Optimization and Modeling in Energy Systems

Optimization and Modeling in Energy Systems

Panos M. Pardalos

Distinguished ProfessorCenter for Applied Optimization

Department of Industrial & Systems EngineeringUniversity of Florida, USA.

Laboratory of Algorithms and Technologies for Networks AnalysisNational Research University Higher School of Economics, Russia.

2013

IntroductionSmart Grid

Hydro-Thermal SchedulingActivities

“activity, operation”

ânèrgeia

1

: energeia

“Air, earth, water and fire are everexisting elements beginning andend of the Universe.”

Empedocles, pre - Socratic philosopher (c. 490 BC - 430 BC)

Optimization in Energy Panos M. Pardalos 2 (102)



Dynamics of global energy systems

Changes in oil and gas production and trade flows (shale gasand oil, new fields in USA / Canada, oil production in Iraq,changes in global economy and geopolitical balance)Renewable energy (solar, wind, biofuels etc.)Focus on energy efficiency / sustainable energy systems (climatechanges)CO2 emissions remain at record highIssues with Fossil Fuel SubsidiesOver 1 billion people have no access to electricityEnergy/Water/Environmental issuesAdvances in Technology/Modeling/Optimization




Introduction

World Energy Outlook 2012 1

© OECD/IEA 2012

A United States oil & gas transformation

US oil and gas production

The surge in unconventional oil & gas production has implications well beyond the United States

Unconventional gas

Conventional gas

Unconventional oil

Conventional oil

mboe/d

5

10

15

20

25

1980 1990 2000 2010 2020 2030 2035

1WEO-2012Optimization in Energy Panos M. Pardalos 4 (102)

http://www.slideshare.net/fullscreen/internationalenergyagency/world-energy-outlook-2012-presentation-to-press/1



Introduction


© OECD/IEA 2012

Iraq oil poised for a major expansion

Iraq oil production

Iraq accounts for 45% of the growth in global production to 2035; by the 2030s it becomes the second-largest global oil exporter, overtaking Russia

1

2

3

4

5

6

7

8

9

2012 2020 2035

mb/d North

Centre

South

Iraq oil exports

1

2

3

4

5

6

7

8

9

2012 2020 2035

mb/d Other

Asia





Introduction


© OECD/IEA 2012

Natural gas: towards a globalised market

Major global gas trade flows, 2010

Rising supplies of unconventional gas & LNG help to diversify trade flows, putting pressure on conventional gas suppliers & oil-linked pricing mechanisms

Major global gas trade flows, 2035





Introduction3 2BSmart Grid Conceptual Model

shown in Figure 3.

Figure 3 – Smart Grid Conceptual Model – Top Level

The conceptual model consists of several domains, each of which contains many applications and actors that are connected by associations, which have interfaces at each end:

• Actors may be devices, computer systems or software programs and/or the organizations that own them. Actors have the capability to make decisions and exchange information with other actors through interfaces.

• Applications are the tasks performed by the actors within the domains. Some applications are performed by a single actor, others by several actors working together.

• Domains group actors to discover the commonalities that will define the interfaces. In general, actors in the same domain have similar objectives. Communications within the same domain may have similar characteristics and requirements. Domains may contain other domains.

Report to NIST on the Smart Grid Interoperability Standards Roadmap June 17, 2009 21

Smart Grid must predict and intelligently respond to the behaviorand actions of power usersElectricity demand is growing worldwideMaking the grid more flexibleSecurity concernsNetwork expansion problems

Energy Systems are InterdependentIncreased use of natural gas for electricity generationLiquified Natural Gas terminalsNatural gas transportation and distribution systemsLong-term planning horizon for expansion planning

Hydro-Thermal SchedulingUncertainties in weather, demand, and pricesScenario reductionCO2 emissions constraints




Outline

1 Introduction

2 Smart GridIslandingReliability AnalysisStochastic Unit Commitment ProblemExpansion Planning

3 Hydro-Thermal Scheduling

4 ActivitiesPublicationsBooksEnergy Systems Journal




IslandingReliability AnalysisStochastic Unit Commitment ProblemExpansion Planning

Outline

1 Introduction








Power Grid Islanding – Background

Recently, the number of massive blackouts has increased.Potential reasons for these blackouts:

system limits, weak conditions, unexpected events, hidden failures,human errors, intentional attacks, natural disasters, etc.two main reasons: security and stability issues

People Location Date(s)2005 Java-Bali Blackout 100M Indonesia 2005-08-181999 Southern Brazil blackout 97M Brazil, south and southeastern 1999-03-112009 Brazil and Paraguay blackout 60M Brazil and Paraguay 2009-11-10/2009-11-11Northeast Blackout of 2003 55M North America, northeastern 2003-08-14/2003-08-152003 Italy blackout 55M Italy 2003-09-28Northeast Blackout of 1965 30M North America, northeastern 1965-11-09





Security of Power Systems

Security refers to the degree of risk in its ability to surviveimminent disturbances (contingencies) without interruption ofcustomer service.

ability to withstand the effects of contingencieskeep the power flows and bus voltages within acceptable limitsdespite changes in load or available resourcesavoidance of cascading outages leading to blackout

Power system security analysisfive states: normal, alert, emergency, extreme emergency andrestorativeplanning and operating criteria: N − 1,N − k contingency analysis;automatic generation control, transmission line switching, loadsheddingSCADA: Supervisory Control And Data Acquisition for securityassessment





Security of Power Systems

Security refers to the degree of risk in its ability to surviveimminent disturbances (contingencies) without interruption ofcustomer service.

ability to withstand the effects of contingencieskeep the power flows and bus voltages within acceptable limitsdespite changes in load or available resourcesavoidance of cascading outages leading to blackout

Power system security analysisfive states: normal, alert, emergency, extreme emergency andrestorativeplanning and operating criteria: N − 1,N − k contingency analysis;automatic generation control, transmission line switching, loadsheddingSCADA: Supervisory Control And Data Acquisition for securityassessment





