-
Introduction Problem Formulation Numerical Examples
Conclusions
Market Power with Combined Heat and
Power Production in the Nordic Energy
System
Vilma Virasjokia Afzal S. Siddiquia,b Behnam Zakeria
Ahti Saloa
aAalto University bStockholm University, e-mail address:
[email protected]
28 October 2020
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Introduction Problem Formulation Numerical Examples
Conclusions
Table of Contents
1 Introduction
2 Problem Formulation
3 Numerical Examples
4 Conclusions
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Introduction Problem Formulation Numerical Examples
Conclusions
Introduction
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Introduction Problem Formulation Numerical Examples
Conclusions
Motivation
Asymmetrically Linked Markets and the Feasible Operating
Region (FOR)
Power
production
(MWh)
Heat
production
(MWhth)
200
200
400
400
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Introduction Problem Formulation Numerical Examples
Conclusions
Related Work
Impact of CHP on System Operations
Wu and Rosen (1999) develop an equilibrium model with CHP
in a conventional power system
Lund et al. (2005) demonstrate how CHP could facilitate
integration of wind power
Chen et al. (2015) examine how CHP with electric boilers and
heat storage can provide system flexibility with VRE
integration by developing a linear model with a convex CHP
FOR
Potential for exertion of market power (Hobbs, 2001)?
Fridolfsson and Tanger̊as (2009) do not find systematically
higher-than-marginal-cost prices in Nord Pool, but there
could
be local market power due to transmission constraints and
strategic use of CHP
Mitridati and Pinson (2016) develop a hierarchical
stochastic
setup to model CHP operations under uncertainty
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Introduction Problem Formulation Numerical Examples
Conclusions
Contribution
Research Objective and Findings
Examine the system-wide effects of CHP given VRE output,
transmission constraints, and market power
How is market power is impacted by CHP? How can market
power can affect district heating (DH) supply?
Equilibrium model: profit-maximising firms and a grid owner
Decoupling CHP increases electricity prices with a higher
impact under perfect competition than under Cournot
oligopoly
CHP facilitates exercise of market power because the
decrease
in revenue from withholding is offset by DH sales
Market power shifts production toward CHP from heat-only
plants because CHP makes it easier to withhold generation
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Introduction Problem Formulation Numerical Examples
Conclusions
Contribution
Numerical Experiments
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
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Introduction Problem Formulation Numerical Examples
Conclusions
Contribution
Numerical Experiments
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
-
Introduction Problem Formulation Numerical Examples
Conclusions
Contribution
Numerical Experiments
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
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Introduction Problem Formulation Numerical Examples
Conclusions
Contribution
Numerical Experiments
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
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Introduction Problem Formulation Numerical Examples
Conclusions
Problem Formulation
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Introduction Problem Formulation Numerical Examples
Conclusions
Setup
Assumptions
Economic and technological attributes
Linearised DC power flow
CHP marginal cost allocated to heat and power proportionally
VRE output: periodic availability and zero marginal costs
Hydropower variations: periodic maximum available capacity
Linear inverse demand: D intn,t − Dslpn,t∑
i ′∈I qpowern,i ′,t
Each firm i owns both conventional and VRE capacity plusCHP
Maximises profit by determining production and sales
Generates electricity at node n and sells it at node n′
Generation is constrained by installed capacity and FOR for
CHP
Heat storage respects capacities and rates
Grid owner
Controls transmission flows in order to maximise revenues
from
congestion charges on net imports
Must respect thermal limits of transmission lines
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Introduction Problem Formulation Numerical Examples
Conclusions
Setup
Equilibrium Model
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Introduction Problem Formulation Numerical Examples
Conclusions
Numerical Examples
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Introduction Problem Formulation Numerical Examples
Conclusions
Data
Transmission Lines
Line Type Positive direction Negative direction
ℓ1 AC 650 450
ℓ2 AC 150 250
ℓ3 AC 600 1000
ℓ4 AC 2145 2095
ℓ5 DC 950 1000
ℓ6 AC 650 450
ℓ7 AC 150 250
ℓ8 AC 600 1000
ℓ9 AC 2145 2095
ℓ10 DC 680 740
ℓ11 DC 590 600
ℓ12 AC 1700 1300
ℓ13 AC 1480 1120
ℓ14 DC 1200 1200
ℓ15 AC 1480 1120
ℓ16 DC 1200 1200
ℓ17 DC 860 1016
ℓ18 AC 750 779
ℓ19 AC 1234 684
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Introduction Problem Formulation Numerical Examples
Conclusions
Data
Generation Costs
Fuel Types Cpowern,u,t Cchp,pyB /C
chp,hyB C
chp,pyE /C
chp,hyE C
heatx
Coal u1, y1, x1 29 - 15 / 30 47Natural gas u2, y2, x2 85 45 / 50
46 / 51 53CCGT u3 47,* - - -Oil u4, y4, x4 78 37 / 51 38 / 52
54Biomass u5, y5, x5 62 - 28 / 26 27Oil shale u6 33 - - -Nuclear u7
9 - - -Hydro u8 *,** - - -Waste y9, x9 - - 25 / 3 24Peat y10, x10 -
- 22 / 25 25Bio oil x11 - - - 51
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Introduction Problem Formulation Numerical Examples
Conclusions
Data
Installed Power-Only Capacity by Producer and at Node (GW)
Node Producer u1 u2 u3 u4 u5 u6 u7 u8 Wind PVSE i1 - - - - - -
3.2 8 0.4 -
i2 - 0.1 0.3 0.3 0.1 - - 2.6 0.2 -i3 - - - - - - 3.7 - - -i4 - -
- - - - 2.6 - - -i5 - - - - - - - 2.9 - -i6 0.1 0.1 0.1 1.8 0.1 - -
3.9 4.8 0.1
FI i5 0.6 - - - - - 1.0 0.8 - -i7 - - - - - - 1.8 - - -i8 0.5 -
- 0.2 - - - 0.4 - -i9 - - - 0.1 - - - - - -i10 - - - - - - - 1.1 -
-i11 - 0.1 - 1.1 - - - 0.8 0.6 -
DK1 i12 - - - - - - - - 0.4 -i1 - - - - - - - - 0.4 -i13 - - - -
- - - - 3.0 0.4
DK2 i12 - - - 0.7 - - - - 0.2 -i13 - - - - - - - - 0.8 0.2
NO i14 - - 0.2 - - - - 9.9 0.2 -i15 - - - - - - - 2.5 - -i16 - -
- - - - - 2.0 - -i17 - - - - - - - 1.8 - -i18 - - - - - - - 1.7 -
-i19 - - - - - - - 1.4 - -i20 - - - - - - - 1.2 - -i21 - - 0.2 0.1
- - - 9.6 0.7 -
EE∗ i22 - 0.2 - - 0.1 1.8 - - 0.3 -LV∗ i22 - 0.5 - - - - - 1.5
0.1 -LT∗ i22 - 1.6 0.5 0.8 - - - 1.0 0.3 -% Available 90 95 80 86
95 90 90 See fig. See fig. See fig.
