European energy equilibrium and decomposition Anes Dallagi EDF - - PowerPoint PPT Presentation

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

European energy equilibrium and decomposition

Anes Dallagi

EDF R&D – OSIRIS – Optimization methods and tools

ICSP 2013 Bergamo, July 12th, 2013

1/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

2/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

3/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

EDF generation side vs. demand side

EDF generation mix : 47 thermal units (fuel, coal and gas turbine) ; 58 nuclear units ; 50 hydro-valleys. Each hydro-valley is a set of interconnected reservoirs (150) and power plants (448). Water stock : 7000hm3 ; 25 withdrawal options ; Other : wind, solar, biomass in significant growth. Demand to be satisfied : Residential, industrial demand and LT contracts ; Provision of energy by committing a number of turbines to be running ; Provision of spinning reserve by committing turbines to be in synchronized condensing mode ; Provision of frequency-keeping services from a selection of turbines ; How to match supply to demand while minimizing costs ?

4/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

EDF generation side vs. demand side

EDF generation mix : 47 thermal units (fuel, coal and gas turbine) ; 58 nuclear units ; 50 hydro-valleys. Each hydro-valley is a set of interconnected reservoirs (150) and power plants (448). Water stock : 7000hm3 ; 25 withdrawal options ; Other : wind, solar, biomass in significant growth. Demand to be satisfied : Residential, industrial demand and LT contracts ; Provision of energy by committing a number of turbines to be running ; Provision of spinning reserve by committing turbines to be in synchronized condensing mode ; Provision of frequency-keeping services from a selection of turbines ; How to match supply to demand while minimizing costs ?

4/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

EDF generation side vs. demand side

EDF generation mix : 47 thermal units (fuel, coal and gas turbine) ; 58 nuclear units ; 50 hydro-valleys. Each hydro-valley is a set of interconnected reservoirs (150) and power plants (448). Water stock : 7000hm3 ; 25 withdrawal options ; Other : wind, solar, biomass in significant growth. Demand to be satisfied : Residential, industrial demand and LT contracts ; Provision of energy by committing a number of turbines to be running ; Provision of spinning reserve by committing turbines to be in synchronized condensing mode ; Provision of frequency-keeping services from a selection of turbines ; How to match supply to demand while minimizing costs ?

4/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

EDF generation side vs. demand side

EDF generation mix : 47 thermal units (fuel, coal and gas turbine) ; 58 nuclear units ; 50 hydro-valleys. Each hydro-valley is a set of interconnected reservoirs (150) and power plants (448). Water stock : 7000hm3 ; 25 withdrawal options ; Other : wind, solar, biomass in significant growth. Demand to be satisfied : Residential, industrial demand and LT contracts ; Provision of energy by committing a number of turbines to be running ; Provision of spinning reserve by committing turbines to be in synchronized condensing mode ; Provision of frequency-keeping services from a selection of turbines ; How to match supply to demand while minimizing costs ?

4/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

5/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

The optimization process : time decomposition

Prices, Demand and technology LT forecast Long-term opti- mization problem Investments, MC forecast, export/import forecast Stochastic approximate model Prices, Demand and inflow MT forecast Mid-term optimi- zation problem Water values Prices, Demand and inflow ST forecast Short-term opti- mization problem Operating schedule Deterministic accurate model

6/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

The optimization process : time decomposition

Prices, Demand and technology LT forecast Long-term opti- mization problem Investments, MC forecast, export/import forecast Stochastic approximate model Prices, Demand and inflow MT forecast Mid-term optimi- zation problem Water values Prices, Demand and inflow ST forecast Short-term opti- mization problem Operating schedule Deterministic accurate model

6/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

The optimization process : time decomposition

Prices, Demand and technology LT forecast Long-term opti- mization problem Investments, MC forecast, export/import forecast Stochastic approximate model Prices, Demand and inflow MT forecast Mid-term optimi- zation problem Water values Prices, Demand and inflow ST forecast Short-term opti- mization problem Operating schedule Deterministic accurate model