Power Grid Islanding

Splitting a large power system into subsystemsmost parts of the system can operate in an acceptable conditioneach grid island is a self-sufficient subnetwork







(1) Islanding for self-healing strategylarge disturbances, such as simultaneous loss of severalgenerating units or major transmission linescatastrophic failure, by vulnerability analysiscontrol actions, to limit the extent of the disturbancefacilities the restoration process:

islanding with low generation-load imbalance in each island;smaller islands with slightly reduced capacity and being restoredquickly;the extent of disruption is limited







(2) Islanding for distributed generation systemrenewable energy resources connected to the existed systemcentralized generation becomes distributed generationan islanding operation occurs when the DG continues supplyingpower into the grid after power from the main utility is interrupted





DC Optimal Power Flow Model

Linear programming model

minPG,PS ,Pij ,θ

∑i∈I

(CGiPGi

+ CSiPSi

)

s.t. Pij = Bij (θi − θj ), ∀(i, j) ∈ L

PGi+

∑j<i

Pji = (PDi− PSi

) +∑j>i

Pij , ∀i ∈ I

− Pijmax≤ Pij ≤ Pijmax

, ∀(i, j) ∈ L

0 ≤ PGi≤ PGi max

, ∀i ∈ I

0 ≤ PSi≤ PDi

, ∀i ∈ I





Objective

The K -islanding problem is to separate a power grid into Kcomponents∪k Ik = I, Ik ∩ Ik ′ = ∅ for k 6= k ′, ik ∈ Ikeach component, an induced graph by IkObjective: minimizing the generating and load shedding cost

minPG,PS ,Pij ,θ,x,y,z

∑i∈I

(CGiPGi

+ CSiPSi

)





Complete Islanding Constraints

DC-OPF constraints:

Pij = Bij (θi − θj )zij , ∀(i, j) ∈ L−Pijmax ≤ Pij ≤ Pijmax , ∀(i, j) ∈ LPGi

+∑

j<i Pji = (PDi − PSi) +

∑j>i Pij , ∀i ∈ I

0 ≤ PGi≤ PGi max

, ∀i ∈ I0 ≤ PSi

≤ PDi , ∀i ∈ I

approximate active power flow on transmission lines

the maximum power flow on each line

the power balance at each bus, where the served satisfied at bus i is (PDi − PSi)

the maximum generating output

the limitation of load shedding by the maximum load

zij = 1, the constraint is the same as in standard DC-OPF since line (i, j) is inside of anisland; zij = 0, the constraint becomes Pij = 0 since line (i, j) is between two islands and isremoved.






Graph partitioning constraints:

K∑k=1

xik = 1, ∀i ∈ V∑i∈I

1gi xik ≥ 1,

∑i∈I

1di xik ≥ 1, ∀k

zij =∑

k

xik xjk , ∀(i, j) ∈ L

every node must belong to exactly one island

every island must have at least one generator and one load consumer

if two buses i and j are in the same island, there exists exactly one k ′(1 ≤ k ≤ K ) such thatxik′ = xjk′ = 1, xik = xjk = 0 for all other ks, and thus zij = 1. Otherwise, zij = 0.







K∑k=1

xik = 1, ∀i ∈ V∑i∈I

1gi xik ≥ 1,

∑i∈I

1di xik ≥ 1, ∀k

zij =∑

k











K∑k=1

xik = 1, ∀i ∈ V∑i∈I

1gi xik ≥ 1,

∑i∈I

1di xik ≥ 1, ∀k

zij =∑

k











K∑k=1

xik = 1, ∀i ∈ V∑i∈I

1gi xik ≥ 1,

∑i∈I

1di xik ≥ 1, ∀k

zij =∑

k










Connectivity constraints to ensure that every island is connected:

single commodity flow model to ensure connectivity

Island 𝑘

Island 𝐾

… …

… Island 1

…

… …

…

…

𝑦𝑘 𝑖𝑘





IEEE-30-Bus System

30 buses, 41 lines6 generatorstotal generation capacity 130 MW, and total load demand 137.5 MW





Complete Islanding

By GI-DC-OPF model

K Root Buses Obj. Islands Real Gen. Gen. Cap. Dem. Sat. Dem.

1 1 22.5 Island 1: 1-30 130.0 130.0 137.5 94.5%2 1,13 22.5 Island 1: 1,2,5-8,15,21-30 70.0 70.0 76.9 91.0%

Island 2: 3,4,9-14,16-20 60.0 60.0 60.6 99.0%3 1,8,13 22.5 Island 1: 1,2,5-7,9-11,14,15,18-24 85.0 85.0 87.3 97.4%

Island 2: 8,25-30 15.0 15.0 16.5 90.9%Island 3: 3,4,12,13,16,17 30.0 30.0 33.7 89.0%

4 1,8,11,13 37.5 Island 1: 1,3,4 10.0 15.0 10.0 100.0%Island 2: 2,5-8.21.22.24-30 55.0 55.0 65.5 84.0%Island 3: 9-11,16,17,19,20 30.0 30.0 30.0 100.0%Island 4: 12-15,18,23 30.0 30.0 32.0 93.8%

5 1,5,8,11,13 37.5 Island 1: 1,3,4 10.0 15.0 10.0 100.0%Island 2: 5,7 15.0 15.0 22.8 65.8%Island 3: 8,25-30 15.0 15.0 16.5 90.9%Island 4: 2,6,9-11,17,19-24 55.0 55.0 55.9 98.4%Island 5: 12-16,18 30.0 30.0 32.3 92.9%

6 1,2,5,8,11,13 112.5 Island 1: 1,3 2.4 15.0 2.4 100.0%Island 2: 2,4 7.6 25.0 7.6 100.0%Island 3: 5,7 15.0 15.0 22.8 65.8%Island 4: 8,25-30 15.0 15.0 16.5 90.9%Island 5: 6,9-11,21-24 30.0 30.0 35.2 85.2%Island 6: 12-20 30.0 30.0 53.0 56.6%





