Rudige r Schultz: On Unit Commitmentcost-td1207.zib.de/sites/default/files/schultz.pdf · Unit...

Rudiger Schultz: On Unit Commitment

Birthplace of Unit Commitment

Step Back in Time for 101 years

ZschornewitzBiggest lignite-fired thermal power station of its time inaugurated.

I Build within one year (Groundbreaking March 24, 1915, First Turbine(16 MW) in Operation December 15, 1915, 1916: 8×16 MW installed)

Step Back in Time for 101 years 1915

Zschornewitz

Biggest lignite-fired thermal power station of its time inaugurated.

Unit Commitment

I is the problem of determining switching and operational decisions,

I for a system of power producing units, over some time horizon,

I so that all relevant technological and economical conditions are met.

1985 VEAG in (East Germany)

2006 Virtual Power Plant

Unit Commitment

Specification (Mixed-Integer Linear Program – When Deterministic)

Unit Commitment for a hydro-thermal system (early VEAG + Vattenfall)

minc>1 ξ1 + c>2 ξ2 : A1ξ1 + A2ξ2 = b, ξ1 ∈ X1, ξ2 ∈ X2

Variables:

I ξ1: start-up/shut-down for thermal units,

I ξ2: all remaining, i.e., power output, pumping/generating in pumped-storage(psp), water levels in psp, auxillary variables for modeling specific effects.

Objective:

I affinely linear fuel costs for operation and piece-wise constant for switchingof thermal units

Constraints:

I connecting units: load balances, reserve balances, ramping

I for individual units: output bounds, minimum up- and down-times, watermanagement in psp,

Variables:

Objective:

Constraints:

Variables:

Objective:

Constraints:

Variables:

Objective:

Constraints:

The Greatest – Unit Commitment UnderUncertainty

Unit Commitment Under UNCERTAINTY

in the 1970ies and 1980ies

Unit Commitment Under UNCERTAINTY

in the 1970ies and 1980ies

Unit Commitment under Uncertainty Fourty Years Ago

I Before deregulation, power producers optimized costs by fuel costminimization, with power demand as major source of uncertainty.

I TV sets consumed more energy than today. Their operation had to beincluded when estimating power demand, at least during certainperiods of the day.

I In the 1970ies and 1980ies Heavyweight Boxing was a very popularspectator sport (Ali, Frazier, Foreman etc.), in West and EastGermany.

I Time zone difference and duration of fight (knock-out: if at all andwhen) produced random variables that were hard to handle ... and(induced) water consumption was uncertain, too!

I Time zone difference and duration of fight (knock-out: if at all andwhen) produced random variables that were hard to handle ...

and(induced) water consumption was uncertain, too!

Specification (continued)

Unit Commitment with random load

f (ξ1, ω) = [c>1 ξ1 + minξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

, ω ∈ Ω

QE(ξ1) :=

[c>1 ξ1 + min

ξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

]P(dω)

Variables:

I ξ1 ∈ X1: start-up/shut-down for thermal units,

I ξ2(ω): all remaining, i.e., power output, pumping/generating inpumped-storage (psp), water levels in psp, auxillary variables for modelingspecific effects.

Objective:

f (ξ1, .) random cost profile for operation and switching of thermal unitsinuced by start-up/shut-down scheme ξ1

QE(ξ1) :=

∫Ωf (ξ1, ω)P(dω)−−− Expected Value – Risk Neutral Model

f (ξ1, ω) = [c>1 ξ1 + minξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

, ω ∈ Ω

QE(ξ1) :=

[c>1 ξ1 + min

ξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

]P(dω)

Variables:

Objective:

QE(ξ1) :=

f (ξ1, ω) = [c>1 ξ1 + minξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

, ω ∈ Ω

QE(ξ1) :=

[c>1 ξ1 + min

ξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

]P(dω)

Variables:

Objective:

QE(ξ1) :=

f (ξ1, ω) = [c>1 ξ1 + minξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

, ω ∈ Ω

QE(ξ1) :=

[c>1 ξ1 + min

ξ2∈X2

c>2 ξ2 : A2ξ2 = b(ω)− A1ξ1

]P(dω)

Variables:

Objective:

QE(ξ1) :=

Unit Commitment under Uncertainty over the Years

c>x + min

q>y : Wy = h(ω)− Tx , y ∈ Y

︸︷︷︸

f (x,ω)

: x ∈ X

I 1985: Load the only quantity with relevant uncertainty -Risk neutral models, only !

f (x , z(ω))− total cost for up/down regime x under random load z(ω)

I 2006: After deregulation omnipresent uncertainty at input (renewables) andoutput sides. - Risk aversion became more and more indispensable !

f (x , z(ω)) −total cost for aquisition x of a vpp under random power in- and outputs z