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Introduction Problem Formulation Numerical Examples
Conclusions
Data
VRE and Hydropower Profiles
5 10 15 20Hour of the day
0
20
40
%
(M) (J) (S) (D)
5 10 15 20Hour of the day
0
50
%
(M) (J) (S) (D)
NO SE FI LV,LTNode
0
50
%
(M) (J) (S) (D)
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Introduction Problem Formulation Numerical Examples
Conclusions
Data
Installed CHP Power Capacity by Producer and at Node (GW)
Node Producer y1E y9E y2B y2E y4B y4E y10E y5E
SE i1 - - - - - - - 0.2i2 - - - - 0.1 - - 0.2i5 0.1 0.1 - - -
0.2 - 0.2i6 0.2 0.2 0.1 0.1 0.1 0.4 - 1.8
FI i5 0.1 - 0.3 - - - 0.1 -i9 0.4 - 0.6 - - - - -i11 0.9 - 0.7
0.2 - 0.1 1.1 0.5
DK1 i12 1.1 - 0.1 0.4 - - - 0.1i1 0.7 - - - - - - -i13 - 0.1 0.5
0.4 - - - -
DK2 i12 1.1 - - 0.3 - - - 0.5i13 0.3 0.1 0.1 0.3 - 0.1 - 0.1
% Available 90 90 95 95 95 95 95 95
Rp-t-hyB/E 0.58 0.28 0.78 1.18 0.78 1.18 0.43 0.43
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Introduction Problem Formulation Numerical Examples
Conclusions
Data
Installed Heat-Only Capacity by Producer and at Node (GW)
Node Producer x1 x9 x2 x4 x10 x5 x11SE∗ i1 - - - - - 0.5 -
i2 - 0.3 0.1 - - 0.2 -i5 0.2 0.4 - - - 0.2 0.1i6 0.6 6.4 0.6 0.6
0.8 12.2 0.6
FI i5 0.2 - 0.6 0.4 - - -i9 0.2 - 1.0 1.2 - - -i11 - - 2.3 5.3
0.9 1.3 -
DK1∗ i13 1.7 1.7 1.5 0.1 - 2.9 -DK2∗ i13 1.1 1.1 1.0 0.1 - 1.9
-NO∗ i14 - 0.3 - - - 0.1 -
i21 - 3.3 0.3 0.1 - 1.2 -
% Available 90 90 95 95 95 95 95
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Introduction Problem Formulation Numerical Examples
Conclusions
Data
Calibration for March Scenario
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Introduction Problem Formulation Numerical Examples
Conclusions
Results
Test Cases and Main Results (in ke)
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
-
Introduction Problem Formulation Numerical Examples
Conclusions
Results
Test Cases and Main Results (in ke)
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
-
Introduction Problem Formulation Numerical Examples
Conclusions
Results
Test Cases and Main Results (in ke)
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
-
Introduction Problem Formulation Numerical Examples
Conclusions
Results
Test Cases and Main Results (in ke)
Model Perfect Competition Cournot Oligopoly
CHP status quo Case 1 (SQ-PC) Case 2 (SQ-CO)
CHP decoupling Case 3 (DE-PC) Case 4 (DE-CO)
Case 1 Case 2 ∆SQ Case 3 Case 4 ∆DE
(SQ-PC) (SQ-CO) (DE-PC) (DE-CO)
SW 87 113 86 777 - 336 84 549 84 237 - 312
CS 57 697 51 439 - 6 258 53 393 48 607 - 4 785
PS 28 578 34 686 6 107 30 136 34 805 4 670
GR 838 652 - 185 1 020 825 -196
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Introduction Problem Formulation Numerical Examples
Conclusions
Results
Impact of CHP Decoupling on Electricity Prices
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Introduction Problem Formulation Numerical Examples
Conclusions
Results
Impact of Market Power on Electricity Prices
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Introduction Problem Formulation Numerical Examples
Conclusions
Results
Analysis of Withholding via CHP (December Scenario)
00-0
202
-04
04-0
606
-08
08-1
010
-12
12-1
414
-16
16-1
818
-20
20-2
222
-00
Hour of the day
-10
-5
0E
lect
ricity
Gen
erat
ion
Cha
nge
(GW
h)
Power-onlyCHP
00-0
202
-04
04-0
606
-08
08-1
010
-12
12-1
414
-16
16-1
818
-20
20-2
222
-00
Hour of the day
-10