Complete Islanding

Figure : IEEE-30-Bus network with two islands

(K = 2, Root Buses 1,13)Optimization in Energy Panos M. Pardalos 19 (102)




Complete Islanding

Figure : IEEE-30-Bus network with three islands

(K = 3, Root Buses 1,8,13)Optimization in Energy Panos M. Pardalos 19 (102)




Complete Islanding

Figure : IEEE-30-Bus network with four islands





Extensions

More considerations and constraints:adding security and stability constraints to our current model. Forexample, the voltage constraints on each bus to ensure they worknormally.adding the generation and load demand balance constraints. Foreach island, there is a limit on load shedding to prevent blackouts.

r(0 < r ≤ 1): total generation of an island should be large than rtimes of its total load consumption, while a small part of unsatisfieddemand is allowed∑

i∈I

PGixik ≥ r ·

∑i∈I

(PDi− PSi

)xik , ∀k

adding physical location constraints. For example, physically closebuses should be divided into one island to reduce transmissioncost.

C = (i, j) ∈ L : bus i and bus j should be in the same island: theset for buses should be within the same island

xik = xjk , ∀(i, j) ∈ C,∀k





Extensions

More considerations and constraints:adding security and stability constraints to our current model. Forexample, the voltage constraints on each bus to ensure they worknormally.adding the generation and load demand balance constraints. Foreach island, there is a limit on load shedding to prevent blackouts.

r(0 < r ≤ 1): total generation of an island should be large than rtimes of its total load consumption, while a small part of unsatisfieddemand is allowed∑

i∈I

PGixik ≥ r ·

∑i∈I

(PDi− PSi

)xik , ∀k

adding physical location constraints. For example, physically closebuses should be divided into one island to reduce transmissioncost.

C = (i, j) ∈ L : bus i and bus j should be in the same island: theset for buses should be within the same island

xik = xjk , ∀(i, j) ∈ C,∀k





Outline

1 Introduction








N − k Contingency – Introduction

Contingency analysis is a key function in the EnergyManagement System (EMS).

a set of unexpected events happening within a short durationfailures of buses (generators, substations, etc) or transmission anddistribution lines

N − 1 contingency, not sufficient to model the application inreality to evaluate the vulnerabilities of power gridsN − k contingency: reflecting a larger variation of vulnerabilities,a substantial computational burden for analysisTwo steps: contingency selection and evaluation





N − k OPF Model

z(δ, σ) = ming,s,f ,θ

∑i∈I

(hi gi + ri si )

s.t. fij = bij (θi − θj )(1 − δi )(1 − δj )(1 − σij ), ∀(i, j) ∈ L

− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑

jfij + (Di − si ), ∀i ∈ I

0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I

δ, σ: selected of k failures on buses and/or linesgenerating and load shedding cost for economic analysis





N − k OPF Model


∑i∈I

(hi gi + ri si )


− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑


0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I

approximate active power flow on transmission lines byconsidering failures on two ending buses and the line





N − k OPF Model


∑i∈I

(hi gi + ri si )


− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑


0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I

line capacity limits





N − k OPF Model


∑i∈I

(hi gi + ri si )


− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑


0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I

power balance at each bus, where the served satisfied at bus i isDi − si





N − k OPF Model


∑i∈I

(hi gi + ri si )


− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑


0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I

generating output limits





N − k OPF Model


∑i∈I

(hi gi + ri si )


− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑


0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I

the limitation of load shedding by the maximum load





Interdiction Analysis

Interdiction: worst case scenario analysisA group of terrorists will attack the power grid with limitedresources to maximize the disruptionSalmeron, Wood, Baldick (2004)

maxδ,σ

ming,s,f ,θ

∑i∈I

(hi gi + ri si )


− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑


0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I∑i∈Iδi +

∑(i,j)∈L

σij = k

δi , σij ∈ 0, 1, ∀i ∈ I, (i, j) ∈ L





Selection of k Buses

assume all failures, or attacks happen on buses,∑

i∈I δi = k .The random failure: selects k buses for failure with equalprobability;The degree based method: selects k buses starting with thehighest degree bus, till the k th highest degree;The maximum-traffic and minimum-traffic methods: themaximum-traffic method selects k buses with highest Tis, whilethe minimum-traffic method selects k buses with lowest Tis,where from OPF model

Ti = |gi − (Di − si )|+∑j:j∈I

|bij (θi − θj )|






The node betweenness method: finds k buses with highest nodebetweenness. The node betweenness for bus i is defined as

CB(i) =∑

s 6=i 6=t∈I

nst (i)nst

where nst is the number of shortest paths from s to t , and nst (i)is the number of shortest paths from s to t that pass through abus i .

Floyd-Warshall algorithm, Johnson’s algorithm, Brandes’ algorithm






The critical node detection problem (CNP) method: detects a setof vertices in a graph whose deletion results in the graph havingthe minimum pairwise connectivity between the remainingvertices. It is is NP-hard and can be formulated as a mixedinteger linear problem like:

min∑

i,j:vi ,vj∈Vxij

s.t. xij + δi + δj ≥ 1, ∀(i, j) ∈ L

xij + xjt − xit ≤ 1, ∀i, j, t ∈ I

xij − xjt + xit ≤ 1, ∀i, j, t ∈ I

− xij + xjt + xit ≤ 1, ∀i, j, t ∈ I∑i:i∈I

δi = k

δi , xij ∈ 0, 1, ∀i, j ∈ I

Arulselvan, Commander, Elefteriadou, Pardalos (2009)





Selection of k Lines

The edge betweenness method: finds k edges with highest edgebetweenness. The edge betweenness is adapted from nodebetweenness, and it can be expressed as

C(i,j)B =∑s,t∈I

nst (i , j)nst

,

where nst is the number of shortest paths from bus s to bus t ,and nst (i , j) is the number of shortest paths from bus s to t thatpass through line (i , j).