6/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

The optimization process : time decomposition

Prices, Demand and technology LT forecast Long-term opti- mization problem Investments, MC forecast, export/import forecast Stochastic approximate model Prices, Demand and inflow MT forecast Mid-term optimi- zation problem Water values Prices, Demand and inflow ST forecast Short-term opti- mization problem Operating schedule Deterministic accurate model

6/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

7/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

8/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

Capacity expansion and asset valuation

What we need ? European capacity expansion plan for the next 20-30 years with a special focus on the French system. Asset and project valuation. What we use ? European unit commitment problem. French unit commitment problem and rest of Europe inverse demand/supply curve. ֒ → Large scale stochastic optimization/equilibrium problems. ֒ → Electricity spot prices forecasts as marginal costs. NPVs computation to valuate assets and investments.

9/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

Capacity expansion and asset valuation

What we need ? European capacity expansion plan for the next 20-30 years with a special focus on the French system. Asset and project valuation. What we use ? European unit commitment problem. French unit commitment problem and rest of Europe inverse demand/supply curve. ֒ → Large scale stochastic optimization/equilibrium problems. ֒ → Electricity spot prices forecasts as marginal costs. NPVs computation to valuate assets and investments.

9/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

10/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

The optimization process : European capacity expansion problem

Demands, Installed and projected capacities European equi- librium model : ≈ 20 linked zones, 1 reservoir and ≈ 100 TU for each Marginal costs NPVs computation European Investment decisions Long-term European capacity expansion problem Optimal LT European energy mix

11/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

The optimization process : European capacity expansion problem

Demands, Installed and projected capacities European equi- librium model : ≈ 20 linked zones, 1 reservoir and ≈ 100 TU for each Marginal costs NPVs computation European Investment decisions Long-term European capacity expansion problem Optimal LT European energy mix

11/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

The optimization process : European capacity expansion problem

Demands, Installed and projected capacities European equi- librium model : ≈ 20 linked zones, 1 reservoir and ≈ 100 TU for each Marginal costs NPVs computation European Investment decisions Long-term European capacity expansion problem Optimal LT European energy mix

11/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

The optimization process : Focus on the French capacity expansion problem

European energy mix Rest of Europe inverse demand function French equili- brium model : 4 reservoirs and ≈ 100 TU Marginal cost NPVs computation French Investment decisions Long-term French capacity expansion problem French energy mix Import/Export Price

12/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

The optimization process : Focus on the French capacity expansion problem

European energy mix Rest of Europe inverse demand function French equili- brium model : 4 reservoirs and ≈ 100 TU Marginal cost NPVs computation French Investment decisions Long-term French capacity expansion problem French energy mix Import/Export Price

12/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

The optimization process : Focus on the French capacity expansion problem

European energy mix Rest of Europe inverse demand function French equili- brium model : 4 reservoirs and ≈ 100 TU Marginal cost NPVs computation French Investment decisions Long-term French capacity expansion problem French energy mix Import/Export Price

12/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion What we need ? What we use ? The optimization process : from multiple to a single zone problem

The optimization process : Focus on the French capacity expansion problem

European energy mix Rest of Europe inverse demand function French equili- brium model : 4 reservoirs and ≈ 100 TU Marginal cost NPVs computation French Investment decisions Long-term French capacity expansion problem French energy mix Import/Export Price

12/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

13/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

14/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Decomposition and splitting methods for network problems : scope

Goals Compute over a simplified pan-European framework, long term (30 years) electricity prices forecasts. Focus on a detailed French system to compute long term (30 years) electricity prices forecasts.

Figure: An example of the pan-European interconnection model

15/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Decomposition and splitting methods for network problems : scope

Goals Compute over a simplified pan-European framework, long term (30 years) electricity prices forecasts. Focus on a detailed French system to compute long term (30 years) electricity prices forecasts.