I 2010: Congestion and capacity management under uncertain in- and outputs

f (x , z(ω)) −x pre-commitment so that renewables’ inflow z compensated with

minimal re-commitment/re-dispatch and without overloading grid components

c>x + min

q>y : Wy = h(ω)− Tx , y ∈ Y

︸︷︷︸

f (x,ω)

: x ∈ X

c>x + min

q>y : Wy = h(ω)− Tx , y ∈ Y

︸︷︷︸

f (x,ω)

: x ∈ X

c>x + min

q>y : Wy = h(ω)− Tx , y ∈ Y

︸︷︷︸

f (x,ω)

: x ∈ X

Unit Commitment – Subjective Comments

Unit Commitment and Economic Dispatch

Unit Commitment (UC) and Economic Dispatch (ED) have aresearch history of more than 50 years.

UC models being large-scale mixed-integer nonlinearoptimization problems solution approaches always have beeninspired by ideas from different subdisciplines of optimization,with permanently adjusting “large-scale” to bigger and biggernumbers.In recent years, the integration of UC into energy optimizationmodels which, themselves, already are large-scale, e.g., powerflow or uncertainty management in production and trading,became a focal research topic.

Unit Commitment (UC) and Economic Dispatch (ED) have aresearch history of more than 50 years.UC models being large-scale mixed-integer nonlinearoptimization problems solution approaches always have beeninspired by ideas from different subdisciplines of optimization,with permanently adjusting “large-scale” to bigger and biggernumbers.

In recent years, the integration of UC into energy optimizationmodels which, themselves, already are large-scale, e.g., powerflow or uncertainty management in production and trading,became a focal research topic.

Unit Commitment (UC) and Economic Dispatch (ED) have aresearch history of more than 50 years.UC models being large-scale mixed-integer nonlinearoptimization problems solution approaches always have beeninspired by ideas from different subdisciplines of optimization,with permanently adjusting “large-scale” to bigger and biggernumbers.In recent years, the integration of UC into energy optimizationmodels which, themselves, already are large-scale, e.g., powerflow or uncertainty management in production and trading,became a focal research topic.

The Early Days

In their UC literature synopsis [Sheble-Fahd1994] review thefirst 25 years of the development and identify approaches someof which later became major pathways in algorithmic unitcommitment.

They address both heuristics, such as Exhaustive Enumeration,Priority Lists, or Simulated Annealing, as well asmathematically rigorous methods from subdisciplines ofoptimization as there are Dynamic Programming,(Mixed-Integer) Linear Programming, Network Flows, andLagrangian Methods.

The computationally more demanding rigorous methods, on theother hand, yield provably optimal solutions or at least lowerbounds allowing for gap estimates between objective functionvalues of the best feasible solution found so far and lowerbounds generated in the course of the algorithm.

The Early Days

Lagrangian Relaxation - as it was

[ShebleFahd1994] grant a “clear consensus presently tendingtoward the Lagrangian Relaxation (LR) over othermethodologies”. Indeed, still today LR offers flexiblepossibilities for relaxing constraints complicating the model,however, at the cost of having to solve repeatedly “closecousins” to the relaxed problem.

Lagrangian Relaxation - as it was

[ShebleFahd1994] grant a “clear consensus presently tendingtoward the Lagrangian Relaxation (LR) over othermethodologies”. Indeed, still today LR offers flexiblepossibilities for relaxing constraints complicating the model,however, at the cost of having to solve repeatedly “closecousins” to the relaxed problem.

The key features of LR applied to UC have been and still are:

(i) relaxation of constraints interlinking units, e.g., loadcoverage or reserve requirements, and arrival at single-unitsubproblems,

(ii) addition of the relaxed constraints, together with Lagrangemultipliers, to the objective, so that the resulting problem iseasier to solve than the original,

(iii) solution of the convex, nonsmooth Lagrangian dual whoseobjective-function value calculation benefits from reduction tosolving single-unit subproblems and whose optimal value formsa lower bound to the optimal value of the UC problem,

(iv) application of Lagrangian heuristics to obtain “promising”feasible primal solutions from the results of the dualoptimization.

Lagrangian Relaxation - as it is

Fueled by improved bundle-trust subgradient methods for theLagrangian dual and by permanent progress in “off-the-shelve”mixed-integer linear programming (MILP) software, up to theadvent of market deregulation, two basic aproaches developedwhich still today are widely used:

(i) LR, often in conjunction with heuristic methods for finding“promising” feasible solutions,(ii) direct solution (by branch-and-bound) of MILPformulations of UC by “off-the-shelve” solvers.

