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Laboratory for Information & Decision Systems (LIDS) MIT September 21st, 2016 Estimation of demand and supply functions for spot electricity prices in JEPX (Japan Electric Power Exchange) Y. Yamada Faculty of Business Sciences University of Tsukuba, Tokyo, Japan E-mail: [email protected] http://www2.gssm.otsuka.tsukuba.ac.jp/staff/yuji/ 1 Acknowledgement Supported by Grant-in-Aid for Scientific Research (A) 16H01833 (PI: Yuji Yamada) from Japan Society for the Promotion of Science (JSPS). Collaborators Naoki Makimoto(a), Setsuya Kurahashi(a) Ryuta Takashima(b), Nobuyuki Yamaguchi(c) Junya Goto(d) (a)Faculty of Business Sciences, University of Tsukuba (b)Faculty of Science and Technology, Tokyo University of Science (c)Faculty of Engineering Division I, Tokyo University of Science (d)Faculty of Science and Engineering, Chuo University 2 What are exchange traded markets? Exchange-traded markets are the one in which all transactions are routed through a central source. Investors Stock exchange Investors 3 Why exchange traded market? Match sell and buy orders efficiently and effectively Example of orders Admissible price and volume Price Sell ≥ 325 215 324 175 5 323 145 5 30 10 322 120 15 321 20 25 321 90 40 320 35 30 320 70 70 319 20 25 319 35 95 318 10 20 318 15 115 317 5 10 317 5 125 316 30 316 155 ≤ 315 55 ≤ 315 210 Price Sell ≥ 325 40 324 30 323 25 322 Buy Buy 4 Visualization of price-volume relationship Volume (shares) 250 200 150 100 Maximum volume 70 50 0 314 316 318 320 322 324 326 Price ($) Admissible price The intersection of price-volume functions provides the maximum volume to be traded in the market 5 Power exchange market Centralized place for selling/buying electricity Sellers Buyers Power generation companies Wholesale companies Electric power exchange Sell orders Buy orders Demanders Supply Power generation companies (sellers) and wholesale companies (buyers) can trade contracts for producing and supplying electricity 6 Japan Electric Power Exchange: JEPX (http://www.jepx.org/) Match sell and buy orders for delivering a fixed amount of electricity (kWh) in nine areas, Hokkaido, Tohoku, Tokyo, Chubu, Hokuriku, Kansai, Chugoku, Shikoku, Kyushu From short term (5min, 30min) to long term (1 week, 1 month) Example of 30 min sales order: Sell 12 units (1 unit = 0.5MWh/0.5h) for 10 yen/kWh in the period of 2:00—2:30 pm on September 21st 7 Japan Electric Power Exchange: JEPX (http://www.jepx.org/) 30 min fixed delivery traded everyday Spot electricity in JEPX - 48 products for 24 hours a day. All the orders are closed at a specified time one day before the delivery. - Transaction price and volume are determined by constructing selling and buying volume-price functions. 