55
Monday
310 56
Tuesday
322 68
Wednesday
324 67
Thursday
326 68
Friday
327 68
Saturday
329 68Figure 6: Weekly dependence of the number of offer and bid layers10onth Numberof offers of bids
January
331 65
February
341 79
March
324 81
April
305 72
May
298 57
June
298 54
July
322 55
August
305 58
September
300 64
October
309 66
November
348 58
December
357 57Figure 7: Monthly dependence of the number of offer and bid layers
Since the maximum market clearing price for the period under review (i.e. from 01.01.2017to 31.12.2017) is 350 e , in all the experiments we restricted ourselves to a maximum priceof 400 e . For the realization of our algorithm we are using the function lsqcurvefit fromMATLAB Optimization Toolbox.First, we download the data from a text file and choose the number of basis function M .After that, we need to divide our problem into M sub-problems. Then each part of the supplycurve must be approximated by one error function.Our first attempt (Method 1) was just to divide y -axis uniformly into M equal intervals (seeFigure 8). However this approach is ineffective, as a huge jump concentrates on itself, keepinguselessly many components.To resolve this problem we created a simple algorithm - Method 2 - that finds the points p , . . . , p M on the y -axis such that our supply curve takes the value exactly p i on some non-trivialinterval (see Figure 9).Then M times we resolve the same optimization problem for the values of the supply curvebetween p i and p i +1 using function lsqcurvefit (see Figure 10). On each part we need to findonly 3 coefficients a i , b i , c i of the function G ( x ) = k (cid:88) i =1 a i (erf( c i · ( x − b i )) + 1) . (4.1) Here, for convenience of representation we are using { erf( c i · ( x − b i )) + 1 } instead of { erf( c i · ( x − b i )) } , as our data values are never negative.The lsqcurvefit function solves nonlinear data-fitting problems in least-squares sense.Suppose that we have data points X N = { x i : i = 1 , , . . . , N } and data values Y N = { y i : i = 1 , , . . . , N } ⊂ R and we want to find a function f such that f ( x i ) ≈ y i , i = 1 , . . . , N. Wecan consider the family of functions { f ( x, p ) : p ∈ R k } , depending of some parameter p ∈ R k .Let p ∈ R k be an “initial guess” such that f ( x i , p ) is reasonably close to y i . The function lsqcurvefit starts at p and finds coefficients p from some neighborhood of p to best fit thedata set Y N : min p (cid:107) f ( x i , p ) − y i (cid:107) . Notice that this function works well only if the number of parameters ( p , . . . , p k ) is not verybig. That is why we are forced to divide our problem into many local problems.For optimizing the numerical procedure we solved some parts of the optimization problemby ourselves: in fact, when the interval [ p i , p i +1 ] contains only one jump, then a i := f ( p i +1 ) − f ( p i ) for any kernel function φ with unit integral. Figure 10: Local interpolation by one error function with lsqcurvefit function12 summary of the results is shown in Table 2. For all experiments we proceed with thedata for period from 01.01.2017 to 31.12.2017. We used different number of basis function toapproximate supply and demand curves, and then compared the equilibrium price, which wasreceived as intersection of approximants ( P appr ), with the correct equilibrium price ( P ). Wedid this for each hour of each day, and then computed the average value of | P − P appr | (Error)for all 8 664 hours of the year and the maximum value of | P − P appr | (Max error).This empirical results show that the accuracy of our approximation is good enough, if weuse 5 basis function for the demand curve and 15 basis function for the supply curve. Then theincrease in the number of functions leads to more time consumption, but the increase of theaccuracy is less significant. Table 2: Results of numerical experimentNumber of functions ResultsFor demand For supply Error Max error Running time5 5 3.9 e e
69 min.5 10 2.2 e e
82 min.5 15 1.5 e e
103 min.5 20 1.3 e e
110 min.5 25 1.2 e e
135 min.5 30 1.2 e e
159 min.5 35 1.2 e e
177 min.5 40 1.2 e e
190 min.5 45 1.2 e e
199 min.5 50 1.2 e e
207 min.10 5 3.9 e e
100 min.10 10 2.1 e e
128 min.10 15 1.4 e e
146 min.10 20 1.2 e e
162 min.10 25 1.1 e e
183 min.10 30 1.1 e e
199 min.10 35 1.0 e e
223 min.10 40 0.98 e e
241 min.10 45 0.98 e e
255 min.10 50 0.98 e e
273 min.
As a last step we analyzed the stability of the coefficients for the case when we approximatethe supply curve with 10 basis functions and the demand curve with 5 basis functions for thesame period of time, as S ( x ) = (cid:88) i =1 A i (erf( C i · ( x − B i )) + 1) and D ( x ) = (cid:88) i =1 E i (erf( K i · ( x − L i )) + 1) . rom Table 3 we can see that these coefficients do not have a stable behavior (namely, maxi-mum values, minimum values and mean values are presented). Although the values attained bythe supply curve are clustered around layers, which correspond to different production technolo-gies, we came to the conclusion that we have no chance to choose these coefficients uniformlyfor all curves, but we need to calculate them for all supply and demand curves. Figure 11: Supply curve approximated with 10 basis functions14able 3: Stability of the coefficientsMin Mean MaxCoefficients for supply curve A
10 14.76981 18 A A A
11 15.53944 22 A
11 16.8968 27.5 A A A
19 29.69132 57.5 A
17 24.48784 48 A
21 25.64777 50Coefficients for demand curve E
12 30.95154 37.5 E
25 34.31039 58.5 E
25 36.24469 50 E
33 40.19715 50 E
50 58.29623 75
We presented a parsimonious way to represent supply and demand curves, using a mesh-free method based on Radial Basis Functions. Using the tools of functional data analysis, weare able to approximate the original curves with far less parameters than the original ones.Namely, in order to approximate piece-wise constant monotone functions, we are using linearcombinations of integrals of Gaussian functions.The real data about supply and demand bids from the Italian day-ahead electricity marketshowed that there is no direct relationship between the number of offer and bid layers and thehour of the day, the day of the week, and the time of the year. Based on this observation, wedecided to choose the same number of basis functions independently of the hour of the day, theday of the week, and the time of the year.The numerical results showed that the accuracy of our approximation is good enough, if weuse 5 basis function for the demand curve and 15 basis function for the supply curve, and thenthe increase in the number of functions leads to more time-consumption, but the increase ofthe accuracy is less significant. cknowledgements The authors thank Enrico Edoli, Marco Gallana, and Emma Perracchione for several usefuldiscussions. The authors wish to thank also Florian Ziel, Carlo Lucheroni, Stefano Marmi,Sergei Kulakov, Enrico Moretto for their comments and suggestions. The authors would liketo thank the participants of the following events: Energy Finance Christmas Workshop (2018)in Bolzano; Quantitative Finance Workshop (2019) in Zurich; Energy Finance Italia Workshopin Milan (2019), the Freiburg-Wien-Zurich Workshop (2019) in Padova.The first author is pursuing her Ph.D. with a fellowship for international students fundedby Fondazione Cassa di Risparmio di Padova e Rovigo (CARIPARO) and acknowledges thesupport of this project. The second author acknowledges financial support from the researchprojects of the University of Padova BIRD172407-2017 "New perspectives in stochastic meth-odsfor finance and energy markets" and BIRD190200/19 "Term Structure Dynamics in InterestRate and Energy Markets: Modelling and Numerics".
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