Energy, Models, Decision-Making (Part I):
Complementarity Modelling in Energy Markets
M. Sc. Vilma Virasjoki
Department of Mathematics and Systems Analysis,
Aalto University School of Science
PHYS-C1380 - Multi-Disciplinary Energy Perspectives
March 2, 2017
The document can be stored and made available to the public on the open internet pages of Aalto University. All other rights are reserved.
Agenda
1. Operations Research & Systems Analysis:
Models and Decision-Making
2. Complementarity Modelling in Energy
Markets
3. Numerical Examples
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Agenda
1. Operations Research & Systems Analysis:
Models and Decision-Making
2. Complementarity Modelling in Energy
Markets
3. Numerical Examples
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Introduction
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Operations Research & Systems Analysis
Operations Research
“… a discipline that deals with the application of advanced analytical methods
to help make better decisions. The terms management science and
analytics are sometimes used as synonyms for operations research.
Employing techniques from other mathematical sciences, such as mathematical
modelling, statistical analysis, and mathematical optimization, operations
research arrives at optimal or near-optimal solutions to complex decisionmaking problems.” 1
Systems Analysis & Systems Thinking
Systems are not the sum of their parts
A holistic approach is needed.
1 INFORMS: https://www.informs.org/About-INFORMS/What-is-Operations-Research
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Energy, Models, and Decision-Making
Energy Markets
The big picture: market efficiency
A complex dynamic system
Interlinked optimisation
problems (e.g. production
decisions)
Regulation, policy
Plant investments &
regulated operations
RE requirements
District heating
Short vs. long planning horizon
Uncertainties involved
Market price of electricity
Demand
Stochastic production (VRE)
Policy changes
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…
Models and Decision-Making
“All Models Are Wrong, Some are Useful”
-George Box, 1976
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Agenda
1. Operations Research & Systems Analysis:
Models and Decision-Making
2. Complementarity Modelling in Energy
Markets
3. Numerical Examples
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Microeconomic Principles: Example
Demand
Supply
Example from: Gabriel, S. A., Conejo, A. J., Fuller, J. D., Hobbs, B. F. and Ruiz, C.: Complementarity Modeling in Energy Markets. Springer, 2013
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Market Equilibrium
The intersection point
A= Consumer surplus, B= Producer surplus
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Market Equilibrium
Social welfare (”market efficiency”) maximisation
max 𝑆𝑊 𝑞
𝑓𝑠−1 𝑞
𝑑𝑆𝑊
֜
=0
𝑑𝑞
֜ 𝑓𝑑−1 𝑞 − 𝑓𝑠−1 𝑞 = 0
𝑓𝑑−1 𝑞
֞ 𝑓𝑑−1 𝑞 = 𝑓𝑠−1 𝑞
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Models for Market Competition
Individual firms as ”players” deciding on production
Perfect competition
Firms are not large/powerful enough to influence the price of a homogenous product
I.
Perfect competition of price-taking firms
Social welfare maximisation: the ”ideal”, monitored by regulators and usually assumed in models
Imperfect competition
M arket power: firms have ability & incentive to impact prices
II.
Monopoly model
Firm uses its 1) knowledge of the inverse demand function to influence the market price
III. Nash Cournot oligopoly
Firm uses its 1) knowledge of the inverse demand function and makes 2) a guess about others’
production to influence the market price
+
Additional approaches, e.g. leader-follower games
NB. These are models, the reality is often perhaps ”something in between”
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Illustrative Example: Comparison of I-III
2,5
Price (p)
2,0
Market equilibria
1,5
Demand function
1,0
Supply (I, perfect competition)
0,5
Supply (II, monopoly)
Supply (III, Nash-Cournot)
0,0
0
20000
40000
60000
80000
Production Quantity (q)
min max
Quantity (q)
Price (p)
Consumer surplus (CS)
Producer surplus (PS)
Social Welfare (SW)
I Perfect competition
75 000
1.25
46 875
40 910
87 785
II Monopoly
51 600
1.64
22 188
51 926
74 114
III Nash-Cournot
68 200
1.36
38 760
46 967
85 728
Example from: Gabriel, S. A., Conejo, A. J., Fuller, J. D., Hobbs, B. F. and Ruiz, C.: Complementarity Modeling in Energy Markets. Springer, 2013
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How to solve problems other than perfect
competition (i.e. several simultaneous
optimisation problems)?
Optimisation
problem
max 𝑆𝑊 𝑞
s.t. ℎ 𝑥 = 0
g 𝑥 ≤0
Perfect competition
E.g. Cournot oligopoly
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Mixed Complementarity Problem (MCP)
How to solve problems other than perfect competition?