8 Example of spot electricity transactions Spot market for 14:00—14:30 Merketer A Merketer B Merketer C 14:00~14:30 14:00~14:30 14:00~14:30 7.00 8.50 9.00 (price) 13 7 5 7.50 8.10 15 6 7.00 7.20 9.00 (price) ▲7 ▲12 ▲25 (vol.) (vol.) (price) Buy order (vol.) Sell order Vol.(,000kWh/h) 30 Buy (A, B) Sell (C) (Transaction price) 8.10yen/kWh (Transaction vol.) 12,000kWh/h 20 Total buy order vol. Total sell order vol. 10 7.00 8.00 9.00 Price (yen/kWh) 9 Volume-Price functions Total sell order vol. Transaction vol. & price Supply & Demand functions Total buy order vol. - Analysis of electricity market structure and price jumps - Optimal strategies for power generation and whole sale companies Partial information is available only such as transaction vol. and price 10 Outline Two approaches for estimating supply & demand functions 1. Problem difficulties and basic idea of our solution 2. Two approaches using additional information Sell & Buy matching rates approach Parametric equations approach 3. Estimation models 4. Empirical results and comparison of the two approaches 5. Concluding remarks 11 Difficulty 1 Two functions cannot be identified separately from the single equilibrium point Supply? or Demand? 12 Difficulty 2 Generation stack has a strong nonlinearity (https://www.e-education.psu.edu/ebf200wd/node/151) Linear models may be inadequate Use spline regression? 13 Difficulty 3 Smoothing spline functions are not monotonic! yn 5 4 3 2 Cubic spline func. 1 Data points 0 1 後藤 , 山田 ( 中央大 , 筑波大 ) 0 1 2 3 4 5 6 xn 14 Use additional information to overcome D1 Total selling and buying order volumes are released by JEPX Total sell order vol. Transaction vol. & price Total buy order vol. 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. Sell matching rate = , Buy matching rate = 𝑇𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. 𝑇𝑇𝑇𝑇𝑇 𝑏𝑏𝑏 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. (BMR) (SMR) 15 Matching rates approach Volume-price functions with sell/buy order volume coordinates Rate-price function with sell/buy order volume per total coordinates Total sell order vol. Total sell order vol. Divide by total to transfer to rate “Volume-price function” “Rate-price function” Multiply total to transfer to vol.16 Parametric equations approach Use parametric representations of supply and demand functions 𝒚 Supply function on 𝒙-𝒚 plane 𝒚 = 𝒇(𝒙) 𝒚 Parametric representation of 𝒛 𝒙, 𝒚 = (𝒉𝒙 𝒛 , 𝒉𝒚 (𝒛)) 𝟎 𝒙 𝟎 𝒙 Find a new variable that reflect on price and volume for supply or demand. 