Fueled by improved bundle-trust subgradient methods for theLagrangian dual and by permanent progress in “off-the-shelve”mixed-integer linear programming (MILP) software, up to theadvent of market deregulation, two basic aproaches developedwhich still today are widely used:(i) LR, often in conjunction with heuristic methods for finding“promising” feasible solutions,

(ii) direct solution (by branch-and-bound) of MILPformulations of UC by “off-the-shelve” solvers.

Fueled by improved bundle-trust subgradient methods for theLagrangian dual and by permanent progress in “off-the-shelve”mixed-integer linear programming (MILP) software, up to theadvent of market deregulation, two basic aproaches developedwhich still today are widely used:(i) LR, often in conjunction with heuristic methods for finding“promising” feasible solutions,(ii) direct solution (by branch-and-bound) of MILPformulations of UC by “off-the-shelve” solvers.

Lagrangian Relaxation - as it will be

Rather than the transition to different time horizons, fromshort via medium to long-term, the new economic environmentin the course of energy market deregulation poses researchnecessities and provides incentive to integrate UC and ED withload flow and uncertainty treatment, [Gabriel-etal2013].

The latter is intended in the widest sense, from handlingstochasticity to topics of mathematical equilibria in the contextof power trading and bidding into power markets. In particular,this means to integrate UC into models which already arecomplex themselves.

Power Flow - Integrating UC and AC Load Flow

This was considered utopic throughout the “Early Days”, butnow became possible by studying the quadratic nonconvex ACload flow equations from the viewpoint of semidefiniteoptimization. In [LavaeiLow2011] after relaxation of the rankcondition the solution to the dual of the remaining convexmodel allows to retrieve a primal solution often meeting therelaxed rank condition, and thus enabling to solve nonconvexpower flow optimization problems to global optimality.

Power Flow - Integrating UC and AC Load Flow

This was considered utopic throughout the “Early Days”, butnow became possible by studying the quadratic nonconvex ACload flow equations from the viewpoint of semidefiniteoptimization. In [LavaeiLow2011] after relaxation of the rankcondition the solution to the dual of the remaining convexmodel allows to retrieve a primal solution often meeting therelaxed rank condition, and thus enabling to solve nonconvexpower flow optimization problems to global optimality.

Power Flow – DC Model and Ohmic Losses

The DC Load Flow Model provides a linear approximation ofits AC counterpart by resorting to linear relations and avoidingvariables in the space of complex numbers, see[Franketal2012a,Franketal2012b]. The Ohmic Lossesapproximation, [Sanchez-MartinRamos1997], provides thepossibility to include power losses within the DC-approximationof an AC power system. Precise modeling of power losses turnsout instrumental in congestion management when loaddispatches or even commitments of units have to be revised toincrease throughput of the grid under increased inflows ofrenewables.

Power Flow – DC Model and Ohmic Losses

The DC Load Flow Model provides a linear approximation ofits AC counterpart by resorting to linear relations and avoidingvariables in the space of complex numbers, see[Franketal2012a,Franketal2012b]. The Ohmic Lossesapproximation, [Sanchez-MartinRamos1997], provides thepossibility to include power losses within the DC-approximationof an AC power system. Precise modeling of power losses turnsout instrumental in congestion management when loaddispatches or even commitments of units have to be revised toincrease throughput of the grid under increased inflows ofrenewables.

Polyhedral Methods

Despite its success in combinatorial optimization, cutting planemethods based on polyhedral studies, either applied directly orenhancing branch-and-bound came to the fore in UC a bit morethan 10 years ago, only. There always have been polytopes

with fancy names ‘‘pretending to be real-life’’ .

However, UC, really a real-life model in this

respect, was a no-name product, which has changed,

fortunately. At this time, market deregulation enforcedthe need of solving UC in a competitive environment underincomplete information. In this way, solving UC problemsbecame a subroutine in the treatment of more complex decisionproblems in electricity supply. Today tight formulations forcrucial model ingredients and for complete polytopes arising inUC are available[LeeLeungMargot2004,Morales-EspanaGentileRamos2015].

Polyhedral Methods

Despite its success in combinatorial optimization, cutting planemethods based on polyhedral studies, either applied directly orenhancing branch-and-bound came to the fore in UC a bit morethan 10 years ago, only. There always have been polytopes

with fancy names ‘‘pretending to be real-life’’ .

However, UC, really a real-life model in this

respect, was a no-name product, which has changed,

fortunately. At this time, market deregulation enforcedthe need of solving UC in a competitive environment underincomplete information. In this way, solving UC problemsbecame a subroutine in the treatment of more complex decisionproblems in electricity supply. Today tight formulations forcrucial model ingredients and for complete polytopes arising inUC are available[LeeLeungMargot2004,Morales-EspanaGentileRamos2015].