Lagrangian function
Complementarity
conditions
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Complementarity Modelling
How to solve problems other than perfect competition?
1. Optimisation
Problem
2. Lagrangian
function
3. KKT
Conditions
1.
2.
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4.
Complementarity
Problem
Complementarity Modelling
KKT (Karush-Kuhn-Tucker) Conditions
1. Optimisation
Problem
2. Lagrangian
function
3. KKT
Conditions
4.
Complementarity
Problem
3.
Complementarity
conditions
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Complementarity Modelling
Summary
To represent the market equilibrium of
Several interacting players (companies, grid owner)
Interacting markets in time (dynamics of storage and power plant ramping)
Interacting markets in place (physical power system)
Primal and dual variables (decisions and shadow prices,
respectively) considered simultaneously
Efficient algorithms
E.g. GAMS Software, solver PATH
Suitable for a variety of energy market structures
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Agenda
1. Operations Research & Systems Analysis:
Models and Decision-Making
2. Complementarity Modelling in Energy
Markets
3. Numerical Examples
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Numerical Examples
1. Market Impacts of Energy Storage
Comparing an energy system with and without
power storage.
What are the impacts under perfect and imperfect
competition?
2. Market Power in Power and Heat Markets
What is the of role combined heat and power
production under imperfect competition?
Does CHP enable less market power to be used?
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Numerical Examples
1. Market Impacts of Energy Storage
Comparing an energy system with and without
power storage.
What are the impacts under perfect and imperfect
competition?
2. Market Power in Power and Heat Markets
What is the of role combined heat and power
production under imperfect competition?
Does CHP enable less market power to be used?
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1. Market Impacts of Energy Storage
Assumptions
Western European grid: Direct current (DC) load flow linearization
Renewables uncertainty: Discrete scenario tree
Virasjoki V., Rocha P., Siddiqui A. S., and Salo A.: “Market Impacts of Energy Storage in a Transmission-Constrained Power System”,
IEEE Transactions on Power Systems, 2016
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1. Market Impacts of Energy Storage
Mathematical Model
Market participants
Price (p)
Producers: maximise profit from
power sales
Power-only, RE
Power storage
Grid owner: maximise profit from
congestion fees
Consumers: inverse demand
function
Inverse
demand
function
(p*,q*)
Social
welfare
Inverse
supply
function
Comparison of
1. Perfect competition (PC)
Quantity (q)
Social welfare maximisation
2. Cournot oligopoly (CO)
Imperfect competition
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1. Market Impacts of Energy Storage
Main Conclusions
Storage may...
1. Smooth prices over time
2. Reduce ramping and ramping costs
3. Alleviate network congestion
4. Increase (and reverse) expected power flows
under market power due to
a) strategic withholding of supply and
b) strategic storage use
5. Increase CO2 emissions under PC
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1. Price-Smoothing Effect
Moving electricity from excess supply to scarcity with storage leads to a
price-smoothing effect between off-peak and peak demand periods
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1. Ramping and network congestion
Producers with storage rely less on ramping their conventional generation at
peak demand, which brings savings on costs. Storage alleviates network
congestion by reducing the expected congestion rent for the grid owner.
1. Expected ramping costs (producers)
2. Expected congestion rent (grid owner)
100
90
180
No Storage
Storage
160
80
120
50
k€
k€
60
100
-12%
80
40
30
-74%
60
40
-80%
10
0
-6%
140
70
20
No Storage
Storage
20
Perfect Competition (PC)
Cournot Oligopoly (CO)
0
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Perfect Competition (PC)
Cournot Oligopoly (CO)
1. Impact of market power on power flows
Expected transmission is reversed under Cournot oligopoly (CO) due to
market power use in n2 (withholding of sales, strategic use of storage).
Dominating transmission directions:
Unchanged from PC:
Reversed from PC:
Expected supply (GWh)
CO
∆ from PC
Node n2
196
-19%
All nodes
517
-14%
Bottlenecks
Expected stored power (GWh)
CO
∆ from PC
Node n2
49
+3%
All nodes
107
-5%
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1. CO2 emissions
Storage may increase CO2 emissions under PC due to efficiency losses and
an increase in coal and CCGT based generation at off-peak storage charging.
Under CO there is no increase in emissions.