17 Parametric equations approach The shapes of two functions are assumed to be indifferent over time 𝒙𝟏 , 𝒚𝟏 = 𝒉𝒙 𝒛𝟏 , 𝒉𝒚 𝒛𝟏 Volume [MWh] 𝒛𝟏 = Total sell order vol. Power price (Yen/kWh) Power price (Yen/kWh) If the total sell order vol. 𝒛𝟏 changes, the demand function may be shifted and a different coordinate point on the supply function is observed. 𝒛𝟐 = Total buy order vol. 𝒙𝟐 , 𝒚𝟐 = 𝒌𝒙 𝒛𝟐 , 𝒌𝒚 𝒛𝟐 Volume [MWh] 18 Apply spline regressions to overcome D2 Cubic smoothing splines: yn = f ( xn ) + ε n , mean(ε n ) = 0 Minimizes the Penalized Residual Sum of Squares (PRSS) PRSS = ∑ {yn − f ( xn )} + λ ∫ { f " (u )} du 2 2 n λ →∞ - λ may be found by minimizing the cross validation sum of squares: - Can be applied to the sum of smooth functions characterized as Generalized Additive Models (GAMs; Hastie and Tibshirani ’90) m ( ) yn = ∑ f i xn(i ) + ε n , i =1 19 Add monotonic constraint to overcome D3 Cubic spline w/o monotonicity: 𝑁 1 𝑓 𝑥 ≔ 𝛾0 + 𝛾1 𝑥 + � 𝜃𝑛 |𝑥 − 𝑥𝑛 |3 12 𝑛=1 𝛾0 , 𝛾1 , 𝜃1 , … , 𝜃𝑛 ∈ ℜ 𝑥1 ≤ 𝑥2 ≤ ⋯ ≤ 𝑥𝑁 : Given data points 𝑁 Objective function: min � 𝑦𝑛 − 𝑓(𝑥𝑛 ) 2 + 𝜆 � 𝑓𝑓𝑓(𝑢) 2 𝑑𝑑 𝛾0 ,𝛾1 ,𝜃𝑛 𝑛=1 Convex QP + Monotonic constraint based on prior information Monotonic cubic spline regression 20 Monotonicity condition Monotonic on [𝑥𝑖 , 𝑥𝑖+1 ) ⇔ 𝑓 ′ 𝑥 = 𝛼𝑖,0 + 𝛼𝑖,1 𝑥 + 𝛼𝑖,2 𝑥 2 ≥ 0 yn 𝑥𝑖 𝑥𝑖+1 5 𝛼𝑖,0 4 3 2 Data points 0 1 0 1 2 3 4 5 6 xn 𝑖 𝑁 1 = 𝛾1 + � 𝜃𝑛 𝑥𝑛2 − � 𝜃𝑛 𝑥𝑛2 4 𝑖 𝑛=1 𝑛=𝑖+1 𝑁 𝛼𝑖,1 1 =− � 𝜃𝑛 𝑥𝑛 − � 𝜃𝑛 𝑥𝑛 2 𝛼𝑖,2 1 = � 𝜃𝑛 − � 𝜃𝑛 4 Cubic spline func. 1 (or ≤ 0) 𝑖 𝑛=1 𝑛=1 𝑁 𝑛=𝑖+1 𝑛=𝑖+1 21 Bertsimas & Popescu (2002) transformation ∀𝑥 ∈ 𝐿, 𝑈 , 𝛼0 + 𝛼1 𝑥 + 𝛼2 𝑥 2 ≥ 0; 𝛽00 = 𝛼0 + 𝐿𝛼1 + 𝐿2 𝛼2 , 𝛽11 + 2𝛽02 = 2𝛼0 + (𝐿 + 𝑈)𝛼1 + 2𝐿𝐿𝛼2 , 𝛽22 = 𝛼0 + 𝑈𝛼1 + 𝑈 2 𝛼2 , 𝛽00 𝑍= 0 𝛽02 0 𝛽11 0 𝛽02 0 𝛽22 ≽0 𝛽00 , 𝛽11 , 𝛽22 ≥ 0, 𝛽𝑖00 + 𝛽𝑖22 ≥ 2 𝛽𝑖00 − 𝛽𝑖22 2 𝛽𝑖02 Second order cone condition 2 22 Monotonic cubic spline regression Gotoh & Yamada (2015): 𝑁 min � 𝑦𝑛 − 𝑓(𝑥𝑛 ) 𝛾0 ,𝛾1 ,𝜃𝑛 s.t. 𝑛=1 2 + 𝜆 � 𝑓𝑓𝑓(𝑢) 2 𝑑𝑑 𝑁 ∑𝑁 𝑛=1 𝜃𝑛 = 0, ∑𝑛=1 𝜃𝑛 𝑥𝑛 = 0, 𝛾1 ± 𝛽𝑖𝑖𝑖 𝑖 𝑁 𝑛=1 𝑛=𝑖+1 𝑁 1 � 𝑥𝑛2 𝜃2 ≥ 0 4 𝑛=1 1 1 2 2 = 𝛾1 + � 𝑑𝑛,𝑖 𝜃𝑛 − � 𝑑𝑛,𝑖 𝜃𝑛 (𝑑𝑛,𝑖 ≔ 𝑥𝑛 − 𝑥𝑖 ) 4 4 𝛽𝑖𝑖𝑖 + 2𝛽𝑖𝑖𝑖 𝑖 𝑁 1 1 = 2𝛾1 + � 𝑑𝑛,𝑖 𝑑𝑛,𝑖+1 𝜃𝑛 − � 𝑑𝑛,𝑖 𝑑𝑛,𝑖+1 𝜃𝑛 2 2 𝑛=1 𝑖 𝑁 𝑛=𝑖+1 1 1 2 2 𝛽𝑖𝑖𝑖 = 𝛾1 + � 𝑑𝑛,𝑖+1 𝜃𝑛 − � 𝑑𝑛,𝑖+1 𝜃𝑛 4 4 𝛽𝑖00 +𝛽𝑖22 2 ≥ 𝑛=1 𝛽𝑖00 −𝛽𝑖22 2 𝛽𝑖02 𝑛=𝑖+1 2 , 𝛽𝑖𝑖𝑖 , 𝛽𝑖𝑖𝑖 , 𝛽𝑖𝑖𝑖 ≥ 0 𝑖 = 1, … , 𝑁 − 1 23 Alternative method using QP Gotoh & Yamada (2015): Monotonic cubic spline regression Solved as a second order cone programming problem Number of constraints tends