Interdiction Analysis

Interdiction: worst case scenario analysisA group of terrorists will attack the power grid with limitedresources to maximize the disruptionSalmeron, Wood, Baldick (2004)

maxδ,σ

ming,s,f ,θ

∑i∈I

(hi gi + ri si )


− Fij ≤ fij ≤ Fij , ∀(i, j) ∈ L∑j

fji + gi =∑


0 ≤ gi ≤ Gi (1 − δi ), ∀i ∈ I

0 ≤ si ≤ Di , ∀i ∈ I∑i∈Iδi +

∑(i,j)∈L

σij = k

δi , σij ∈ 0, 1, ∀i ∈ I, (i, j) ∈ L





Case Study

1 2 3 4 5 6 7 8 9 102000

3000

4000

5000

6000

7000

8000

9000

10000

The Number of Failures on Buses

Gen

erat

ing

and

Load

She

ddin

g C

ost

RandomDegreeBetweennessMax TrafficMin TrafficCNPInterdiction

Figure : Generating and load shedding cost vs. failed buses (RTS-96 System)





Case Study

1 2 3 4 5 6 7 8 9 102000

3000

4000

5000

6000

7000

The Number of Failures on Lines

Gen

erat

ing

and

Load

She

ddin

g C

ost

BetweennessInterdiction

Figure : Generating and load shedding cost vs. failed lines (RTS-96 System)





Conclusions

random failures, degree attack, maximum-traffic attack,minimum-traffic attack and betweenness attackthe critical node detection method only works with thecontingencies consisting of failures only on busesthe interdiction model can select contingencies consisting bothbuses and linesinterdiction always select the most crucial componentsthe maximum-traffic method and critical node detection methodselect the second most crucial buses, while the minimum-trafficmethod finds the least crucial ones





Outline

1 Introduction








Stochastic Unit Commitment Problem – Introduction

Unit Commitment in Electrical Power GenerationA key optimization problem in power system operations and control(short-term)Minimizing total generation costTechnical constraints (minimum on/off, ramping, reserves, capacity,etc.)Mixed Integer Nonlinear Programs with binary variable for on/offstatus

Approaches to Unit CommitmentsPriority listDynamic programmingLagrangian relaxationBranch-and-bound based MILP algorithmsBenders decomposition, etc.





Introduction

Uncertainties related to the Deregulation of Power MarketUncertainties due to the High Penetration of Renewable EnergyUncertainties of infrastructure stability (generator, transmissionline, failure)Generalized Unit commitment

Commitment of unitsEconomic DispatchOperating ReservesPower Transmission





Introduction

Approaches to handle uncertainties in unit commitment problemsReserve requirements for operating unitsStochastic programming models (two-stage and multi-stage)Robust Optimization models (given uncertainty levels)





Features of the model

Two-stage stochastic optimization problem for day-aheadschedulingFirst stage unit commitmentSecond stage economic dispatch and power transmissionNetwork constraints and power loss calculationChance constraints for risk controlBounds on risk measures such as VaR and CVaR





Piecewise linear fuel cost function

𝐹(𝑝)

𝑝

Δ1 0

𝑐1

𝑐2

𝑐3

𝑐4

𝑐5

Δ2 Δ3 Δ4 Δ5

Figure : Piecewise Linear Approximation of the Fuel Cost FunctionOptimization in Energy Panos M. Pardalos 38 (102)




Risk Controlling (Why and How?)

Why pay for things which are very unlikely to happen?Extremely “bad” scenarios with very small probability.Probabilistic constraints:

Determine how “bad” it is;Determine how “unlikely” it is.

Limiting VaR (Value at Risk) or CVaR (Conditional Value at Risk).





Take Risk and Have It Under Control

Suppose L(x,y) is the loss function of random variable x and decisionvariable y .

Value at Risk (VaR):

VaRθ = inf l ∈ R : P(L(x , y) ≥ l) ≤ 1− θ

Conditional value at Risk (CVaR):

CVaRθ = E (L(x , y)|L(x , y) ≥ VaRθ)





VaR and CVaR





Cost V.S. Risk

Figure : Minimal Cost V.S. Limit of CVaR

CVaR-85%

CVaR-90%

CVaR-95%

CVaR-99%

Cost





Handling a large number of scenarios in the SMIP

The problem becomes difficult to solve when there is a largenumber of scenariosThe structure makes the problem highly decomposableUsing Benders decomposition to approximate the second stageSecond stage becomes many separable problems in parallel





Master Problem

MinT∑

t=1

∑g∈G

(SUgtvgt + SDgtwgt )

s.t. (u, v ,w) ∈ Uy t (ξ) ≥ 0, ∀t ∈ T , ξ ∈ Ξ,

ηt +∑ξ∈Ξ

Pr(ξ)

1− θ y t (ξ) ≤ φ,∀t ∈ T .

Optimality cuts

Feasibility cuts





Generating Optimality Cuts

Need to solve the following subproblem of scenario ξ,

MinT∑

t=1

∑g∈G

Fi

(pξgt

)s.t. (p(ξ), s(ξ), f (ξ), d(ξ), β(ξ)) ∈ F(u)

d ti (ξ) + x t

i (ξ) ≥ Dti (ξ), t = 1, . . . ,T , ∀i ∈ N,∑

i∈N

Lti (ξ)x t

i (ξ) ≤ ηt + y t (ξ), ∀t ∈ T ,

The optimal dual solution can help constructing an Optimality cut.