-5
0
Ele
ctric
ity G
ener
atio
nC
hang
e (G
Wh)
Power-onlyCHP (decoupled)
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Introduction Problem Formulation Numerical Examples
Conclusions
Results
Impact of Market Power on DH Sector and CO2 Emissions
-0.2
0
0.2
CO
2 E
mis
sion
s
Cha
nge
(Gg)
(M) (J) (S) (D)
-5
0
5
Hea
t Gen
erat
ion
Cha
nge
(GW
hth
)
CHP-heat Heat-only Total DH CO2
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Introduction Problem Formulation Numerical Examples
Conclusions
Conclusions
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Introduction Problem Formulation Numerical Examples
Conclusions
Conclusions
Summary
High penetration of VRE in Nordic countries may encourage
expansion of CHP capacity
CHP can intensify producers’ market power by reducing the
opportunity cost of withholding output
Market power may shift DH production to CHP from
heat-only because of the additional leverage it provides
during
peak hours
Even with fixed DH sales, market power may impact CO2emissions
from the DH sector due to production mix changes
Further policy analysis with VRE uncertainty and bi-level
model for optimal CHP investment
-
Mathematical Appendix
Mathematical Appendix
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
minΓ
∑t∈T
∑n∈N
[−
(Dintn,t − D
slpn,t
∑i′∈I
qpowern,i′,t
− ωn,t)
qpowern,i,t − Pheatn Q
heatn,i,t +
∑u∈Un,i
(Cpowern,u,t − ωn,t
)gpowern,i,u,t
+∑
x∈Xn,i
Cheatx gheatn,i,x,t +
∑y∈Yn,i
(Cchp,py − ωn,t
)gchp,pn,i,y,t +
∑y∈Yn,i
Cchp,hy gchp,hn,i,y,t − ωn,t
∑e∈E
gen,i,t
](1)
s.t.∑
n∈Nqpowern,i,t −
∑n∈N
∑u∈Un,i
gpowern,i,u,t −∑
n∈N
∑y∈Yn,i
gchp,pn,i,y,t −∑
n∈N
∑e∈E
gen,i,t = 0(δpoweri,t
), ∀t (2)
Qheatn,i,t −∑
x∈Xn,i
Etransn gheatn,i,x,t −
∑y∈Yn,i
Etransn gchp,hn,i,y,t + r
inn,i,t − r
outn,i,t = 0
(δheatn,i,t
), ∀n, t (3)
gchp,pn,i,yB,t
− Rp-t-hyB
gchp,hn,i,yB,t
= 0
(ϕBn,i,yB,t
), ∀n, yB ∈ YBn,i , t (4)
Rp-t-hyE
gchp,hn,i,yE,t
− gchp,pn,i,yE,t
≤ 0(ϕE,p-t-h
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (5)
Rf-t-hyE
gchp,hn,i,yE,t
+ Rf-t-pyE
gchp,pn,i,yE,t
− FyE ≤ 0(ϕE,ub-fuel
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (6)
Rheat-onlyn,i Qheatn,i,t −
∑x∈Xn,i
Etransn gheatn,i,x,t ≤ 0
(δheat-onlyn,i,t
), ∀n, t (7)
gpowern,i,u,t − TtGpowern,i,u ≤ 0
(λpowern,i,u,t
), ∀n, u ∈ Un,i , t (8)
gchp,hn,i,y,t − TtGchp,hn,i,y ≤ 0
(λchp,hn,i,y,t
), ∀n, y ∈ Yn,i , t (9)
gheatn,i,x,t − TtGheatn,i,x ≤ 0
(λheatn,i,x,t
), ∀n, x ∈ Xn,i , t (10)
gen,i,t − TtAen,tG
en,i = 0
(λ
en,i,t
), ∀n, e, t (11)
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
minΓ
∑t∈T
∑n∈N
[−
(Dintn,t − D
slpn,t
∑i′∈I
qpowern,i′,t
− ωn,t)
qpowern,i,t − Pheatn Q
heatn,i,t +
∑u∈Un,i
(Cpowern,u,t − ωn,t
)gpowern,i,u,t
+∑
x∈Xn,i
Cheatx gheatn,i,x,t +
∑y∈Yn,i
(Cchp,py − ωn,t
)gchp,pn,i,y,t +
∑y∈Yn,i
Cchp,hy gchp,hn,i,y,t − ωn,t
∑e∈E
gen,i,t
](1)
s.t.∑
n∈Nqpowern,i,t −
∑n∈N
∑u∈Un,i
gpowern,i,u,t −∑
n∈N
∑y∈Yn,i
gchp,pn,i,y,t −∑
n∈N
∑e∈E
gen,i,t = 0(δpoweri,t
), ∀t (2)
Qheatn,i,t −∑
x∈Xn,i
Etransn gheatn,i,x,t −
∑y∈Yn,i
Etransn gchp,hn,i,y,t + r
inn,i,t − r
outn,i,t = 0
(δheatn,i,t
), ∀n, t (3)
gchp,pn,i,yB,t
− Rp-t-hyB
gchp,hn,i,yB,t
= 0
(ϕBn,i,yB,t
), ∀n, yB ∈ YBn,i , t (4)
Rp-t-hyE
gchp,hn,i,yE,t
− gchp,pn,i,yE,t
≤ 0(ϕE,p-t-h
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (5)
Rf-t-hyE
gchp,hn,i,yE,t
+ Rf-t-pyE
gchp,pn,i,yE,t
− FyE ≤ 0(ϕE,ub-fuel
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (6)
Rheat-onlyn,i Qheatn,i,t −
∑x∈Xn,i
Etransn gheatn,i,x,t ≤ 0