to be large for our problem Alternative method: Monotonicity approximation Apply cubic spline regression w/o monotonic constraint first Approximate the regression function with a set of strictly increasing points by solving QP 24 Data used for analysis Data period: 2005/8/8−2014/6/10 JEPX spot price, Transaction volume, Total sell/buy order volumes (48 products a day) Convert 24 hours values per day: 24 hour values are computed by taking the average between the two values, i.e., 0:00−0:30am and 0:30−1:00am, 1:00−1:30am and 1:30−2:00am, and so on. Control variables: Temperature index for 24 hours in the same period (Population weighted average of selected areas in Japan) Day and holiday dummy variables Day trend variable (long term linear trend) 25 Time series data (daily average, min, max) JEPX system price (daily average, min and max 60 【Yen/1kWh】 50 40 average min max JEPX system price (Yen/kWh) 30 20 10 0 2005/08/08 2008/05/24 2013/09/30 Trading volume (daily average, min and max 2500 2000 2011/03/11 average min max Transaction volume (MWh/h) 【MWh/h】 1500 1000 500 0 2005/08/08 2008/05/24 【Celsius】 2013/09/30 2011/03/11 2013/09/30 Temperature (daily average, min and max 40 30 2011/03/11 average min max 20 10 0 -10 2005/08/08 2008/05/24 Temperature (℃) 26 Estimation of GAMs using matching rates 𝑆𝑡 𝑚 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. , 𝑇𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. - Construct GAMs of 𝑃𝑡 Pt = f (m) Pt = g (m) 𝑃𝑡 𝑇𝑡 𝑚 𝑚 𝑚 𝐵𝑡 𝑚 w.r.t. 𝑆𝑡 (S )+ h (T )+ α (B ( ) )+ h (T ( ) )+ α (m ) t (m ) f t m t m g t = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. , 𝑇𝑇𝑇𝑇𝑇 𝑏𝑏𝑏 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. f 𝑚 or 𝐵𝑡 𝑚 【Day 𝑡 time 𝑚 values】 using control variables × t + ∑ j =0 β (f m, j) × Dummy j + ε (fm,t) 7 (m ) (m ) × + β × + ε t Dummy ∑ j =0 g , j g j g ,t 7 : Day 𝑡 time 𝑚 price, 𝐷𝐷𝐷𝐷𝐷0−6 : Day/Holiday dummy 𝑚 𝑚 : Day 𝑡 time 𝑚 temperature index, 𝜀𝑓,𝑡 , 𝜀𝑔,𝑡 : Residuals - Monotonicity approximation may be applied for 𝑓 (𝑚) , 𝑔(𝑚) by solving quadratic programing (𝑚 will be omitted hereafter) 27 Simultaneous estimation model In the electricity market, supply depends on demand and vice versa. Introduce a control variable of demand (or supply) for the sellmatching rate (or the buy-matching rate) model Pt ≅ f (St ) + h f (Tt ) + α f × t + ∑ j =0 β f , j × Dummy j + g (Bt ) 6 Pt ≅ g (Bt ) + hg (Tt ) + α g × t + ∑ j =0 β g , j × Dummy j + f (St ) 6 Pt = f (St ) + g (Bt ) + h(Tt ) + α × t + ∑ j =0 β j × Dummy j + ε t 6 Systematic term - Height = systematic + demand or supply factors - Scale of 𝑥-axis ⇒ Adjusted by multiplying total volume 28 Estimation results for spline functions 𝑆𝑡 𝑚 𝑃𝑡 = 𝑓 𝑆𝑡 + 𝑔 𝐵𝑡 + ℎ 𝑇𝑡 + 𝛼 × 𝑡 + ∑𝑗 𝛽𝑗 𝐷𝐷𝐷𝐷𝑦𝑗 +𝜀𝑡 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. , 𝑇𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. 