Figure: An example of the French-European splitting model

15/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Decomposition and splitting methods for network problems : the model

We model the problem as a multistage stochastic optimal control problem : min

p,x,u,f

E  

T−1

• t=0

 

z∈Z

cz(pz,t) +

• e∈E

le(fe,t)     , s.t pz,t + uz,t +

• e∈z+

fe,t = dz,t +

• e∈z−

fe,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, xz,t+1 = xz,t − uz,t + iz,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, pz,t, uz,t Ft, ∀z ∈ Z, ∀t = 0, . . . , T − 1, fe,t Ft, ∀e ∈ E, ∀t = 0, . . . , T − 1. ֒ → Only one wise European optimizer. ֒ → Non separable by zones because of network constraints. ֒ → Non separable by time because of storage units. ֒ → Non separable by scenario because of non-anticipativity. ֒ → Perfect information for all actors (no specific zonal information constraints).

16/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Decomposition and splitting methods for network problems : the model

We model the problem as a multistage stochastic optimal control problem : min

p,x,u,f

E  

T−1

• t=0

 

z∈Z

cz(pz,t) +

• e∈E

le(fe,t)     , s.t pz,t + uz,t +

• e∈z+

fe,t = dz,t +

• e∈z−

fe,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, xz,t+1 = xz,t − uz,t + iz,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, pz,t, uz,t Ft, ∀z ∈ Z, ∀t = 0, . . . , T − 1, fe,t Ft, ∀e ∈ E, ∀t = 0, . . . , T − 1. ֒ → Only one wise European optimizer. ֒ → Non separable by zones because of network constraints. ֒ → Non separable by time because of storage units. ֒ → Non separable by scenario because of non-anticipativity. ֒ → Perfect information for all actors (no specific zonal information constraints).

16/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Decomposition and splitting methods for network problems : the model

We model the problem as a multistage stochastic optimal control problem : min

p,x,u,f

E  

T−1

• t=0

 

z∈Z

cz(pz,t) +

• e∈E

le(fe,t)     , s.t pz,t + uz,t +

• e∈z+

fe,t = dz,t +

• e∈z−

fe,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, xz,t+1 = xz,t − uz,t + iz,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, pz,t, uz,t Ft, ∀z ∈ Z, ∀t = 0, . . . , T − 1, fe,t Ft, ∀e ∈ E, ∀t = 0, . . . , T − 1. ֒ → Only one wise European optimizer. ֒ → Non separable by zones because of network constraints. ֒ → Non separable by time because of storage units. ֒ → Non separable by scenario because of non-anticipativity. ֒ → Perfect information for all actors (no specific zonal information constraints).

16/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Decomposition and splitting methods for network problems : the model

We model the problem as a multistage stochastic optimal control problem : min

p,x,u,f

E  

T−1

• t=0

 

z∈Z

cz(pz,t) +

• e∈E

le(fe,t)     , s.t pz,t + uz,t +

• e∈z+

fe,t = dz,t +

• e∈z−

fe,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, xz,t+1 = xz,t − uz,t + iz,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, pz,t, uz,t Ft, ∀z ∈ Z, ∀t = 0, . . . , T − 1, fe,t Ft, ∀e ∈ E, ∀t = 0, . . . , T − 1. ֒ → Only one wise European optimizer. ֒ → Non separable by zones because of network constraints. ֒ → Non separable by time because of storage units. ֒ → Non separable by scenario because of non-anticipativity. ֒ → Perfect information for all actors (no specific zonal information constraints).

16/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Decomposition and splitting methods for network problems : the model

We model the problem as a multistage stochastic optimal control problem : min

p,x,u,f

E  

T−1

• t=0

 

z∈Z

cz(pz,t) +

• e∈E

le(fe,t)     , s.t pz,t + uz,t +

• e∈z+

fe,t = dz,t +

• e∈z−

fe,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, xz,t+1 = xz,t − uz,t + iz,t, ∀z ∈ Z, ∀t = 0, . . . , T − 1, pz,t, uz,t Ft, ∀z ∈ Z, ∀t = 0, . . . , T − 1, fe,t Ft, ∀e ∈ E, ∀t = 0, . . . , T − 1. ֒ → Only one wise European optimizer. ֒ → Non separable by zones because of network constraints. ֒ → Non separable by time because of storage units. ֒ → Non separable by scenario because of non-anticipativity. ֒ → Perfect information for all actors (no specific zonal information constraints).