(δheat-onlyn,i,t
), ∀n, t (7)
gpowern,i,u,t − TtGpowern,i,u ≤ 0
(λpowern,i,u,t
), ∀n, u ∈ Un,i , t (8)
gchp,hn,i,y,t − TtGchp,hn,i,y ≤ 0
(λchp,hn,i,y,t
), ∀n, y ∈ Yn,i , t (9)
gheatn,i,x,t − TtGheatn,i,x ≤ 0
(λheatn,i,x,t
), ∀n, x ∈ Xn,i , t (10)
gen,i,t − TtAen,tG
en,i = 0
(λ
en,i,t
), ∀n, e, t (11)
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
minΓ
∑t∈T
∑n∈N
[−
(Dintn,t − D
slpn,t
∑i′∈I
qpowern,i′,t
− ωn,t)
qpowern,i,t − Pheatn Q
heatn,i,t +
∑u∈Un,i
(Cpowern,u,t − ωn,t
)gpowern,i,u,t
+∑
x∈Xn,i
Cheatx gheatn,i,x,t +
∑y∈Yn,i
(Cchp,py − ωn,t
)gchp,pn,i,y,t +
∑y∈Yn,i
Cchp,hy gchp,hn,i,y,t − ωn,t
∑e∈E
gen,i,t
](1)
s.t.∑
n∈Nqpowern,i,t −
∑n∈N
∑u∈Un,i
gpowern,i,u,t −∑
n∈N
∑y∈Yn,i
gchp,pn,i,y,t −∑
n∈N
∑e∈E
gen,i,t = 0(δpoweri,t
), ∀t (2)
Qheatn,i,t −∑
x∈Xn,i
Etransn gheatn,i,x,t −
∑y∈Yn,i
Etransn gchp,hn,i,y,t + r
inn,i,t − r
outn,i,t = 0
(δheatn,i,t
), ∀n, t (3)
gchp,pn,i,yB,t
− Rp-t-hyB
gchp,hn,i,yB,t
= 0
(ϕBn,i,yB,t
), ∀n, yB ∈ YBn,i , t (4)
Rp-t-hyE
gchp,hn,i,yE,t
− gchp,pn,i,yE,t
≤ 0(ϕE,p-t-h
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (5)
Rf-t-hyE
gchp,hn,i,yE,t
+ Rf-t-pyE
gchp,pn,i,yE,t
− FyE ≤ 0(ϕE,ub-fuel
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (6)
Rheat-onlyn,i Qheatn,i,t −
∑x∈Xn,i
Etransn gheatn,i,x,t ≤ 0
(δheat-onlyn,i,t
), ∀n, t (7)
gpowern,i,u,t − TtGpowern,i,u ≤ 0
(λpowern,i,u,t
), ∀n, u ∈ Un,i , t (8)
gchp,hn,i,y,t − TtGchp,hn,i,y ≤ 0
(λchp,hn,i,y,t
), ∀n, y ∈ Yn,i , t (9)
gheatn,i,x,t − TtGheatn,i,x ≤ 0
(λheatn,i,x,t
), ∀n, x ∈ Xn,i , t (10)
gen,i,t − TtAen,tG
en,i = 0
(λ
en,i,t
), ∀n, e, t (11)
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
minΓ
∑t∈T
∑n∈N
[−
(Dintn,t − D
slpn,t
∑i′∈I
qpowern,i′,t
− ωn,t)
qpowern,i,t − Pheatn Q
heatn,i,t +
∑u∈Un,i
(Cpowern,u,t − ωn,t
)gpowern,i,u,t
+∑
x∈Xn,i
Cheatx gheatn,i,x,t +
∑y∈Yn,i
(Cchp,py − ωn,t
)gchp,pn,i,y,t +
∑y∈Yn,i
Cchp,hy gchp,hn,i,y,t − ωn,t
∑e∈E
gen,i,t
](1)
s.t.∑
n∈Nqpowern,i,t −
∑n∈N
∑u∈Un,i
gpowern,i,u,t −∑
n∈N
∑y∈Yn,i
gchp,pn,i,y,t −∑
n∈N
∑e∈E
gen,i,t = 0(δpoweri,t
), ∀t (2)
Qheatn,i,t −∑
x∈Xn,i
Etransn gheatn,i,x,t −
∑y∈Yn,i
Etransn gchp,hn,i,y,t + r
inn,i,t − r
outn,i,t = 0
(δheatn,i,t
), ∀n, t (3)
gchp,pn,i,yB,t
− Rp-t-hyB
gchp,hn,i,yB,t
= 0
(ϕBn,i,yB,t
), ∀n, yB ∈ YBn,i , t (4)
Rp-t-hyE
gchp,hn,i,yE,t
− gchp,pn,i,yE,t
≤ 0(ϕE,p-t-h
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (5)
Rf-t-hyE
gchp,hn,i,yE,t
+ Rf-t-pyE
gchp,pn,i,yE,t
− FyE ≤ 0(ϕE,ub-fuel
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (6)
Rheat-onlyn,i Qheatn,i,t −
∑x∈Xn,i
Etransn gheatn,i,x,t ≤ 0
(δheat-onlyn,i,t
), ∀n, t (7)
gpowern,i,u,t − TtGpowern,i,u ≤ 0
(λpowern,i,u,t
), ∀n, u ∈ Un,i , t (8)
gchp,hn,i,y,t − TtGchp,hn,i,y ≤ 0
(λchp,hn,i,y,t
), ∀n, y ∈ Yn,i , t (9)
gheatn,i,x,t − TtGheatn,i,x ≤ 0
(λheatn,i,x,t
), ∀n, x ∈ Xn,i , t (10)
gen,i,t − TtAen,tG
en,i = 0
(λ
en,i,t
), ∀n, e, t (11)
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
minΓ
∑t∈T
∑n∈N
[−
(Dintn,t − D
slpn,t
∑i′∈I
qpowern,i′,t
− ωn,t)
qpowern,i,t − Pheatn Q
heatn,i,t +
∑u∈Un,i
(Cpowern,u,t − ωn,t
)gpowern,i,u,t
+∑
x∈Xn,i
Cheatx gheatn,i,x,t +
∑y∈Yn,i
(Cchp,py − ωn,t
)gchp,pn,i,y,t +
∑y∈Yn,i
Cchp,hy gchp,hn,i,y,t − ωn,t
∑e∈E
gen,i,t
](1)
s.t.∑
n∈Nqpowern,i,t −
∑n∈N
∑u∈Un,i
gpowern,i,u,t −∑
n∈N
∑y∈Yn,i
gchp,pn,i,y,t −∑
n∈N
∑e∈E
gen,i,t = 0(δpoweri,t
), ∀t (2)
Qheatn,i,t −∑
x∈Xn,i
Etransn gheatn,i,x,t −
∑y∈Yn,i
Etransn gchp,hn,i,y,t + r
inn,i,t − r
outn,i,t = 0
(δheatn,i,t
), ∀n, t (3)
gchp,pn,i,yB,t
