optimal reserve offer deliverability in co- optimized ... · 1 july 6, 2016 danmarsks tekniske...

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July 6, 2016 Danmarsks Tekniske Universitet, Lyngby, Denmark

EES-UETP Course on Uncertainty in Electricity Markets and System Operation

Optimal reserve offer deliverability in co-optimized electricity markets: A two-stage

robust optimization approach

José M. Arroyo* Power and Energy Analysis and Research Lab (PEARL)

Departamento de Ingeniería Eléctrica, Electrónica, Automática y Comunicaciones Universidad de Castilla – La Mancha

*In collaboration with Noemi G. Cobos, Universidad de Castilla – La Mancha, Spain, and Alexandre Street, Pontifical Catholic University of Rio de Janeiro, Brazil

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Contents

• Introduction

• Robust model

• Solution approach

• Numerical results

• Conclusions

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Introduction

• Power system operation is exposed to uncertainty sources

Fluctuations in nodal net injections

(demand, renewable-based generation)

Loss of system components

• Capability to withstand uncertainty realizations

implemented via reserves

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Reserves

• Different types of reserves according to practical operating manuals

• One general classification:

Synchronized or spinning reserves ⇒ Provided by units that are scheduled on

Non-synchronized or non-spinning reserves ⇒ Provided even by units that are scheduled off

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Trading reserves

• Reserves are considered as tradable commodities in a similar way to energy

• Market participants are allowed to submit reserve offers

• Typical reserve offer ⇒ Pair quantity-price • Market operator is responsible for scheduling

energy and reserves

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Trading reserves

Producer i

Energy & reserve offer

Producer n


Market operator (ISO) Market-clearing

procedure

Producer 1


Awarded offers and bids

Market-clearing prices

··· ···

··· ···

Energy bid & reserve offer

Consumer 1


Consumer j


Consumer n

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Trading reserves

• Current market implementations:

Sequential markets for energy and reserves (Europe) ⇒ Simple albeit suboptimal

Co-optimized electricity markets (Greece, USA) ⇒ Energy and reserves are jointly cleared

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Goal of the co-optimization

• Determination of the awarded levels of energy and reserve offers that:

Maximize declared social welfare

Comply with operational limits

Guarantee power balance under the normal

state and under any uncertainty realization

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Problem definition

• Two different categories of system states:

Normal state ⇒ Forecast uncertainty realizations

States under uncertainty

• Straightforward solution ⇒ Replication of

operational constraints (contingency-constrained model)

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Pre-contingency constraints

Contingency-constrained model

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑐(𝑥, 𝑟)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:

𝑔(𝑥, 𝑟) ≤ 0

𝑔𝑘(𝑥, 𝑟, 𝑥𝑘) ≤ 0 ∀𝑘 ∈ 𝒞

• Large-scale mixed-integer program

Social welfare (energy and reserves)

Post-contingency constraints

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• Operational constraints (pre- and post-contingency):

Generation limits

Reserve limits

Network-related constraints ⇒ dc load flow

• Mixed-integer linear programming

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• Uncertainty realizations (nodal net injections and component outages) are characterized as contingencies indexed by 𝑘 within the contingency set 𝒞

𝐷𝑏𝑡𝑘 ⇒ Realizations of nodal net injections

𝐴𝑖𝑡𝑘 , 𝐴𝑙𝑡𝑘 ⇒ 0/1 parameters modeling the

unavailability/availability of generators and lines

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• Contingency set 𝒞 determined by:

Plausible realizations of nodal net injections

𝐷𝑏𝑡𝑘 ∈ D𝑏𝑡

Security criterion (n – 1, n – 2, n – K, n – KG

– KL)

𝒇 ��𝐴𝑖𝑡𝑘 �𝑖∈𝐼 , �𝐴𝑙𝑡

𝑘 �𝑙∈ℒ� ≤ 𝟎

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Contingency-constrained model Practical issues

• Problem dimension depends on the number of

states

• Contingency set 𝒞 is infinitely large

• Intractable model in practice

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Practical solution

• Pre-specification of system reserve requirements based on engineering judgment

• Operation under uncertainty realization typically disregarded ⇒ Use of forecast values

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Practical solution for energy and reserve co-optimization

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑐(𝑥, 𝑟)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: 𝑔(𝑥, 𝑟) ≤ 0 𝑔𝑟𝑒𝑞(𝑟) ≤ 0 • Drawback ⇒ Feasible deployment of reserve

offers is not guaranteed even for sufficient reserve requirements!!