Generating feasibility Cuts

Need to solve the following violation testing problem of scenario ξ,

MinT∑

t=1

αt +∑g∈G

δti

s.t. (p(ξ), s(ξ), f (ξ), d(ξ), β(ξ)) ∈ F(u)

δti +

∑g∈Gi

sgt (ξ) ≥ RSit (ξ), t ∈ T , ∀i ∈ N,

d ti (ξ) + x t

i (ξ) ≥ Dti (ξ), t = 1, . . . ,T , ∀i ∈ N,

−αt +∑i∈N

Lti (ξ)x t

i (ξ) ≤ ηt + y t (ξ), ∀t ∈ T ,

The optimal dual solution is actually an extreme ray of the dual to thesubproblem, which helps construct a feasibility cut.





Conclusions & Future Research

Stochastic security constrained unit commitment with CVaRconstraintBenders’ decomposition for large number of scenarioMore extensive numerical experimentsModeling of quick-start generatorsModeling of demand response, etc.Long-term power system expansion planning with embeddedstochastic unit commitmentNew emerging technologies, in particular in solar energy,can drastically change the dynamics of energy systems.





Outline

1 Introduction








Electricity and Natural Gas Network ExpansionProblem

Considers demand for both electricity and gasAccounts for expansion of

LNG terminalsGas Distribution NetworkElectricity Network

Captures uncertainty in the forecasted future electricity and gasdemand





Natural Gas Introduction

Natural gas release less green house gas than oil and coal whilegiving fair amount of energy when burnt.More and more gas fired power plants are built to protectenvironment. (Give the rise to Stochastic Unit Commitmentproblems.)From 1980 to 2007, the world’s demand of natural gas hasdoubled, 52.9 to 108 trillion cubic feet (EIA 2010).The demand is predicted to increase by 44% more until 2035 toaround 156 TCF (EIA 2010).Natural gas-fired electricity production increases by 2.1 % peryear from 3.9 trillion kilowatthours in 2007 to 6.8 trillionkilowatthours in 2035 (EIA 2010)





Natural Gas Introduction

Natural gas release less green house gas than oil and coal whilegiving fair amount of energy when burnt.More and more gas fired power plants are built to protectenvironment. (Give the rise to Stochastic Unit Commitmentproblems.)From 1980 to 2007, the world’s demand of natural gas hasdoubled, 52.9 to 108 trillion cubic feet (EIA 2010).The demand is predicted to increase by 44% more until 2035 toaround 156 TCF (EIA 2010).Natural gas-fired electricity production increases by 2.1 % peryear from 3.9 trillion kilowatthours in 2007 to 6.8 trillionkilowatthours in 2035 (EIA 2010)





World Gas Consumption Trend

Figure : World gas consumption in billion cubic feet (DOE)





Electricity Generation

Figure : Electricity generation capacity additions by fuel type, 2009-2035(gigawats) (EIA)





World’s Natural Gas Reserves

According to EIA International Energy Outlook 2010,World average RTP (Reserve To Production) ratio is about 60years.Central and South America RTP is about 46 years.72 and 68 years are for Russia and Africa.More than 100 years for Middle East.US production rate is about 20 TCF per year and its estimatedreserves are about 1747.47 TCF. (RTP: 87 years.)

In national level, how to analyze the whole electricity and naturalgas systems by considering transmission networks and LNGlocations together?





Our Modeling Aims

System Level (linear model with line reverse for gas network)Capable of handling different demand and supply patternsGas transportation network expansion and LNG location due toimbalanced reserves and different economic growths in theworld.Electricity generation and transmission network capacityexpansion to satisfy growing demand.Meet the electricity and gas demands minimizing the costs.





The CVaR Risk Management Constraints

∑(i,j)∈A+

i

fij (ξ)−∑

(j,i)∈A−i

(1− lji )fji (ξ) + si (ξ) = dGi (ξ) + dGP

i (ξ)− µi (ξ),

∀i ∈ NG \ NR , ξ ∈ Ξ

µi (ξ) = 0, ∀i ∈ NG \ NR , ξ ∈ Ξ

µi ≥ µi (ξ) ≥ 0, ∀i ∈ NR , ξ ∈ Ξ∑i∈NG

µi (ξ) ≤ η + w(ξ), ∀ξ ∈ Ξ

w(ξ) ≥ 0, ∀ξ ∈ Ξ

η +∑ξ∈Ξ

Pr(ξ)

1− ζ w(ξ) ≤ φ





Problem Formulation

The resulting problem is a two stage stochastic programFirst stage corresponds to investment decisionsSecond stage corresponds to operational constraints (generationand transmission decisions)Use Benders decomposition to solve the problem





Restricted Master Problem[RMP]:

Min∑

(i,j)∈AG

∑k∈Kij

ckij α

kij +

∑i∈NLNG

∑k∈Ki

cki β

ki

+∑

(i,j)∈AELC

pijxij +∑

i∈NGEN

∑k∈Ki

r ki yk

i +∑ξ∈Ξ

Prob(ξ)π(ξ)

s.t.

uij = uij +∑k∈Kij

∆ki,jα

kij , ∀(i , j) ∈ AG,

vi = v i +∑k∈Ki

∆ki β

ki , ∀i ∈ NLNG,

Gi = gmaxi

+∑k∈Ki

∆ki yk

i , ∀i ∈ NGEN ,

π(ξ) + a1k (ξ)u + a2

k (ξ)v + a3k (ξ)x + a4

k (ξ)G ≥ a5k (ξ), ∀ ∈ K (ξ), ξ ∈ Ξ,

αkij ∈ 0,1, ∀k ∈ Kij , (i , j) ∈ A, βk

i ∈ 0,1, ∀k ∈ Ki , i ∈ NLNG,

xij ∈ 0,1, ∀(i , j) ∈ AELC , yki ∈ 0,1, ∀k ∈ Ki , i ∈ NGEN





SubproblemsThe subproblem of scenario ξ, given the first stage solution(u, v , x , y ):[SP(ξ)]:

Min Generation and Transportation Costsubject toFlow balance Constraints,Physical Limits Constraints,Risk Management constraints.