− Rp-t-hyB
gchp,hn,i,yB,t
= 0
(ϕBn,i,yB,t
), ∀n, yB ∈ YBn,i , t (4)
Rp-t-hyE
gchp,hn,i,yE,t
− gchp,pn,i,yE,t
≤ 0(ϕE,p-t-h
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (5)
Rf-t-hyE
gchp,hn,i,yE,t
+ Rf-t-pyE
gchp,pn,i,yE,t
− FyE ≤ 0(ϕE,ub-fuel
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (6)
Rheat-onlyn,i Qheatn,i,t −
∑x∈Xn,i
Etransn gheatn,i,x,t ≤ 0
(δheat-onlyn,i,t
), ∀n, t (7)
gpowern,i,u,t − TtGpowern,i,u ≤ 0
(λpowern,i,u,t
), ∀n, u ∈ Un,i , t (8)
gchp,hn,i,y,t − TtGchp,hn,i,y ≤ 0
(λchp,hn,i,y,t
), ∀n, y ∈ Yn,i , t (9)
gheatn,i,x,t − TtGheatn,i,x ≤ 0
(λheatn,i,x,t
), ∀n, x ∈ Xn,i , t (10)
gen,i,t − TtAen,tG
en,i = 0
(λ
en,i,t
), ∀n, e, t (11)
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
minΓ
∑t∈T
∑n∈N
[−
(Dintn,t − D
slpn,t
∑i′∈I
qpowern,i′,t
− ωn,t)
qpowern,i,t − Pheatn Q
heatn,i,t +
∑u∈Un,i
(Cpowern,u,t − ωn,t
)gpowern,i,u,t
+∑
x∈Xn,i
Cheatx gheatn,i,x,t +
∑y∈Yn,i
(Cchp,py − ωn,t
)gchp,pn,i,y,t +
∑y∈Yn,i
Cchp,hy gchp,hn,i,y,t − ωn,t
∑e∈E
gen,i,t
](1)
s.t.∑
n∈Nqpowern,i,t −
∑n∈N
∑u∈Un,i
gpowern,i,u,t −∑
n∈N
∑y∈Yn,i
gchp,pn,i,y,t −∑
n∈N
∑e∈E
gen,i,t = 0(δpoweri,t
), ∀t (2)
Qheatn,i,t −∑
x∈Xn,i
Etransn gheatn,i,x,t −
∑y∈Yn,i
Etransn gchp,hn,i,y,t + r
inn,i,t − r
outn,i,t = 0
(δheatn,i,t
), ∀n, t (3)
gchp,pn,i,yB,t
− Rp-t-hyB
gchp,hn,i,yB,t
= 0
(ϕBn,i,yB,t
), ∀n, yB ∈ YBn,i , t (4)
Rp-t-hyE
gchp,hn,i,yE,t
− gchp,pn,i,yE,t
≤ 0(ϕE,p-t-h
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (5)
Rf-t-hyE
gchp,hn,i,yE,t
+ Rf-t-pyE
gchp,pn,i,yE,t
− FyE ≤ 0(ϕE,ub-fuel
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (6)
Rheat-onlyn,i Qheatn,i,t −
∑x∈Xn,i
Etransn gheatn,i,x,t ≤ 0
(δheat-onlyn,i,t
), ∀n, t (7)
gpowern,i,u,t − TtGpowern,i,u ≤ 0
(λpowern,i,u,t
), ∀n, u ∈ Un,i , t (8)
gchp,hn,i,y,t − TtGchp,hn,i,y ≤ 0
(λchp,hn,i,y,t
), ∀n, y ∈ Yn,i , t (9)
gheatn,i,x,t − TtGheatn,i,x ≤ 0
(λheatn,i,x,t
), ∀n, x ∈ Xn,i , t (10)
gen,i,t − TtAen,tG
en,i = 0
(λ
en,i,t
), ∀n, e, t (11)
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
minΓ
∑t∈T
∑n∈N
[−
(Dintn,t − D
slpn,t
∑i′∈I
qpowern,i′,t
− ωn,t)
qpowern,i,t − Pheatn Q
heatn,i,t +
∑u∈Un,i
(Cpowern,u,t − ωn,t
)gpowern,i,u,t
+∑
x∈Xn,i
Cheatx gheatn,i,x,t +
∑y∈Yn,i
(Cchp,py − ωn,t
)gchp,pn,i,y,t +
∑y∈Yn,i
Cchp,hy gchp,hn,i,y,t − ωn,t
∑e∈E
gen,i,t
](1)
s.t.∑
n∈Nqpowern,i,t −
∑n∈N
∑u∈Un,i
gpowern,i,u,t −∑
n∈N
∑y∈Yn,i
gchp,pn,i,y,t −∑
n∈N
∑e∈E
gen,i,t = 0(δpoweri,t
), ∀t (2)
Qheatn,i,t −∑
x∈Xn,i
Etransn gheatn,i,x,t −
∑y∈Yn,i
Etransn gchp,hn,i,y,t + r
inn,i,t − r
outn,i,t = 0
(δheatn,i,t
), ∀n, t (3)
gchp,pn,i,yB,t
− Rp-t-hyB
gchp,hn,i,yB,t
= 0
(ϕBn,i,yB,t
), ∀n, yB ∈ YBn,i , t (4)
Rp-t-hyE
gchp,hn,i,yE,t
− gchp,pn,i,yE,t
≤ 0(ϕE,p-t-h
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (5)
Rf-t-hyE
gchp,hn,i,yE,t
+ Rf-t-pyE
gchp,pn,i,yE,t
− FyE ≤ 0(ϕE,ub-fuel
n,i,yE,t
), ∀n, yE ∈ YEn,i , t (6)
Rheat-onlyn,i Qheatn,i,t −
∑x∈Xn,i
Etransn gheatn,i,x,t ≤ 0
(δheat-onlyn,i,t
), ∀n, t (7)
gpowern,i,u,t − TtGpowern,i,u ≤ 0
(λpowern,i,u,t
), ∀n, u ∈ Un,i , t (8)
gchp,hn,i,y,t − TtGchp,hn,i,y ≤ 0
(λchp,hn,i,y,t
), ∀n, y ∈ Yn,i , t (9)
gheatn,i,x,t − TtGheatn,i,x ≤ 0
(λheatn,i,x,t
), ∀n, x ∈ Xn,i , t (10)
gen,i,t − TtAen,tG
en,i = 0
(λ
en,i,t
), ∀n, e, t (11)
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
gpowern,i,u,t − gpowern,i,u,t−1 − TtR
power-upu G
powern,i,u ≤ 0
(ρpower-upn,i,u,t
), ∀n, u ∈ Un,i , t (12)
gpowern,i,u,t−1 − gpowern,i,u,t − TtR
power-downu G
powern,i,u ≤ 0
(ρpower-downn,i,u,t