16/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

17/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time ). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized : the information of the rest of the network is summarized on a unique random process qz.

18/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time ). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized : the information of the rest of the network is summarized on a unique random process qz.

18/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time ). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized : the information of the rest of the network is summarized on a unique random process qz.

18/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time ). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized : the information of the rest of the network is summarized on a unique random process qz.

18/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time ). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized : the information of the rest of the network is summarized on a unique random process qz.

18/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The demand side

For each zone z ∈ Z, we define Cz : L2(RT−1) → R that maps the algebraic import of Europe from zone z to its mean operating cost : Cz(qz) = min

p,x,u,f

E  

T−1

• t=0

 

• z′∈Z\{z}

cz′(pz′,t) +

• e∈E

le(fe,t)     , s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t, [−λz,t] pz′,t + uz′,t +

• e∈z′+

fe,t = dz′,t +

• e∈z′+

fe,t, ∀z′ ∈ Z\{z}, xz′,t+1 = xz′,t − uz′,t + iz′,t, ∀z′ ∈ Z\{z}, pz′,t, uz′,t, fe,t Ft ∀z′ ∈ Z\{z}, ∀e ∈ E. ֒ → We assume that we have the appropriate optimization tools to solve this problem (simplified model) for a given export random process. ֒ → We can also obtain marginal costs as output : −λz,t ∈ ∂Cz(qz).

19/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The demand side

For each zone z ∈ Z, we define Cz : L2(RT−1) → R that maps the algebraic import of Europe from zone z to its mean operating cost : Cz(qz) = min

p,x,u,f

E  

T−1

• t=0

 

• z′∈Z\{z}

cz′(pz′,t) +

• e∈E

le(fe,t)     , s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t, [−λz,t] pz′,t + uz′,t +

• e∈z′+

fe,t = dz′,t +

• e∈z′+

fe,t, ∀z′ ∈ Z\{z}, xz′,t+1 = xz′,t − uz′,t + iz′,t, ∀z′ ∈ Z\{z}, pz′,t, uz′,t, fe,t Ft ∀z′ ∈ Z\{z}, ∀e ∈ E. ֒ → We assume that we have the appropriate optimization tools to solve this problem (simplified model) for a given export random process. ֒ → We can also obtain marginal costs as output : −λz,t ∈ ∂Cz(qz).

19/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The demand side

For each zone z ∈ Z, we define Cz : L2(RT−1) → R that maps the algebraic import of Europe from zone z to its mean operating cost : Cz(qz) = min

p,x,u,f

E  

T−1

• t=0

 

• z′∈Z\{z}

cz′(pz′,t) +

• e∈E

le(fe,t)     , s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t, [−λz,t] pz′,t + uz′,t +

• e∈z′+

fe,t = dz′,t +

• e∈z′+

fe,t, ∀z′ ∈ Z\{z}, xz′,t+1 = xz′,t − uz′,t + iz′,t, ∀z′ ∈ Z\{z}, pz′,t, uz′,t, fe,t Ft ∀z′ ∈ Z\{z}, ∀e ∈ E. ֒ → We assume that we have the appropriate optimization tools to solve this problem (simplified model) for a given export random process. ֒ → We can also obtain marginal costs as output : −λz,t ∈ ∂Cz(qz).

19/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The optimization/equilibrium problem

Optimization problem The overall optimization problem can be written as : min

qz Fz(qz) + Cz(qz)

Equilibrium problem Define the inverse supply curve S−1

z

(qz) = ∂Fz(qz) : at which price zone z is willing to sell its extra production to neighboring zones. Define the inverse demand curve D−1

z

(qz) = −∂Cz(qz) : at which price all zones z′ ∈ Z\{z} are willing to by zone z production. An equilibrium is such that : ∃(qz, λz) ∈ L2(RT−1) × L2(RT−1), λz ∈ S−1

z

(qz) ∩ D−1

z

(qz). We can prove equivalence under mild assumptions.