Uncertainty

In the presence of uncertainty, UC either lives in anon-competitive or competitive environment.The former concerns the time before, the latter sincederegulation of energy markets. Before deregulation load hasbeen the dominating uncertainty prone entity,[TakritiBirgeLong1996].After deregulation UC-relevant sources of uncertainty havespread considerably: power input from renewables, power pricesdetermined by bidding into power exchanges, competitors’actions at electricity markets.

Uncertainty

In the presence of uncertainty, UC either lives in anon-competitive or competitive environment.

The former concerns the time before, the latter sincederegulation of energy markets. Before deregulation load hasbeen the dominating uncertainty prone entity,[TakritiBirgeLong1996].After deregulation UC-relevant sources of uncertainty havespread considerably: power input from renewables, power pricesdetermined by bidding into power exchanges, competitors’actions at electricity markets.

Uncertainty

In the presence of uncertainty, UC either lives in anon-competitive or competitive environment.The former concerns the time before, the latter sincederegulation of energy markets. Before deregulation load hasbeen the dominating uncertainty prone entity,[TakritiBirgeLong1996].

After deregulation UC-relevant sources of uncertainty havespread considerably: power input from renewables, power pricesdetermined by bidding into power exchanges, competitors’actions at electricity markets.

Uncertainty

In the presence of uncertainty, UC either lives in anon-competitive or competitive environment.The former concerns the time before, the latter sincederegulation of energy markets. Before deregulation load hasbeen the dominating uncertainty prone entity,[TakritiBirgeLong1996].After deregulation UC-relevant sources of uncertainty havespread considerably: power input from renewables, power pricesdetermined by bidding into power exchanges, competitors’actions at electricity markets.

Yet, UC is understood in a broader context than before. Itrather is the scheduling of decentralized power supply with itssmall generating facilities than commitment of thermal let alonenuclear generation units.While the mathematical apparatus is fairly well developed forexogenous uncertainty, the situation is completely different forendogenous uncertainty, i.e., with decision dependentprobability distributions.In case uncertainty is captured by probability measures,stochastic integer programming offers methodology for handlingUC, both algorithmically and regarding structuralunderstanding,[TakritiBirgeLong1996,CarøeSchultz1998,Schultz2003].

References

I bertocchietal2011 Bertocchi, M.I., Consigli, G., Dempster, M.A.H. (Eds.) :Stochastic optimization methods in finance and energy, Springer, New York,2011.

I caroeschultz1998 Carøe, C.C.; Schultz R.: A Two-Stage Stochastic Programfor Unit Commitment Under Uncertainty in a Hydro-Thermal PowerSystem, ZIB Preprint 98-11, Konrad-Zuse-Zentrum fur Informationstechnik(ZIB) Berlin, 1998.

I franketal2012a Frank, S., Steponavice, I., Rebennack, S.: A primer onoptimal power flow: A bibliographic survey - (i) formulations anddeterministic methods, Energy Systems 3(2012), 221-258.

I franketal2012b Frank, S., Steponavice, I., Rebennack, S.: A primer onoptimal power flow: A bibliographic survey - (ii) non-deterministic andhybrid methods, Energy Systems 3(2012), 259-289.

I gabrieletal2013 Gabriel, S.A., Conejo, A.J., Fuller, J.D., Hobbs, B.F., Ruiz,C.: Complementarity Modeling in Energy Markets, Springer, New York2013.

I lavaeilow2011 Lavaei, J., Low, S. H. : Zero duality gap in optimal power flowproblem, IEEE Transactions on Power Systems 27(2011), 92-107.

I leeleungmargot2004 Lee, J., Leung, J., Margot, F.: Min-up/min-downpolytopes, Discrete Optimization 1(2004), 7785.

I moralesespanaetal2015 Morales-Espana, G., Gentile, C, Ramos, A.: TightMIP formulations of the power-based unit commitment problem, ORSpectrum (to appear 2015).

I sanchezmartinramos1997 Sanchez-Martin, P., Ramos, A.: . Modelingtransmission ohmic losses in a stochastic bulk production cost model,Instituto de Investigacion Tecnologica, Universidad Pontificia Comillas,Madrid, 1997.

I schultz2003 Schultz, R.: Stochastic programming with integer variables,Mathematical Programming 97(2003), 285-309.

I sheblefahd1994 Sheble, G.B., Fahd, G.N.: Unit commitment literaturesynopsis, IEEE Transactions on Power Systems 9(1994), 128-135.

I takritibl1996 Takriti, S., Birge, J.R., Long, E.: A stochastic model for theunit commitment problem, IEEE Transactions on Power Systems 11(1996),1497-1508.

Zheng, Wang, Liu: Stochastic Optimization for Unit Commitment - A Review,

IEEE Transactions on Power Systems 30(4) (2015), 1913-1924.

Rudige r Schultz: On Unit Commitmentcost-td1207.zib.de/sites/default/files/schultz.pdf · Unit...

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