Download Estimation of demand and supply functions for spot electricity prices

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