20/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The optimization/equilibrium problem

Optimization problem The overall optimization problem can be written as : min

qz Fz(qz) + Cz(qz)

Equilibrium problem Define the inverse supply curve S−1

z

(qz) = ∂Fz(qz) : at which price zone z is willing to sell its extra production to neighboring zones. Define the inverse demand curve D−1

z

(qz) = −∂Cz(qz) : at which price all zones z′ ∈ Z\{z} are willing to by zone z production. An equilibrium is such that : ∃(qz, λz) ∈ L2(RT−1) × L2(RT−1), λz ∈ S−1

z

(qz) ∩ D−1

z

(qz). We can prove equivalence under mild assumptions.

20/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The optimization/equilibrium problem

Optimization problem The overall optimization problem can be written as : min

qz Fz(qz) + Cz(qz)

Equilibrium problem Define the inverse supply curve S−1

z

(qz) = ∂Fz(qz) : at which price zone z is willing to sell its extra production to neighboring zones. Define the inverse demand curve D−1

z

(qz) = −∂Cz(qz) : at which price all zones z′ ∈ Z\{z} are willing to by zone z production. An equilibrium is such that : ∃(qz, λz) ∈ L2(RT−1) × L2(RT−1), λz ∈ S−1

z

(qz) ∩ D−1

z

(qz). We can prove equivalence under mild assumptions.

20/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

21/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft, and the overall production cost as : F(q) =

• z∈Z

Fz(qz). ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time and scenario). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized.

22/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft, and the overall production cost as : F(q) =

• z∈Z

Fz(qz). ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time and scenario). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized.

22/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft, and the overall production cost as : F(q) =

• z∈Z

Fz(qz). ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time and scenario). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized.

22/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft, and the overall production cost as : F(q) =

• z∈Z

Fz(qz). ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time and scenario). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized.

22/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The supply side

For each zone z ∈ Z, we define Fz : L2(RT−1) → R that maps the algebraic export of one zone to its mean cost : Fz(qz) = min

pz ,xz ,uz

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = dz,t + qz,t, [λz,t] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft, and the overall production cost as : F(q) =

• z∈Z

Fz(qz). ֒ → We assume that we have the appropriate optimization tools to solve this problem (detailed model) for a given export random process. ֒ → Dynamic programming based methods could be used (separable cost function by time and scenario). ֒ → We can also obtain marginal costs as output : λz,t ∈ ∂Fz(qz). ֒ → The problem is decentralized.

22/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The demand side

We define C : L2(RT−1)(♯Z) → R that maps the exports on each node to their mean transportation cost : C(q) = min

f

E  

T−1

• t=0

 

e∈E

le(fe,t)     , s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t, ∀z ∈ Z, fe,t Ft. ֒ → A TSO is asking for the zone’s algebraic export to dispatch it into the network. ֒ → We assume that this problem is easy to solve : no time coupling constraints.

23/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The demand side

We define C : L2(RT−1)(♯Z) → R that maps the exports on each node to their mean transportation cost : C(q) = min

f

E  

T−1

• t=0

 

e∈E

le(fe,t)     , s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t, ∀z ∈ Z, fe,t Ft. ֒ → A TSO is asking for the zone’s algebraic export to dispatch it into the network. ֒ → We assume that this problem is easy to solve : no time coupling constraints.

23/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The demand side

We define C : L2(RT−1)(♯Z) → R that maps the exports on each node to their mean transportation cost : C(q) = min

f

E  

T−1

• t=0

 

e∈E

le(fe,t)     , s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t, ∀z ∈ Z, fe,t Ft. ֒ → A TSO is asking for the zone’s algebraic export to dispatch it into the network. ֒ → We assume that this problem is easy to solve : no time coupling constraints.