), ∀n, u ∈ Un,i , t (13)
gchp,hn,i,y,t − gchp,hn,i,y,t−1 − TtR
chp-upy G
chp,hn,i,y ≤ 0
(ρchp-upn,i,y,t
), ∀n, y ∈ Yn,i , t (14)
gchp,hn,i,y,t−1 − gchp,hn,i,y,t − TtR
chp-downy G
chp,hn,i,y ≤ 0
(ρchp-downn,i,y,t
), ∀n, y ∈ Yn,i , t (15)
gheatn,i,x,t − gheatn,i,x,t−1 − TtR
heat-upx G
heatn,i,x ≤ 0
(ρheat-upn,i,x,t
), ∀n, x ∈ Xn,i , t (16)
gheatn,i,x,t−1 − gheatn,i,x,t − TtR
heat-downx G
heatn,i,x ≤ 0
(ρheat-downn,i,x,t
), ∀n, x ∈ Xn,i , t (17)
rston,i,t − (1 − Edec)Tt rston,i,t−1 − E
effr inn,i,t + routn,i,t = 0
(βbaln,i,t
), ∀n, t (18)
r inn,i,t − TtEinEmaxn,i ≤ 0
(βinn,i,t
), ∀n, t (19)
routn,i,t − TtEoutEmaxn,i ≤ 0
(βoutn,i,t
), ∀n, t (20)
Eminn,i ≤ rston,i,t ≤ E
maxn,i
(βlbn,i,t , β
ubn,i,t
), ∀n, t (21)
gpowern,i,u,t ≥ 0 ∀n, u ∈ Un,i , t; gchp,pn,i,y,t , g
chp,hn,i,y,t ≥ 0 ∀n, y ∈ Yn,i , t; g
heatn,i,x,t ≥ 0 ∀n, x ∈ Xn,i , t;
gen,i,t , qpowern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t ≥ 0 ∀n, t (22)
Γ := {gpowern,i,u,t , gchp,pn,i,y,t , g
chp,hn,i,y,t , g
heatn,i,x,t , g
en,i,t , q
powern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t}
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
gpowern,i,u,t − gpowern,i,u,t−1 − TtR
power-upu G
powern,i,u ≤ 0
(ρpower-upn,i,u,t
), ∀n, u ∈ Un,i , t (12)
gpowern,i,u,t−1 − gpowern,i,u,t − TtR
power-downu G
powern,i,u ≤ 0
(ρpower-downn,i,u,t
), ∀n, u ∈ Un,i , t (13)
gchp,hn,i,y,t − gchp,hn,i,y,t−1 − TtR
chp-upy G
chp,hn,i,y ≤ 0
(ρchp-upn,i,y,t
), ∀n, y ∈ Yn,i , t (14)
gchp,hn,i,y,t−1 − gchp,hn,i,y,t − TtR
chp-downy G
chp,hn,i,y ≤ 0
(ρchp-downn,i,y,t
), ∀n, y ∈ Yn,i , t (15)
gheatn,i,x,t − gheatn,i,x,t−1 − TtR
heat-upx G
heatn,i,x ≤ 0
(ρheat-upn,i,x,t
), ∀n, x ∈ Xn,i , t (16)
gheatn,i,x,t−1 − gheatn,i,x,t − TtR
heat-downx G
heatn,i,x ≤ 0
(ρheat-downn,i,x,t
), ∀n, x ∈ Xn,i , t (17)
rston,i,t − (1 − Edec)Tt rston,i,t−1 − E
effr inn,i,t + routn,i,t = 0
(βbaln,i,t
), ∀n, t (18)
r inn,i,t − TtEinEmaxn,i ≤ 0
(βinn,i,t
), ∀n, t (19)
routn,i,t − TtEoutEmaxn,i ≤ 0
(βoutn,i,t
), ∀n, t (20)
Eminn,i ≤ rston,i,t ≤ E
maxn,i
(βlbn,i,t , β
ubn,i,t
), ∀n, t (21)
gpowern,i,u,t ≥ 0 ∀n, u ∈ Un,i , t; gchp,pn,i,y,t , g
chp,hn,i,y,t ≥ 0 ∀n, y ∈ Yn,i , t; g
heatn,i,x,t ≥ 0 ∀n, x ∈ Xn,i , t;
gen,i,t , qpowern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t ≥ 0 ∀n, t (22)
Γ := {gpowern,i,u,t , gchp,pn,i,y,t , g
chp,hn,i,y,t , g
heatn,i,x,t , g
en,i,t , q
powern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t}
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
gpowern,i,u,t − gpowern,i,u,t−1 − TtR
power-upu G
powern,i,u ≤ 0
(ρpower-upn,i,u,t
), ∀n, u ∈ Un,i , t (12)
gpowern,i,u,t−1 − gpowern,i,u,t − TtR
power-downu G
powern,i,u ≤ 0
(ρpower-downn,i,u,t
), ∀n, u ∈ Un,i , t (13)
gchp,hn,i,y,t − gchp,hn,i,y,t−1 − TtR
chp-upy G
chp,hn,i,y ≤ 0
(ρchp-upn,i,y,t
), ∀n, y ∈ Yn,i , t (14)
gchp,hn,i,y,t−1 − gchp,hn,i,y,t − TtR
chp-downy G
chp,hn,i,y ≤ 0
(ρchp-downn,i,y,t
), ∀n, y ∈ Yn,i , t (15)
gheatn,i,x,t − gheatn,i,x,t−1 − TtR
heat-upx G
heatn,i,x ≤ 0
(ρheat-upn,i,x,t
), ∀n, x ∈ Xn,i , t (16)
gheatn,i,x,t−1 − gheatn,i,x,t − TtR
heat-downx G
heatn,i,x ≤ 0
(ρheat-downn,i,x,t
), ∀n, x ∈ Xn,i , t (17)
rston,i,t − (1 − Edec)Tt rston,i,t−1 − E
effr inn,i,t + routn,i,t = 0
(βbaln,i,t
), ∀n, t (18)
r inn,i,t − TtEinEmaxn,i ≤ 0
(βinn,i,t
), ∀n, t (19)
routn,i,t − TtEoutEmaxn,i ≤ 0
(βoutn,i,t
), ∀n, t (20)
Eminn,i ≤ rston,i,t ≤ E
maxn,i
(βlbn,i,t , β
ubn,i,t
), ∀n, t (21)
gpowern,i,u,t ≥ 0 ∀n, u ∈ Un,i , t; gchp,pn,i,y,t , g