𝐵𝑡 𝑚 Estimated spline functions = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. , 𝑇𝑇𝑇𝑇𝑇 𝑏𝑏𝑏 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. 【Day 𝑡 time 𝑚 values】 Approximated functions 29 Estimation results for spline functions 𝑃𝑡 = 𝑓 𝑆𝑡 + 𝑔 𝐵𝑡 + ℎ 𝑇𝑡 + 𝛼 × 𝑡 + ∑𝑗 𝛽𝑗 𝐷𝐷𝐷𝐷𝑦𝑗 +𝜀𝑡 Estimated spline functions Approximated functions ※ The larger SMR, the higher the price. The larger the BMR, the lower the price. 30 Estimation results for temperature trends 𝑃𝑡 = 𝑓 𝑆𝑡 + 𝑔 𝐵𝑡 + ℎ 𝑇𝑡 + 𝛼 × 𝑡 + ∑𝑗 𝛽𝑗 𝐷𝐷𝐷𝐷𝑦𝑗 +𝜀𝑡 Trends for 9am—7pm Trends for 9pm—7am ※ The relation between price and temperature may be explained by the demand of using of air conditioning 31 Construction of parametric equations Parametric equations for price and volume - Construct GAMs of 𝑃𝑡 or 𝑉𝑡 w.r.t. total volumes 𝑉�𝑆,𝑡 and 𝑉�𝐵,𝑡 𝑃𝑡 = ℎ𝑝 𝑉�𝐵,𝑡 + 𝑘𝑝 𝑉�𝑆,𝑡 + 𝑢𝑝 𝑇𝑡 + 𝛼𝑝 × 𝑡 + ∑7𝑗=1 𝛽𝑗𝑝 ⋅ 𝐷𝐷𝐷𝐷𝐷𝑗 +𝜀𝑝,𝑡 𝑉𝑡 = ℎ𝑣 𝑉�𝐵,𝑡 + 𝑘𝑣 𝑉�𝑆,𝑡 + 𝑢𝑣 𝑇𝑡 + 𝛼𝑣 × 𝑡 + ∑7𝑗=1 𝛽𝑗𝑗 ⋅ 𝐷𝐷𝐷𝐷𝐷𝑗 +𝜀𝑣,𝑡 𝑉�𝑆,𝑡 : total sell-order volume, Supply: 𝑦1 = ℎ�𝑝 𝑧1 + 𝑘�𝑝 𝑉�𝑆,𝑡 + 𝑒̂𝑝,𝑡 , � � � � 𝑥1 = ℎ𝑣 𝑧1 + 𝑘𝑣 𝑉𝑆,𝑡 + 𝑒̂𝑣,𝑡 𝑉�𝐵,𝑡 : total buy-order volume Demand: 𝑦2 = ℎ�𝑝 � 𝑥2 = ℎ�𝑣 𝑉�𝐵,𝑡 + 𝑘�𝑝 𝑧2 + 𝑒̂𝑝,𝑡 𝑉�𝐵,𝑡 + 𝑘�𝑣 𝑧2 + 𝑒̂𝑣,𝑡 𝑒̂𝑝,𝑡 : = 𝑢𝑝 𝑇𝑡 + 𝛼𝑝 × 𝑡 + ∑7𝑗=1 𝛽𝑗𝑝 ⋅ 𝐷𝐷𝐷𝐷𝐷𝑗 +𝜀𝑝,𝑡 𝑒̂𝑣,𝑡 : = 𝑢𝑣 𝑇𝑡 + 𝛼𝑣 × 𝑡 + ∑7𝑗=1 𝛽𝑗𝑗 ⋅ 𝐷𝐷𝐷𝐷𝐷𝑗 +𝜀𝑣,𝑡 32 Monotonicity conditions Supply function Monotonically increasing 𝜕ℎ�𝑝 𝜕ℎ�𝑣 >0 >0 , 𝜕𝑧1 𝜕𝑧1 Buy order vol.↗Price↗ Demand function Monotonically decreasing 𝜕𝑘�𝑣 𝜕𝑘�𝑝 >0 <0 , 𝜕𝑧2 𝜕𝑧2 Buy order vol.↗Trans. vol.↗ Sell order vol.↗Price↘ Sell order vol.↗Trans. vol.↗ Apply monotonicity approximation and compare the followings: Linear vs. Nonlinear models? Original spline functions vs. Monotonicity approximation Matching rates vs. Parametric equations approaches 33 Comparison of coefficients of determination, 𝑅2 - Linear model: Replace spline functions with linear functions except functions of temperature 𝑅2 for price equations 𝑅2 for volume equations ※ Effect of nonlinearity may be higher for price equations 34 Comparison of errors Mean square error (MSE) yn 2 2 𝜀𝑣,𝑡 + 𝜀𝑝,𝑡 (𝑃�𝑡 , 𝑉�𝑡 ) Error improvement ratio (EIR) (𝑃𝑡 , 𝑉𝑡 ) 𝑉𝑡 − 𝑉�𝑡 = 𝜀𝑣,𝑡 𝑃𝑡 − 𝑃�𝑡 = 𝜀𝑝,𝑡 MSE = 1 2 ∑(𝜀𝑣,𝑡 𝑁 2 + 𝜀𝑝,𝑡 ) xn EIR ≔ 𝐌𝐌𝐌 𝐟𝐟𝐟 𝐧𝐧𝐧𝐧𝐧𝐧𝐧𝐧𝐧 𝐌𝐌𝐌 𝐟𝐟𝐟 𝐥𝐥𝐥𝐥𝐥𝐥 ※ Higher improvement in the day time Stronger effect of nonlinearity? 