23/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The optimization/equilibrium problem

Optimization problem The overall optimization problem can be written as : min

q

• z∈Z

Fz(qz) + C(q) Equilibrium problem Define the inverse supply curves S−1

z

(qz) = ∂Fz(qz) : at which price zone z is willing to sell its production to the TSO. Define the inverse demand curve D−1(q) = −∂C(q) : at which price the TSO is willing to by the exports of each zone in order to insure network integrity. An equilibrium is such that : ∃(q, λ) ∈ L2(RT−1)(♯Z) × L2(RT−1)(♯Z), λ ∈

• ×z∈ZS−1

z

(qz)

• ∩ D−1(q).

We can prove equivalence under mild assumptions.

24/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The optimization/equilibrium problem

Optimization problem The overall optimization problem can be written as : min

q

• z∈Z

Fz(qz) + C(q) Equilibrium problem Define the inverse supply curves S−1

z

(qz) = ∂Fz(qz) : at which price zone z is willing to sell its production to the TSO. Define the inverse demand curve D−1(q) = −∂C(q) : at which price the TSO is willing to by the exports of each zone in order to insure network integrity. An equilibrium is such that : ∃(q, λ) ∈ L2(RT−1)(♯Z) × L2(RT−1)(♯Z), λ ∈

• ×z∈ZS−1

z

(qz)

• ∩ D−1(q).

We can prove equivalence under mild assumptions.

24/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

The optimization/equilibrium problem

Optimization problem The overall optimization problem can be written as : min

q

• z∈Z

Fz(qz) + C(q) Equilibrium problem Define the inverse supply curves S−1

z

(qz) = ∂Fz(qz) : at which price zone z is willing to sell its production to the TSO. Define the inverse demand curve D−1(q) = −∂C(q) : at which price the TSO is willing to by the exports of each zone in order to insure network integrity. An equilibrium is such that : ∃(q, λ) ∈ L2(RT−1)(♯Z) × L2(RT−1)(♯Z), λ ∈

• ×z∈ZS−1

z

(qz)

• ∩ D−1(q).

We can prove equivalence under mild assumptions.

24/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Splitting algorithm

Given a minimization problem : minq F(q) + C(q), splitting methods aims at finding a primal dual solution as an equilibrium : λ ∈ (∂F(q)) ∩ (−∂C(q)) . begin Choose arbitrarily q0. for k = 0, . . . , ∞ do

1

Find λk+1

z

∈ ∂Fz(qk

z ) 2

Compute qk+1

z

= arg minqz Cz(qz) +

• λk+1, qz
• +

1 2ρk

• qz − qk

z

• 2

For a given export quantity qk

z , compute the price λk+1 each zone z is wishing to

receive. ֒ → Problem easy to solve. For a given export price λk+1 compute the quantity qk+1

z

, The TSO will ask for (quantity not too far rom the previous one). ֒ → Problem also easy to solve.

25/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Splitting algorithm

Given a minimization problem : minq F(q) + C(q), splitting methods aims at finding a primal dual solution as an equilibrium : λ ∈ (∂F(q)) ∩ (−∂C(q)) . begin Choose arbitrarily q0. for k = 0, . . . , ∞ do

1

Find λk+1

z

∈ ∂Fz(qk

z ) 2

Compute qk+1

z

= arg minqz Cz(qz) +

• λk+1, qz
• +

1 2ρk

• qz − qk

z

• 2

For a given export quantity qk

z , compute the price λk+1 each zone z is wishing to

receive. ֒ → Problem easy to solve. For a given export price λk+1 compute the quantity qk+1

z

, The TSO will ask for (quantity not too far rom the previous one). ֒ → Problem also easy to solve.

25/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Splitting algorithm

Given a minimization problem : minq F(q) + C(q), splitting methods aims at finding a primal dual solution as an equilibrium : λ ∈ (∂F(q)) ∩ (−∂C(q)) . begin Choose arbitrarily q0. for k = 0, . . . , ∞ do

1

Find λk+1

z

∈ ∂Fz(qk

z ) 2

Compute qk+1

z

= arg minqz Cz(qz) +

• λk+1, qz
• +

1 2ρk

• qz − qk

z

• 2

For a given export quantity qk

z , compute the price λk+1 each zone z is wishing to

receive. ֒ → Problem easy to solve. For a given export price λk+1 compute the quantity qk+1

z

, The TSO will ask for (quantity not too far rom the previous one). ֒ → Problem also easy to solve.