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Contents

• Introduction

• Robust model



• Conclusions

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Motivation for robust optimization

• Exact albeit intractable contingency-constrained model is essentially a worst-case problem

• Equivalent to a penalized contingency-constrained model

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Motivation for robust optimization Equivalent penalized model

• Nodal power imbalance is allowed ⇒ Slack

variables

• Penalty term in the objective function associated with the worst-case system power imbalance

• Equivalence guaranteed by a sufficiently large

penalty coefficient

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Motivation for robust optimization Equivalent penalized model

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑐(𝑥, 𝑟) + 𝐶𝐼𝑚𝑎𝑥𝑘∈𝒞(𝑒𝑇∆𝑃𝑘)


𝑔(𝑥, 𝑟) ≤ 0

𝑔𝑘(𝑥, 𝑟, 𝑥𝑘 ,∆𝑃𝑘) ≤ 0 ∀𝑘 ∈ 𝒞

• Also a large-scale mixed-integer program

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Two-stage adaptive robust optimization

• Worst-case optimization setting ⇒ Suitable for ensuring feasibility under all uncertainty realizations

• No accurate probabilistic information is required ⇒ Suitable for renewable-based generation

• Problem size not dependent on the level of accuracy for uncertainty ⇒ Convenient for tractability purposes

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Two-stage adaptive robust optimization

• First stage:

Decisions that immunize against uncertainty realizations

• Second stage:

Worst-case uncertainty realizations and corresponding corrective actions

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Uncertainty characterization

• Uncertainty realizations are represented by uncertainty-related decision variables

𝑑𝑏𝑡 ⇒ Nodal net injections

𝑎𝑖𝑡𝐺 , 𝑎𝑙𝑡𝐿 ⇒ Loss of generators and lines

• Non-dependent on index 𝑘!!

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Uncertainty characterization

• Feasibility space of uncertainty-related decision variables ⇒ Pre-specified uncertainty set U

𝑎𝑖𝑡𝐺 , 𝑎𝑙𝑡𝐿 , 𝑑𝑏𝑡 ∈ U

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Uncertainty set

• Upper and lower bounds for uncertainty-related decision variables ⇒ Based on historical data or practical aspects

• Uncertainty budget ⇒ Limit on the

conservativeness of the solution based on engineering judgement

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Uncertainty set: Nodal net injections

• Cardinality model:

𝑑𝑏𝑡𝑓 − ∆𝑏𝑡𝑑𝑛≤ 𝑑𝑏𝑡 ≤ 𝑑𝑏𝑡

𝑓 + ∆𝑏𝑡𝑢𝑝

� �𝑚𝑎𝑥�0,𝑑𝑏𝑡 − 𝑑𝑏𝑡

𝑓 �∆𝑏𝑡𝑢𝑝 +

𝑚𝑎𝑥�0,𝑑𝑏𝑡𝑓 − 𝑑𝑏𝑡�

∆𝑏𝑡𝑑𝑛� = 𝐾

𝑏∈B𝑢

• Tradeoff between accuracy and complexity

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• Polyhedral uncertainty sets:

Worst-case uncertainty realization ⇒ Extreme of the polytope

Equivalent discrete model based on binary

variables

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• Binary-variable-based cardinality model:

𝑑𝑏𝑡 = 𝑑𝑏𝑡

𝑓 + ∆𝑏𝑡𝑢𝑝𝑎𝑏𝑡

𝑢𝑝 − ∆𝑏𝑡𝑑𝑛𝑎𝑏𝑡𝑑𝑛

� �𝑎𝑏𝑡𝑢𝑝 + 𝑎𝑏𝑡𝑑𝑛� = 𝐾

𝑏∈B𝑢

𝑎𝑏𝑡𝑢𝑝,𝑎𝑏𝑡𝑑𝑛 ∈ {0,1}

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Uncertainty set: Component outages

• Straightforward use of 0/1 variables 𝑎𝑖𝑡𝐺 and 𝑎𝑙𝑡𝐿 • Uncertainty budget ⇒ Security criterion

• Binary-variable-based cardinality model:

𝑎𝑖𝑡𝐺 ,𝑎𝑙𝑡𝐿 ∈ {0,1}

𝒇 ��𝑎𝑖𝑡𝐺 �𝑖∈𝐼 , {𝑎𝑙𝑡𝐿 }𝑙∈ℒ� ≤ 𝟎

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Robust counterpart

• Contingency constraints are replaced with an optimization problem to characterize the worst case

• Worst case ⇒ Maximum damage (system power imbalance) associated with ALL contingencies implicitly modeled by 𝑎𝑏𝑡

𝑢𝑝, 𝑎𝑏𝑡𝑑𝑛, 𝑎𝑖𝑡𝐺 , 𝑎𝑙𝑡𝐿 ,

• Robust counterpart ⇒ Trilevel program

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Trilevel robust counterpart

Maximize social welfare

Determine: On/off decisions

Power and reserves schedule

Determine:

Maximize the system power imbalance given the scheduled power and reserves

Nodal net injections, unavailable components

Pre-contingency schedule

Worst-case contingency

Minimize the system power imbalance given the scheduled power and reserves and the worst-case contingency Operator’s

reaction Determine: Corrective actions

First stage

Second stage

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subject to:

su

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• Two lowermost optimization problems replace post-contingency constraints ⇒ Mixed-integer trilevel program

• Penalty term in the upper-level objective function is optimized in the two lowermost problems

• Two lowermost optimization problems ⇒ Max-

min optimization with a linear lower-level problem

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Contents

• Introduction

• Robust model



• Conclusions

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Suitable solution approaches

• Decomposition-based methods involving the iterative solution of a master problem and a max-min subproblem: Benders decomposition

Column-and-constraint generation algorithm

• Optimal solution to the master problem and

the max-min subproblem ⇒ Finite convergence to global optimality

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Column-and-constraint generation Master problem

• Relaxation of the trilevel counterpart

• Iteratively tighter by the addition of operating

constraints built with information from the subproblem

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Column-and-constraint generation Master problem

𝑀𝑖𝑛𝑖𝑚𝑖𝑧𝑒α,𝑥,𝑟, 𝑥𝑢,𝑚,∆𝑃𝑢,𝑚 𝑐(𝑥, 𝑟) + 𝐶𝐼α


𝑔(𝑥, 𝑟) ≤ 0

α ≥ 𝑒𝑇∆𝑃𝑢,𝑚; ∀𝑚 ∈ M

𝑔𝑢�𝑥, 𝑟,𝑎(𝑚), 𝑥𝑢,𝑚,∆𝑃𝑢,𝑚� ≤ 0; ∀𝑚 ∈ M

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Column-and-constraint generation Max-min subproblem

• Two lowermost levels for given 𝑥 and 𝑟 from

the master problem

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒𝑎 δ(𝑥, 𝑟,𝑎)


𝑓𝑑𝑖𝑠(𝑎) ≤ 0

δ(𝑥, 𝑟,𝑎) = 𝑚𝑖𝑛𝑥𝑢, ∆𝑃𝑢 (𝑒𝑇∆𝑃𝑢)

𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜: su

𝑔𝑢(𝑥, 𝑟, 𝑎, 𝑥𝑢, ∆𝑃𝑢) ≤ 0

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Column-and-constraint generation Max-min subproblem

• Transformation to a single-level mixed-integer

linear equivalent

Dual of the lower-level problem renders the original max-min a max-max ⇒ Maximization problem (strong duality theorem)

Integer algebra results for the resulting bilinear terms

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Dual of the lower-level problem

δ(𝑥, 𝑟,𝑎) = 𝑚𝑎𝑥π ℎ𝑢,𝑑𝑢𝑎𝑙(π, 𝑥, 𝑟,𝑎)


𝑔𝑢,𝑑𝑢𝑎𝑙(π,𝑎) ≤ 0

• At the optimum (strong duality theorem):

(𝑒𝑇∆𝑃𝑢) = ℎ𝑢,𝑑𝑢𝑎𝑙(π, 𝑥, 𝑟,𝑎)

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Single-level equivalent for the subproblem

𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒𝑎,π ℎ𝑢,𝑑𝑢𝑎𝑙(π, 𝑥, 𝑟, 𝑎)

subject to: 𝑓𝑑𝑖𝑠(𝑎) ≤ 0

𝑔𝑢,𝑑𝑢𝑎𝑙(π,𝑎) ≤ 0

• Bilinear terms in ℎ𝑢,𝑑𝑢𝑎𝑙(π, 𝑥, 𝑟,𝑎) ⇒ Well-known linearization scheme

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Column-and-constraint generation

MASTER PROBLEM

Power and reserves schedule

Add a set of lower-level constraints

SUBPROBLEM

Worst-case contingency

Upper bound for social welfare

Lower bound for social welfare

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Contents

• Introduction

• Robust model



• Conclusions

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Case studies

• Illustrative three-bus, three-line, single-period

example

• IEEE 118-bus system, 24 hourly periods

• CPLEX 12.6 under GAMS 24.2

• 4 Intel Xeon E7-4820 processors, 2.00 GHz, 768 GB of RAM

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Illustrative single-period example

Bus 3

~ ~

~ 130 MW

Unit 3

Forecast generation = 10 MW

Bus 1

Pmax = 30 MW

Forecast generation = 65 MW

Pmax = 100 MW

20 MW

Pmax = 100 MW

Bus 2

Unit 2 Unit 1

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Illustrative single-period example

Unit 𝑃�,𝑅� 𝑃 𝐶𝑣 𝐶𝑓 𝐶𝑢𝑝 𝐶𝑑𝑛 (MW) (MW) ($/MWh) ($) ($/MW) ($/MW)

1 50 2.4 50.0 10.0 1.0 0.25 2 50 2.4 03.0 00.5 0.3 0.02 3 50 2.4 19.9 05.0 0.7 0.10

• All line reactances equal to 0.63 p.u.

• ±40% wind power fluctuation (±30 MW), no

component outages

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Illustrative single-period example Results

Unit

Under no uncertainty

Under uncertainty Robust model Conventional model

(MW)

(MW)

(MW)

(MW)

(MW)

(MW)

(MW)

1 50 52.4 53.6 50 52.4 52.6 50 2 50 48.8 40.2 30 48.8 41.2 30 3 25 23.8 26.2 20 23.8 26.2 20

Cost ($) 653.0 778.1 777.4

• Convergence in 3 iterations requiring less than 1 second

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Illustrative single-period example Worst-case operation

Bus 3

~ ~

~ 130 MW

p3 = 50 MW

Wind generation = 6 MW

Bus 1

p23 = 30 MW


p13 = 11 MW

20 MW

p21 = 19 MW

Bus 2

p2 = 49 MW

p1 = 6 MW

Bus 3

~ ~

~ 130 MW

p3 = 50 MW


Bus 1

p23 = 30 MW


p13 = 10.5 MW

20 MW

p21 = 19.5 MW

Bus 2

p2 = 49.5 MW

p1 = 5 MW

Robust solution Conventional solution

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IEEE 118-bus system

• 118 buses, 186 lines, 54 dispatchable units, 10 wind farms (25.8% wind penetration), 24 periods

• ±20% wind power fluctuation

• 66 contingencies (all dispatchable generators + 12 critical lines)

• 0.01-MW stopping criterion

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IEEE 118-bus system Results

• Convergence in 8 iterations, 2.2 min/iteration

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IEEE 118-bus system Infeasibility of reserve requirements

K Worst power Imbalance (MW)

51 1980.3 52 2675.5 53 3049.5 54 3479.7 55 3908.8 56 4328.4 57 4718.0 58 5107.6 59 5492.2 10 5863.4

• Feasibility ⇒ 0.1%-6.8% cost increase

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Contents

• Introduction

• Robust model



• Conclusions

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Conclusions

• Robust optimization is suitable for co-optimized electricity markets under uncertainty

• Reserve deliverability is achieved with moderate reduction of social welfare

• Optimality is attained in a few iterations

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Further research

• Incorporation of correlated uncertainty sources ⇒ Alternative uncertainty sets

• Consideration of more sophisticated operational models (ac load flow, line switching)

• Analysis of alternative solution approaches to avoid the dual-based transformation

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End of the presentation

Thanks for your attention!

PEARL: http://www3.uclm.es/area/pearl

optimal reserve offer deliverability in co- optimized ... · 1 july 6, 2016 danmarsks tekniske...

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