Conclusions and Future Work

Mixed integer programming modelElectricity and Gas Network expansion planningLNG terminal locationGenerators Capabilities

Risk constraints (VaR / CVaR)MILP implemented in C++ and solved by CPLEXr 12.2,decomposition algorithm is being developedNumerical comparison for large-scale problems




Outline

1 Introduction







Hydro-Thermal Power Systems

Figure : Itaipu, Brazil

Figure : Coal plant

Figure : Gas plant




Water Balance




The World is Uncertain!?

...linear programming methods (to) be extended to include the case ofuncertain demands for the problem of optimal allocation of a carrierfleet to airline routes to meet an anticipated demand distribution...

George B. DantzigLinear Programming under UncertaintyManagement Science, 1:3 & 4, 197–206, 1955




What Exactly is Uncertain?

Such an energy system is subject to different uncertainties:stochastic fuel prices,

stochastic electricity demand,

stochastic (water) inflows,

and in the liberalized market in addition also:stochastic electricity spot prices,

stochastic CO2 prices.




What Exactly is Uncertain?

Such an energy system is subject to different uncertainties:stochastic fuel prices,

stochastic electricity demand,

stochastic (water) inflows,

and in the liberalized market in addition also:stochastic electricity spot prices,

stochastic CO2 prices.




What is the Problem?

Figure : Hydro Scheduling Tradeoff




Multi-Stage Stochastic Optimization

z := min c1(u1) + minEω2∈Ω2

[ct(ut (ωt )

)+ . . .+

+ minEωt∈Ωt

[ct(ut (ωt )

)]+ . . .+

+ minEωT∈ΩT

[cT(uT (ωT )

)]. . .

](1)




Linear Operational Constraints

Electricity Demandload blocks

Hydroreservoir security constraintslimits on total outflowpeak modulation constraints in run-of-the-river plantsrun-of-the-river plants generationirrigation for hydro reservoirsinitial fill-up of reservoirstailwater elevationrisk aversion




Linear Operational Constraints (cont’d)

Thermalpiecewise linear costmust-run thermal plantsfuel consumption limitsfuel consumption rate limitminimum generation constraint for a set of thermal plantsmultiple fuelsunit commitment

Generation Reservespinning reservegeneration reserve




Linear Operational Constraints (cont’d)

Power Transmission Networkinterconnection modellinearized power flow modeltransmission losses

Natural Gas Networkproduction limitspipeline flow limitssupply and demand balance




Assumptions

The considered energy system has the following characteristics:

1 hydro-dominated power system,

2 mid-term to long-term optimization horizon,

3 all operational constraints can be linearized, and




Is the “Hydro-Thermal Scheduling World” Linear?

No!

...but piecewise linear is a very good approximation!

D.D. Wolf and Y. SmeersThe Gas Transmission Problem Solved by an Extension of the Simplex AlgorithmManagement Science, 46, 1454–1465, 2000

R. Rubio-Barros, D. Ojeda-Esteybar, and A. Vargas,Energy Carrier Networks: Interactions and Integrated Operational PlanningHandbook of Networks in Power Systems, P.M. Pardalos, S. Rebennack, M.V.F. Pereira,N. Iliadis, and A. Sorokin (ed.), Springer, to appear




Is the “Hydro-Thermal Scheduling World” Linear?

No!

...but piecewise linear is a very good approximation!

D.D. Wolf and Y. SmeersThe Gas Transmission Problem Solved by an Extension of the Simplex AlgorithmManagement Science, 46, 1454–1465, 2000

R. Rubio-Barros, D. Ojeda-Esteybar, and A. Vargas,Energy Carrier Networks: Interactions and Integrated Operational PlanningHandbook of Networks in Power Systems, P.M. Pardalos, S. Rebennack, M.V.F. Pereira,N. Iliadis, and A. Sorokin (ed.), Springer, to appear




Fundamental Modeling

Assumption: Perfect competition(absence of market power)

centrally dispatched = market-based dispatch

allows fundamental modeling of prices

G. Gross and D. FinlayGeneration supply bidding in perfectly competitive electricity marketsComputational & Mathematical Organizations Theory, 6, 83–98, 2000

P. Lino, L.A.N. Barroso, M.V.F. Pereira, R. Kelman, and M.H.C. FampaBid-Based Dispatch of Hydrothermal Systems in Competitive MarketsAnnals of Operations Research, 120, 81–97, 2003




Solution Methods

Classification with respect to inflow uncertainty methodology:

1 deterministic models,

2 scenario-based methods,

3 sampling-based methods.

W. YehReservoir management and operations models: A state of the art reviewWater Resources Research, 21, 1797–1818, 1985

J. LabadieOptimal operation of multireservoir systems: State-of-the-art reviewJournal of Water Resources Planning and Management, 130, 93–111, 2004




Scenario-Based Methods

IdeaScenario-based methods generate up-front a set of realizations ofthe random space. The realizations are then used to generate theextensive form of the stochastic program. These are then typicallysolved exactly.

typically LP problems(very) large-scale mathematical programssolution quality depends on the approximation of the realizationsto the original, stochastic programreach limitations for multi-stage problems“scenario tree” or “fan”




Scenario-Based Methods (cont’d)

Advantage: various uncertainties (correlated and uncorrelated)can be incorporated into the model; e.g., hydro inflows, electricityspot prices, contract prices, electricity demand, and fuel prices

scenario generation; (Wallace and co-workers)

J. Dupacova, G. Consigli, and S. WallaceScenarios for multistage stochastic programsAnnals of Operations Research, 100, 25–53, 2000

K. Høyland and S. W. WallaceGenerating Scenario Trees for Multistage Decision ProblemsManagement Science, 47, 295–307, 2001

K. Høyland, M. Kaut and S.W. WallaceA heursitics for generating scenario trees for multistage decision problemsComputational Optimization and Applications, 23, 169–185, 2003





Advantage: various uncertainties (correlated and uncorrelated)can be incorporated into the model; e.g., hydro inflows, electricityspot prices, contract prices, electricity demand, and fuel prices

scenario generation; (Wallace and co-workers)

J. Dupacova, G. Consigli, and S. WallaceScenarios for multistage stochastic programsAnnals of Operations Research, 100, 25–53, 2000

K. Høyland and S. W. WallaceGenerating Scenario Trees for Multistage Decision ProblemsManagement Science, 47, 295–307, 2001

K. Høyland, M. Kaut and S.W. WallaceA heursitics for generating scenario trees for multistage decision problemsComputational Optimization and Applications, 23, 169–185, 2003





LimitationsIn order to capture the correlation among the inflows of the reservoirs,a large scenario tree may be required, leading to very large scaledeterministic equivalent programs.