chp,hn,i,y,t ≥ 0 ∀n, y ∈ Yn,i , t; g
heatn,i,x,t ≥ 0 ∀n, x ∈ Xn,i , t;
gen,i,t , qpowern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t ≥ 0 ∀n, t (22)
Γ := {gpowern,i,u,t , gchp,pn,i,y,t , g
chp,hn,i,y,t , g
heatn,i,x,t , g
en,i,t , q
powern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t}
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
gpowern,i,u,t − gpowern,i,u,t−1 − TtR
power-upu G
powern,i,u ≤ 0
(ρpower-upn,i,u,t
), ∀n, u ∈ Un,i , t (12)
gpowern,i,u,t−1 − gpowern,i,u,t − TtR
power-downu G
powern,i,u ≤ 0
(ρpower-downn,i,u,t
), ∀n, u ∈ Un,i , t (13)
gchp,hn,i,y,t − gchp,hn,i,y,t−1 − TtR
chp-upy G
chp,hn,i,y ≤ 0
(ρchp-upn,i,y,t
), ∀n, y ∈ Yn,i , t (14)
gchp,hn,i,y,t−1 − gchp,hn,i,y,t − TtR
chp-downy G
chp,hn,i,y ≤ 0
(ρchp-downn,i,y,t
), ∀n, y ∈ Yn,i , t (15)
gheatn,i,x,t − gheatn,i,x,t−1 − TtR
heat-upx G
heatn,i,x ≤ 0
(ρheat-upn,i,x,t
), ∀n, x ∈ Xn,i , t (16)
gheatn,i,x,t−1 − gheatn,i,x,t − TtR
heat-downx G
heatn,i,x ≤ 0
(ρheat-downn,i,x,t
), ∀n, x ∈ Xn,i , t (17)
rston,i,t − (1 − Edec)Tt rston,i,t−1 − E
effr inn,i,t + routn,i,t = 0
(βbaln,i,t
), ∀n, t (18)
r inn,i,t − TtEinEmaxn,i ≤ 0
(βinn,i,t
), ∀n, t (19)
routn,i,t − TtEoutEmaxn,i ≤ 0
(βoutn,i,t
), ∀n, t (20)
Eminn,i ≤ rston,i,t ≤ E
maxn,i
(βlbn,i,t , β
ubn,i,t
), ∀n, t (21)
gpowern,i,u,t ≥ 0 ∀n, u ∈ Un,i , t; gchp,pn,i,y,t , g
chp,hn,i,y,t ≥ 0 ∀n, y ∈ Yn,i , t; g
heatn,i,x,t ≥ 0 ∀n, x ∈ Xn,i , t;
gen,i,t , qpowern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t ≥ 0 ∀n, t (22)
Γ := {gpowern,i,u,t , gchp,pn,i,y,t , g
chp,hn,i,y,t , g
heatn,i,x,t , g
en,i,t , q
powern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t}
-
Mathematical Appendix
Optimisation Problems
Firm i ∈ I
gpowern,i,u,t − gpowern,i,u,t−1 − TtR
power-upu G
powern,i,u ≤ 0
(ρpower-upn,i,u,t
), ∀n, u ∈ Un,i , t (12)
gpowern,i,u,t−1 − gpowern,i,u,t − TtR
power-downu G
powern,i,u ≤ 0
(ρpower-downn,i,u,t
), ∀n, u ∈ Un,i , t (13)
gchp,hn,i,y,t − gchp,hn,i,y,t−1 − TtR
chp-upy G
chp,hn,i,y ≤ 0
(ρchp-upn,i,y,t
), ∀n, y ∈ Yn,i , t (14)
gchp,hn,i,y,t−1 − gchp,hn,i,y,t − TtR
chp-downy G
chp,hn,i,y ≤ 0
(ρchp-downn,i,y,t
), ∀n, y ∈ Yn,i , t (15)
gheatn,i,x,t − gheatn,i,x,t−1 − TtR
heat-upx G
heatn,i,x ≤ 0
(ρheat-upn,i,x,t
), ∀n, x ∈ Xn,i , t (16)
gheatn,i,x,t−1 − gheatn,i,x,t − TtR
heat-downx G
heatn,i,x ≤ 0
(ρheat-downn,i,x,t
), ∀n, x ∈ Xn,i , t (17)
rston,i,t − (1 − Edec)Tt rston,i,t−1 − E
effr inn,i,t + routn,i,t = 0
(βbaln,i,t
), ∀n, t (18)
r inn,i,t − TtEinEmaxn,i ≤ 0
(βinn,i,t
), ∀n, t (19)
routn,i,t − TtEoutEmaxn,i ≤ 0
(βoutn,i,t
), ∀n, t (20)
Eminn,i ≤ rston,i,t ≤ E
maxn,i
(βlbn,i,t , β
ubn,i,t
), ∀n, t (21)
gpowern,i,u,t ≥ 0 ∀n, u ∈ Un,i , t; gchp,pn,i,y,t , g
chp,hn,i,y,t ≥ 0 ∀n, y ∈ Yn,i , t; g
heatn,i,x,t ≥ 0 ∀n, x ∈ Xn,i , t;
gen,i,t , qpowern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t ≥ 0 ∀n, t (22)
Γ := {gpowern,i,u,t , gchp,pn,i,y,t , g
chp,hn,i,y,t , g
heatn,i,x,t , g
en,i,t , q
powern,i,t , r
ston,i,t , r
inn,i,t , r
outn,i,t}
-
Mathematical Appendix
Optimisation Problems
Grid Owner and Market Clearing
minθnAC,t ,fℓ,t
−∑t∈T
∑n∈N
ωn,t∑ℓ∈L
Mℓ,nTt fℓ,t (23)
s.t.∑
nAC∈NACTtHℓAC,nACθnAC,t − Tt fℓAC,t = 0
(ηℓAC,t
), ∀ℓAC ∈ LAC, t (24)
Tt fℓ,t − TtK ℓ ≤ 0(µℓ,t
), ∀ℓ, t (25)
− Tt fℓ,t − TtK ℓ ≤ 0(µ
ℓ,t
), ∀ℓ, t (26)
TtSnAC θnAC,t = 0(γnAC,t
), ∀nAC ∈ N AC, t (27)
θnAC,t and fℓ,t u.r.s., ∀nAC ∈ N AC, ℓ, t (28)
∑i∈I
qpowern,i ,t −∑i∈I
∑u∈Un,i
g powern,i ,u,t −∑i∈I
∑y∈Yn,i
g chp,pn,i ,y,t
−∑i∈I
∑e∈E