52
Gg CO2
200
+2.2% +1.2%
51
50
+0.0%
+3.0%
49
Gg CO2
250
No Storage
Storage
Benchmark
150
100
48
47
46
45
PC, No Storage
CO, No Storage
PC, Storage
CO, Storage
50
44
0
Perfect Competition (PC)
Cournot oligopoly (CO)
43
t5
t6
t7
Time
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t8
Numerical Examples
1. Market Impacts of Energy Storage
Comparing an energy system with and without
power storage.
What are the impacts under perfect and imperfect
competition?
2. Market Power in Power and Heat Markets
What is the of role combined heat and power
production under imperfect competition?
Does CHP enable less market power to be used?
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2. Market Power in Power and Heat Markets
Background and Assumptions
Nordic System: Direct current (DC) load flow linearization + DC lines
District heating within the nodes
Deregulated
Regulated
Power
Market
District
Heating (DH)
Supply
Market
Power?
Power
Production
Combined Heat
and Power
(CHP)
Production
Heat-Only
Production
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2. Market Power in Power and Heat Markets
Mathematical Model
Market participants
Price (p)
Producers: maximise profit from
power and heat sales
Power-only, RE, CHP, Heat-only
Heat storage
Grid owner: maximise profit from
congestion fees
Consumers: inverse demand
function
Inverse
demand
function
(p*,q*)
Social
welfare
Inverse
supply
function
Comparison of
1. Perfect competition (social
welfare maximisation)
2. Cournot oligopoly (imperfect
competition)
Quantity (q)
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2. Market Power in Power and Heat Markets
Main Conclusions
1. Market power can shift DH supply from CHP to
heat-only plants, or vice versa
From CHP to heat-only, if power production is being withheld to
increase power prices
To CHP from heat-only, if power price is “high enough” that the
resulting power sales offset the price decrease
2. CHP can have a small intensifying impact on
market power (i.e. producers’ ability & incentive to
increase prices)
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2. Market power impacts on CHP and
district heating (DH) operations
Power markets: supply withholding. Similar logic for CHP
Slight shift in DH
supply from CHP to heat-only plants (March, December). Not necessarily
(June, September), e.g. if power price attractive to produce more CHP power.
Market Power Impact (GWhth)
7,0
Heat generation (GWhth)
5,0
3,0
1,0
-1,0
March
June
September
December
-3,0
-5,0
-7,0
CHP heat generation
Heat-only generation
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Total DH generation
2. CHP vs. CHP as ”power & ”heat-only”
Market power impact on power prices is slightly higher with real, status quo
CHP than when the capacity is decoupled.
Market Power Impact on Power Prices
Price Change (€/MWh)
6
5
4
3
2
1
0
March
June
Power Price Increase (CHP)
September
December
Power Price Increase (CHP decoupled)
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Discussion & Summary
Model usefulness and limitations
“Some models are useful”: Understanding the dynamics,
market design, policy impacts, strategic behaviour…
Stylised and aggregated form of the model, data and network
Relatively short studied time frame (4 – 24 hours)
Energy market modelling
Perfect competition: Social welfare maximisation
Complementarity modelling: To represent the equilibrium
with multiple optimisation problems (e.g. multiple producers)
Primal variables (decisions), Dual variables (prices)
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Thank you!
Vilma Virasjoki, M.Sc. (Tech)
Doctoral Student
[email protected]
Systems Analysis Laboratory,
Department of Mathematics and Systems Analysis,
Aalto University School of Science
http://sal.aalto.fi/en/personnel/vilma.virasjoki/contact
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Selected References
Fridolfsson, S. O., and Tangerås, T. P.: Market Power in the Nordic Electricity Wholesale Market: A Survey of the
Empirical Evidence. Energy Policy, 2009
Gabriel, S. A., Conejo, A. J., Fuller, J. D., Hobbs, B. F. and Ruiz, C.: Complementarity Modeling in Energy
Markets. Springer, 2013
Hobbs, B. F.: Linear Complementarity Models of Nash-Cournot Competition in Bilateral and POOLCO Power
Markets. IEEE Transactions on Power Systems, 2001
Joskow, P. L.: Lessons Learned from Electricity Market Liberalization. The Energy Journal, 2008
NordREG: Nordic market report - development in the Nordic electricity market, Technical report, Nordic Energy
Regulators, 2014
Virasjoki V., Rocha P., Siddiqui A. S., and Salo A.: “Market Impacts of Energy Storage in a TransmissionConstrained Power System”, IEEE Transactions on Power Systems, 2016
Zakeri B., Virasjoki V., Syri S., Connolly D., Mathiesen B. V., and Welsch M.: “Impact of Germany's energy
transition on the Nordic power market - A market-based multi-region energy system model”, Energy, 2016
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