S.-E. Fleten and S.W. WallaceDelta-Hedging a Hydropower Plant Using Stochastic Programmingin “Optimization in the Energy Industry,” J. Kallrath, P.M. Pardalos, S. Rebennack, andM. Scheidt (ed.), Springer, series Energy Systems, 1, 507–524, 2009




Sampling-Based Methods

IdeaSampling-based methods generate samples of the random spaceon-the-fly and solve the resulting problems approximately.

typically Dynamic Programming methodsstatistical convergence resultsmay possess “Curse of Dimensionality”very popular for hydro-thermal scheduling




Sampling-Based Methods

The major lines of research for sampling-based methods towardshydro-thermal scheduling is driven by the methods of

Stochastic Dynamic Programming (SDP)Stochastic Dual Dynamic Programming (SDDP)

B.F. Lamond and A. BoukhtoutaOptimizing long-term hydro-power production using markov decision processesInternational Transactions in Operational Research, 3, 223–241, 1996




Solution Methods

When solving the One Stage Dispatch Problem, one encounters (atleast) the following two challenges:

1 the (conditioned) distribution of ω is not known and expected tobe continuous, and

2 One Stage Dispatch Problem cannot be solved computationallyfor the whole continuum of reservoir levels vt .




Solution Methods

When solving the One Stage Dispatch Problem, one encounters (atleast) the following two challenges:

1 the (conditioned) distribution of ω is not known and expected tobe continuous, and

2 One Stage Dispatch Problem cannot be solved computationallyfor the whole continuum of reservoir levels vt .




Solution Methods (cont’d)

Stochastic Dynamic Programming (SDP) and Stochastic DualDynamic Programming (SDDP) overcome these two challenges inthe following way:

1 These inflows are modeled as a linear autoregressive model viaa continuous Markov Process.

2 The set of reservoir levels is discretized into M values. Thefunction zt is then approximated either via

interpolation of the M points (in SDP), or viaextrapolation of the M points (in SDDP).




SDP: Expected Future Cost Interpolation

solve “backwards” in timeM forward samplesL backward openingsdiscretize storage values vector into N1 valuesdiscretize inflows into N2 values




SDP: Challenges

1 “curse of dimensionality”(N1 · N2)I states in each stage

2 static discretization of state space

3 lack of solution quality measure




SDP: Challenges (cont’d)

...The situation with respect to stochastic dynamic programming isthat there are, as yet, no widely applicable computational devisesother than discrete dynamic programming (DDP). Because of theircurse of dimensionality, [...] DDP is not adequate for solving manywater resource problems of interest. The largest numerical stochasticdynamic programming solutions [...] are for problems having at mosttwo or three state variables....

S. YakowitzDynamic programming applications in water resourcesWater Resources Research, 18:4, 673–696, 1982




SDDP: Expected Future Cost Extrapolation

use information of dual to underestimate future cost function“Benders cuts”backwards pass: zforward Monte Carlo simulation: zstop when convergence criteriabis satisfied




SDDP: Strength

1 no curse of dimensionality

2 state space is discretized dynamically

3 statistical solution quality measure

M.V.F. PereiraOptimal stochastic operations scheduling of large hydroelectric systemsInternational Journal of Electrical Power & Energy Systems, 11, 161–169, 1989

M.V.F. Pereira and L.M.V.G. PintoMulti-stage stochastic optimization applied to energy planningMathematical Programming, 52, 359–375, 1991




SDP vs. SDDP

Figure : Approximation of FCF: SDP versus SDDP




Oil Price Scenarios

This is a tree; neither a Markov process nor a Markov Chain




Scenario Tree

Figure : Scenario Tree with 4 stages




Unifying Two Worlds: ‘Tree’ vs. ‘Sampling’

Scenario tree ‘on top’ of the stochastic (dual) dynamicprogramming.

For each stage t and state, we need to solve St one stagedispatch problems.

Instead of M cuts for the future cost function, we obtain M · St .

Computational complexity increases with the size of the tree.

Works also in combination with electricity demand uncertainty.




Unifying Two Worlds: ‘Tree’ vs. ‘Sampling’

Scenario tree ‘on top’ of the stochastic (dual) dynamicprogramming.

For each stage t and state, we need to solve St one stagedispatch problems.

Instead of M cuts for the future cost function, we obtain M · St .

Computational complexity increases with the size of the tree.

Works also in combination with electricity demand uncertainty.




Case Study I: Panama

FortunaMin. Storage [hm3]: 4.67

Max. Storage [hm3]: 172.30

Ø Production [ MWm3/sec.