g en,i ,t −∑ℓ∈L
TtMℓ,nfℓ,t = 0 (ωn,t , u.r.s.) , ∀n, t (29)
-
Mathematical Appendix
Optimisation Problems
Grid Owner and Market Clearing
minθnAC,t ,fℓ,t
−∑t∈T
∑n∈N
ωn,t∑ℓ∈L
Mℓ,nTt fℓ,t (23)
s.t.∑
nAC∈NACTtHℓAC,nACθnAC,t − Tt fℓAC,t = 0
(ηℓAC,t
), ∀ℓAC ∈ LAC, t (24)
Tt fℓ,t − TtK ℓ ≤ 0(µℓ,t
), ∀ℓ, t (25)
− Tt fℓ,t − TtK ℓ ≤ 0(µ
ℓ,t
), ∀ℓ, t (26)
TtSnAC θnAC,t = 0(γnAC,t
), ∀nAC ∈ N AC, t (27)
θnAC,t and fℓ,t u.r.s., ∀nAC ∈ N AC, ℓ, t (28)
∑i∈I
qpowern,i ,t −∑i∈I
∑u∈Un,i
g powern,i ,u,t −∑i∈I
∑y∈Yn,i
g chp,pn,i ,y,t
−∑i∈I
∑e∈E
g en,i ,t −∑ℓ∈L
TtMℓ,nfℓ,t = 0 (ωn,t , u.r.s.) , ∀n, t (29)
-
Mathematical Appendix
Optimisation Problems
Grid Owner and Market Clearing
minθnAC,t ,fℓ,t
−∑t∈T
∑n∈N
ωn,t∑ℓ∈L
Mℓ,nTt fℓ,t (23)
s.t.∑
nAC∈NACTtHℓAC,nACθnAC,t − Tt fℓAC,t = 0
(ηℓAC,t
), ∀ℓAC ∈ LAC, t (24)
Tt fℓ,t − TtK ℓ ≤ 0(µℓ,t
), ∀ℓ, t (25)
− Tt fℓ,t − TtK ℓ ≤ 0(µ
ℓ,t
), ∀ℓ, t (26)
TtSnAC θnAC,t = 0(γnAC,t
), ∀nAC ∈ N AC, t (27)
θnAC,t and fℓ,t u.r.s., ∀nAC ∈ N AC, ℓ, t (28)
∑i∈I
qpowern,i ,t −∑i∈I
∑u∈Un,i
g powern,i ,u,t −∑i∈I
∑y∈Yn,i
g chp,pn,i ,y,t
−∑i∈I
∑e∈E
g en,i ,t −∑ℓ∈L
TtMℓ,nfℓ,t = 0 (ωn,t , u.r.s.) , ∀n, t (29)
-
Mathematical Appendix
Optimisation Problems
Grid Owner and Market Clearing
minθnAC,t ,fℓ,t
−∑t∈T
∑n∈N
ωn,t∑ℓ∈L
Mℓ,nTt fℓ,t (23)
s.t.∑
nAC∈NACTtHℓAC,nACθnAC,t − Tt fℓAC,t = 0
(ηℓAC,t
), ∀ℓAC ∈ LAC, t (24)
Tt fℓ,t − TtK ℓ ≤ 0(µℓ,t
), ∀ℓ, t (25)
− Tt fℓ,t − TtK ℓ ≤ 0(µ
ℓ,t
), ∀ℓ, t (26)
TtSnAC θnAC,t = 0(γnAC,t
), ∀nAC ∈ N AC, t (27)
θnAC,t and fℓ,t u.r.s., ∀nAC ∈ N AC, ℓ, t (28)
∑i∈I
qpowern,i ,t −∑i∈I
∑u∈Un,i
g powern,i ,u,t −∑i∈I
∑y∈Yn,i
g chp,pn,i ,y,t
−∑i∈I
∑e∈E
g en,i ,t −∑ℓ∈L
TtMℓ,nfℓ,t = 0 (ωn,t , u.r.s.) , ∀n, t (29)
-
Mathematical Appendix
Optimisation Problems
Grid Owner and Market Clearing
minθnAC,t ,fℓ,t
−∑t∈T
∑n∈N
ωn,t∑ℓ∈L
Mℓ,nTt fℓ,t (23)
s.t.∑
nAC∈NACTtHℓAC,nACθnAC,t − Tt fℓAC,t = 0
(ηℓAC,t
), ∀ℓAC ∈ LAC, t (24)
Tt fℓ,t − TtK ℓ ≤ 0(µℓ,t
), ∀ℓ, t (25)
− Tt fℓ,t − TtK ℓ ≤ 0(µ
ℓ,t
), ∀ℓ, t (26)
TtSnAC θnAC,t = 0(γnAC,t
), ∀nAC ∈ N AC, t (27)
θnAC,t and fℓ,t u.r.s., ∀nAC ∈ N AC, ℓ, t (28)
∑i∈I
qpowern,i ,t −∑i∈I
∑u∈Un,i
g powern,i ,u,t −∑i∈I
∑y∈Yn,i
g chp,pn,i ,y,t
−∑i∈I
∑e∈E
g en,i ,t −∑ℓ∈L
TtMℓ,nfℓ,t = 0 (ωn,t , u.r.s.) , ∀n, t (29)
-
Mathematical Appendix
Optimisation Problems
Grid Owner and Market Clearing
minθnAC,t ,fℓ,t
−∑t∈T
∑n∈N
ωn,t∑ℓ∈L
Mℓ,nTt fℓ,t (23)
s.t.∑
nAC∈NACTtHℓAC,nACθnAC,t − Tt fℓAC,t = 0
(ηℓAC,t
), ∀ℓAC ∈ LAC, t (24)
Tt fℓ,t − TtK ℓ ≤ 0(µℓ,t
), ∀ℓ, t (25)
− Tt fℓ,t − TtK ℓ ≤ 0(µ
ℓ,t
), ∀ℓ, t (26)
TtSnAC θnAC,t = 0(γnAC,t
), ∀nAC ∈ N AC, t (27)
θnAC,t and fℓ,t u.r.s., ∀nAC ∈ N AC, ℓ, t (28)
∑i∈I
qpowern,i ,t −∑i∈I
∑u∈Un,i
g powern,i ,u,t −∑i∈I
∑y∈Yn,i
g chp,pn,i ,y,t
−∑i∈I
∑e∈E
g en,i ,t −∑ℓ∈L
TtMℓ,nfℓ,t = 0 (ωn,t , u.r.s.) , ∀n, t (29)
-
Mathematical Appendix
Optimisation Problems
Grid Owner and Market Clearing
minθnAC,t ,fℓ,t
−∑t∈T
∑n∈N
ωn,t∑ℓ∈L
Mℓ,nTt fℓ,t (23)
s.t.∑
nAC∈NACTtHℓAC,nACθnAC,t − Tt fℓAC,t = 0
(ηℓAC,t
), ∀ℓAC ∈ LAC, t (24)
Tt fℓ,t − TtK ℓ ≤ 0(µℓ,t
), ∀ℓ, t (25)
− Tt fℓ,t − TtK ℓ ≤ 0(µ
ℓ,t
), ∀ℓ, t (26)
TtSnAC θnAC,t = 0(γnAC,t
), ∀nAC ∈ N AC, t (27)
θnAC,t and fℓ,t u.r.s., ∀nAC ∈ N AC, ℓ, t (28)
∑i∈I
qpowern,i ,t −∑i∈I
∑u∈Un,i
g powern,i ,u,t −∑i∈I
∑y∈Yn,i
g chp,pn,i ,y,t
−∑i∈I
∑e∈E
g en,i ,t −∑ℓ∈L
TtMℓ,nfℓ,t = 0 (ωn,t , u.r.s.) , ∀n, t (29)
IntroductionMotivationRelated WorkContribution
Problem FormulationSetup
Numerical ExamplesDataResults
ConclusionsConclusions
AppendixMathematical AppendixOptimisation Problems