35 Comparison of the two approaches Parametric equations approach - GAMs of 𝑃𝑡 or 𝑉𝑡 w.r.t. total volumes 𝑉�𝑆,𝑡 and 𝑉�𝐵,𝑡 𝑃𝑡 = ℎ𝑝 𝑉�𝐵,𝑡 + 𝑘𝑝 𝑉�𝑆,𝑡 + 𝑢𝑝 𝑇𝑡 + 𝛼𝑝 × 𝑡 + ∑7𝑗=1 𝛽𝑗𝑝 ⋅ 𝐷𝐷𝐷𝐷𝐷𝑗 +𝜀𝑝,𝑡 𝑉𝑡 = ℎ𝑣 𝑉�𝐵,𝑡 + 𝑘𝑣 𝑉�𝑆,𝑡 + 𝑢𝑣 𝑇𝑡 + 𝛼𝑣 × 𝑡 + ∑7𝑗=1 𝛽𝑗𝑗 ⋅ 𝐷𝐷𝐷𝐷𝐷𝑗 +𝜀𝑣,𝑡 𝑉�𝑆,𝑡 : total sell-order volume, 𝑉�𝐵,𝑡 : total buy-order volume Matching rates approach - GAM of 𝑃𝑡 w.r.t. matching rates 𝑆𝑡 and 𝐵𝑡 𝑆𝑡 𝑃𝑡 = 𝑓 𝑆𝑡 + 𝑔 𝐵𝑡 + ℎ 𝑇𝑡 + 𝛼𝑝 × 𝑡 + ∑7𝑗=1 𝛽𝑗𝑝 ⋅ 𝐷𝐷𝐷𝐷𝐷𝑗 +𝜀𝑝,𝑡 𝑚 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. , 𝑇𝑇𝑇𝑇𝑇 𝑠𝑠𝑠𝑠 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. 𝐵𝑡 𝑚 = 𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇𝑇 𝑣𝑣𝑣. , 𝑇𝑇𝑇𝑇𝑇 𝑏𝑏𝑏 𝑜𝑜𝑜𝑜𝑜 𝑣𝑣𝑣. 【Day 𝑡 time 𝑚 values】 36 Comparison using empirical data Difference of 𝑅2 between the two approaches w/ or w/o monotonicity approximation GAMs of 𝑃𝑡 w.r.t. matching rates GAMs of 𝑃𝑡 w.r.t. total volumes ※ Monotonicity approximation error is at most 5% in terms of 𝑅2 . 𝑅2 s are higher for parametric equations approach. 37 Comparison of t-values 𝑡-values of coefficients computed by replacing linear functions: t-values w.r.t. matching rates t-values w.r.t. total volumes ※ All the t-values are significant for matching rates, whereas there are several cases being non-significant for parametric equations approach. 38 Estimated supply and demand functions The date when the 9am price takes its median in 2005/8/8−2014/6/10 Parametric equations approach Matching rates approach ※ For the matching rates approach, the x-axis scale is adjusted that the maximum volumes provide the total sell/buy order volumes. 39 Estimated supply and demand functions The date when the 9am price takes its minimum in 2005/8/8−2014/6/10 Parametric equations approach Matching rates approach ※ For the parametric equations approach, the supply and demand functions shift according to the control variables and residuals 40 Estimated supply and demand functions The date when the 9am price takes its maximum in 2005/8/8−2014/6/10 Parametric equations approach Matching rates approach ※ Sensitivity of the price to the change of demand can be observed for both models 41 Concluding remarks Two approaches for estimating JEPX spot supply & demand functions 1. Sell & Buy matching rate approach 2. Parametric equations approach A nonparametric regression may be applied to take nonlinearity into account Generalized additive model (GAM) Construction of supply and demand functions: Although the parametric equations approach achieves higher 𝑅2 s, estimated demand functions may be non-monotonic in some cases. Those by matching rates are monotonic and t-values are significant. 42