]: 6.67

Capacity [MW]: 300.00

Canjilone

Min. Storage [hm3]: 34.63



]: 1.02


EstrellaMin. Storage [hm3]: 0.06



]: 3.05


Los ValleMin. Storage [hm3]: –

Max. Storage [hm3]: –


]: 2.36


Bayano

Min. Storage [hm3]: 1784.71



]: 0.42


Figure : Hydro-electric system of Panama




Case Study I: Panama (cont’d)

Demand [GWh]

100

200

300

400

500

600

700

800

900

+1.75%

+1.00%

+0.50%

-0.50%

-1.00%

-1.75%

Reference

January

February

March

April

May

June

July

August

September

October

November

December

Figure : Electricity demand scenarios.Optimization in Energy Panos M. Pardalos 91 (102)



Case Study I: Results

Expected Value Solution (EVS)= “ignore demand uncertainty and use expected electricity demand”= $158.653 million

Value of Stochastic Solution (VSS)= EVS - stochastic solution value= $158.653 - $157,374 = $1.279 [million]= 0.81% EVS

Value of Perfect Information= “how much am I willing to pay”= $157.374 - $156.721 [million] = $653,429




Case Study II: Costa Rica

ArenalMin. Storage [hm3]: 722.75



]: 1.62


CorobiciMin. Storage [hm3]: –



]: 1.78


SandillalMin. Storage [hm3]: 0.00


Ø Production [ MWm3/sec. ]: 0.32


Rio MachoMin. Storage [hm3]: –



]: 3.73

Capacity [MW]: 120

CachiMin. Storage [hm3]: 3.53



]: 1.90

Capacity [MW]: 105

Bot Joya

Min. Storage [hm3]: –


Ø Production [ MWm3/sec. ]: 0.93


Figure : Hydro-electric reservoir system of Costa Rica; excluding additional24 run-of-the river plants




Case Study II: Results

Expected Value Solution (EVS)= “ignore demand uncertainty and use expected electricity demand”= $102.089 million

Value of Stochastic Solution (VSS)= EVS - stochastic solution value= $102.089 - $100,947 = $1.142 [million]= 1.12% EVS

Value of Perfect Information= “how much am I willing to pay”= $100,947 - $100,032 [million] = $915,870




PublicationsBooksEnergy Systems Journal

Outline

1 Introduction








Publications – Journal Articles & Book Chapters

S. Rebennack, N. Iliadis, J. Kallrath, and P. M. PardalosShort Term Portfolio Optimization for Discrete Power Plant DispatchingIEEE PES GM proceedings, Calgary, Canada, pp. 1-6, 2009.

S. Rebennack, N. Iliadis, M. V.F. Pereira, and P. M. Pardalos,Electricity and CO2 Emissions System Prices Modeling and OptimizationIEEE PowerTech conference proceedings, Bucharest, Romania, pp. 1-6, 2009

Q. P. Zheng, S. Rebennack, N. Iliadis, and P. M. PardalosOptimization Models in the Natural Gas IndustryHandbook of Power Systems I, pp. 121-148, 2010.

S. Rebennack, J. Kallrath, and P. M. PardalosEnergy Portfolio Optimization for Electric Utilities: Case Study for GermanyEnergy, Natural Resources and Environmental Economics, pp. 221-246, 2010.





Publications – Journal Articles & Book Chapters

Q. P. Zheng and P. M. PardalosStochastic and Risk Management Models and Solution Algorithm for Gas TransmissionNetwork Expansion and LNG Terminal Location PlanningJournal of Optimization Theory and Applications, vol. 147, pp. 337–357, 2010.

S. Rebennack, B. Flach, M. V.F. Pereira, and P. M. Pardalos,Stochastic Hydro–Thermal Scheduling under CO2 Emission ConstraintsIEEE Transactions in Power Systems, Vol. 27, No. 1, pp. 58-68, 2011.

N. Fan, H. Xu, F. Pan, and P.M. PardalosEconomic analysis of the N-k power grid contingency selection and evaluation by graphalgorithms and interdiction methodsEnergy Systems, Vol. 2 No. 3, pp. 313-324, 2011.

N. Fan, D. Izraelevitz, F. Pan, and P.M. PardalosA mixed integer programming approach for optimal power grid intentional islandingEnergy Systems, Vol. 3 No. 1, pp. 77-93, 2012





Publications – Books

Electrical Power Unit Commitment: Models and AlgorithmsQ. P. Zheng and P. M. PardalosSpringer, to appear in 2013

Handbook of Wind Power SystemsV. Pappu, S. Rebennack, P. M. Pardalos, N. Iliadis, M. V. F. Pereira (eds.)Springer, to appear in 2013

Handbook of CO2 in Power SystemQ. P. Zheng, S. Rebennack, P. M. Pardalos, N. Iliadis, M. V. F. Pereira (eds.)Springer 2012





Publications – Books

Handbook of Networks in Power SystemsA. Sorokin, S. Rebennack, P. M. Pardalos, N. Illiadis, M. V. F. Pereira (eds.)Two volumes, Springer 2012

Energy, Natural Resources and Environmental EconomicsE. Bjorndal, M. Bjorndal, P. M. Pardalos, M. Ronnqvist, (eds.)Springer 2010

Handbook of Power SystemsS. Rebennack, P. M. Pardalos, M. V. F. Pereira, N. Illiadis (eds.)Springer 2010

Optimization in the Energy IndustryJ. Kallrath, P. M. Pardalos, S. Rebennack, and M. Scheidt (eds.)Springer 2009





Energy Systems Journal

Energy Systems JournalOptimization, Modeling, Simulation, and Economic AspectsEditor-in-Chief: Panos M. PardalosPublished by Springer

Applies mathematical programming, control, and economic approachesto energy systems topics, and is especially relevant in light of challengesfacing humanity





Heraclitus

jnatoi jnhtoÐ, jnhtoÐ jntatoi, zwntec tìn âkeÐ-nwn jnaton, tìn dè âkeÐnwn bÐon tejnewtec

1

“Mortals are immortals and immortals aremortals, the one living the others’ death anddying the others’ life.”

“They say that Euripides gave Socrates acopy of Heraclitus’ book and asked himwhat he thought of it. He replied: “What Iunderstand is splendid; and I think what Idon’t understand is so too - but it would takea Delian diver to get to the bottom of it.”(Diogenes Laertius, Lives of Philosophers, II22).

Heraclitus of Ephesus (c.535 BC - 475 BC)





The END!

http://www.ise.ufl.edu/pardalos/http://nnov.hse.ru/en/latna/

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