25/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Splitting algorithm

Given a minimization problem : minq F(q) + C(q), splitting methods aims at finding a primal dual solution as an equilibrium : λ ∈ (∂F(q)) ∩ (−∂C(q)) . begin Choose arbitrarily q0. for k = 0, . . . , ∞ do

1

Find λk+1

z

∈ ∂Fz(qk

z ) 2

Compute qk+1

z

= arg minqz Cz(qz) +

• λk+1, qz
• +

1 2ρk

• qz − qk

z

• 2

For a given export quantity qk

z , compute the price λk+1 each zone z is wishing to

receive. ֒ → Problem easy to solve. For a given export price λk+1 compute the quantity qk+1

z

, The TSO will ask for (quantity not too far rom the previous one). ֒ → Problem also easy to solve.

25/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

Splitting algorithm

Given a minimization problem : minq F(q) + C(q), splitting methods aims at finding a primal dual solution as an equilibrium : λ ∈ (∂F(q)) ∩ (−∂C(q)) . begin Choose arbitrarily q0. for k = 0, . . . , ∞ do

1

Find λk+1

z

∈ ∂Fz(qk

z ) 2

Compute qk+1

z

= arg minqz Cz(qz) +

• λk+1, qz
• +

1 2ρk

• qz − qk

z

• 2

For a given export quantity qk

z , compute the price λk+1 each zone z is wishing to

receive. ֒ → Problem easy to solve. For a given export price λk+1 compute the quantity qk+1

z

, The TSO will ask for (quantity not too far rom the previous one). ֒ → Problem also easy to solve.

25/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

26/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (1/4)

We initialize the algorithm with N allocation scenarios for each zone z. These allocations represent the algebraic extra production in each zone/country (exports). (qz)[0] scenario 1 scenario 2 . . . scenario N These starting allocations have to be feasible and could be obtained by considering that there is no power exchanges between the zones.

27/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Marginal cost scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Marginal cost scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Marginal cost scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Marginal cost scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Marginal cost scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Marginal cost scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (2/4)

(p1, u1, x1) c1(p1) (p2, u2, x2) c2(p2) (p3, u3, x3) c3(p3) (p1, u1, x1) c4(p4) (q1)[k] (q2)[k] (q3)[k] (q4)[k] (λ1)[k] (λ2)[k] (λ3)[k] (λ4)[k] Allocation scenarios Solving sub- problems using approximate SDP and simulating the scenarios Marginal cost scenarios Network coordi- nation problem (q1)[k+1] (q2)[k+1] (q3)[k+1] (q4)k+1 New allocation scenarios

28/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (3/4)

For a given allocation (qz,t)[k] the subproblem of zone z : Fz(qz) = min

pz ,xz ,uz ∈L2

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = (qz,t)[k], xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. The allocations are not guaranteed to be Markovian processes. Thus we fix a Markovian observation variable : yz,t+1 = hj

t(yz,t, wt+1),

∀j, ∀t, and a model linking the allocations to this variable : E

• (qz,t)[k]
• yz,t
• = (Qz,t)[k](yz,t),

∀j, ∀t.

29/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (3/4)

For a given allocation (qz,t)[k] the subproblem of zone z : Fz(qz) = min

pz ,xz ,uz ∈L2

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = (qz,t)[k], xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. The allocations are not guaranteed to be Markovian processes. Thus we fix a Markovian observation variable : yz,t+1 = hj

t(yz,t, wt+1),

∀j, ∀t, and a model linking the allocations to this variable : E

• (qz,t)[k]
• yz,t
• = (Qz,t)[k](yz,t),

∀j, ∀t.

29/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (3/4)

For a given allocation (qz,t)[k] the subproblem of zone z : Fz(qz) = min

pz ,xz ,uz ∈L2

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = E

• (qz,t)[k]
• yz,t
• ,

xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. The allocations are not guaranteed to be Markovian processes. Thus we fix a Markovian observation variable : yz,t+1 = hj

t(yz,t, wt+1),

∀j, ∀t, and a model linking the allocations to this variable : E

• (qz,t)[k]
• yz,t
• = (Qz,t)[k](yz,t),

∀j, ∀t.

29/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (3/4)

For a given allocation (qz,t)[k] the subproblem of zone z : Fz(qz) = min

pz ,xz ,uz ∈L2

E T−1

• t=0

cz(pz,t)

• ,

s.t pz,t + uz,t = (Qz,t)[k](yz,t), = ⇒ (λz,t)[k] xz,t+1 = xz,t − uz,t + iz,t, pz,t, uz,t Ft. The allocations are not guaranteed to be Markovian processes. Thus we fix a Markovian observation variable : yz,t+1 = hj

t(yz,t, wt+1),

∀j, ∀t, and a model linking the allocations to this variable : E

• (qz,t)[k]
• yz,t
• = (Qz,t)[k](yz,t),

∀j, ∀t.

29/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (4/4)

For given allocation scenarios (qz,t)k and marginal costs (λz,t)k, the network coordination problem : min

q,f

E  

T−1

• t=0
• e∈E

le(fe,t) +

• z∈Z

λz,t, qz,t + 1 2ρ[k]

• qz,t − (qz,t)[k]
• 2

  , s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t. The network is seen as interconnected markets with prices given by the zonal marginal costs. The problem is separable by time steps and scenarios. Then we get the new allocation scenarios (qz,t)[k+1].

30/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (4/4)

For given allocation scenarios (qz,t)k and marginal costs (λz,t)k, the network coordination problem :

T−1

• t=0

E

• min

qt,ft

• e∈E

le(fe,t) +

• z∈Z

λz,t, qz,t + 1 2ρ[k]

• qz,t − (qz,t)k
• 2

, s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t. The network is seen as interconnected markets with prices given by the zonal marginal costs. The problem is separable by time steps and scenarios. Then we get the new allocation scenarios (qz,t)[k+1].

30/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Stochastic computing issues (4/4)

For given allocation scenarios (qz,t)k and marginal costs (λz,t)k, the network coordination problem :

T−1

• t=0

E

• min

qt,ft

• e∈E

le(fe,t) +

• z∈Z

λz,t, qz,t + 1 2ρ[k]

• qz,t − (qz,t)k
• 2

, s.t

• e∈z−

fe,t −

• e∈z+

fe,t = qz,t. The network is seen as interconnected markets with prices given by the zonal marginal costs. The problem is separable by time steps and scenarios. Then we get the new allocation scenarios (qz,t)[k+1].

30/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Plan

1

Some facts

2

Time decomposition

3

The long-term problems What we need ? What we use ? The optimization process : from multiple to a single zone problem

4

Decomposition and splitting methods for network problems Scope and modeling issues First splitting scheme : one producer / one consumer Second splitting scheme : ♯Z producers / one consumer

5

The splitting algorithm : stochastic computing issues

6

Conclusion

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• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

Conclusion

From the numerical point of view this splitting method is hard to tune. How can we choose a good ρ[k] ? Notice that the decomposition techniques in a stochastic framework needs special care to the information constraint. In our case having less information can be closer to reality than the perfect knowledge of a “wise” pan-European optimizer. Is there any relation with equilibrium problems ?

32/33

• A. Dallagi

European energy equilibrium and decomposition

Some facts Time decomposition The long-term problems Decomposition and splitting methods for network problems The splitting algorithm : stochastic computing issues Conclusion

European energy equilibrium and decomposition

Anes Dallagi

EDF R&D – OSIRIS – Optimization methods and tools

ICSP 2013 Bergamo, July 12th, 2013

33/33

• A. Dallagi

European energy equilibrium and decomposition