Optimization Based Power Generation Scheduling

1

Optimization Based Optimization Based Power Generation SchedulingPower Generation Scheduling

Xiaohong GuanXiaohong Guan

Tsinghua / Xian Jiaotong UniversityTsinghua / Xian Jiaotong University

2

In this talk

Introduction to Power Generation Scheduling Motivations and backgroundDifficultiesCurrent approaches

Problem FormulationSolution Based on Lagrangian RelaxationNumerical Testing ResultsIssues of Homogenous Units and

Resolution

3

Power Generation Scheduling(Unit commitment)

Background and MotivationsMany generating units in a power system connected

through transmission network to supply demandScheduling unit on/off and generation levels to meet

the system demand, reserve and individual constraints

Minimizing the total generation cost Potential for significant cost savings: over 10

millions of US dollars per year for a large generation company

Hot research topics for decades

4

DifficultiesComplicated discrete and continuous and dynamic

constraints of individual units hybrid dynamics and constraints

System wide constraints coupling the operation of individual units

NP-hard mixed integer programming problem: extremely difficult to obtain the optimal schedule

Integrated with bidding problems in the market environment


5

Current approaches Priority list and other heuristics little control on

schedule quality Enumeration such as branch and

boundDynamic programming Bender’s decomposition

Problem of computationalcomplexity

Lagrangian relaxation our approach


6

Lagrangian Relaxation Based Scheduling Algorithms

Relax system wide constraints to form a two level optimization problems

Solve low level individual subproblems with much less efforts

Update Lagrange multipliers at the high level Modify dual solution into a feasible schedule Quantitative estimate of solution quality since the

dual cost is the lower bound of the primal cost

7

’

Dual Cost

Primal Cost

Duality Gap and Solution Quality

8

In this talk



Resolution

9

Problem Formulation

Objective function

Subject to

C, with C =)(),(

mintptp hjti

T

t

I

iitititi txStpC

1 1

))(())((

),()()(11

tPtptp d

J

jhj

I

iti

–System demand

• System wide constraints:

–Reserve requirements

),()))((())((11

tPtwprtpr r

J

jjhjhj

I

ititi

generation level

reserve contribution

generation range

minimum generation

generation capacity

• Individual unit constraints (thermal units):

–Discrete state transitions

)()()1( tutxtx iii if 0>)()( tutx ii

)()1( tutx ii if 0<)()( tutx ii

discrete decision variable of unit i at time t, “1” for up, “-1” for down

u ti ( ):

– Operating regions of thermal units

–Minimum up/down time

)()()( tptptp tititi 0)( txiif

0)( tptiif x ti ( ) 0

1)( tui if ii tx )(1

1)( tui if 1)(- txii

• Ramping constraint continuous dynamics

)()1()( tptptp tititi

feasible region of )1( tpti

t+1

minimum generation

maximum generation

)(tpti

t

– Operating regions of hydro units

–Reservoir level limit

)()()( twtwtw jjj or 0)( tw j

VtVV )(

–Initial and terminal reservoir levels

0)1( VV TVTV )(

–Reservoir dynamics

)()()()()1( ttwtwtVtV d B

• Individual unit constraints (hydro units):

–Hydraulic coupling

P2

R2

R3

R4

P3

R5

P4

R6

R7

P5

C1

P1

R1

-- Power Plant

-- Reservoir

-- Canal

C6

C4

C5

C8

C9

Sea

R3

C7

C3

C10

C2

16

In this talk



Resolution

17

Lagrangian Relaxation

• Lagrangian function

t i

itititi txStpCL ))(()(

jhj

itid tptptPt )()()()(

)()()()(

jhj

itir trtrtPt

Lagrangian relaxation frameworkLagrangian relaxation framework

Update Multipliers

18

Subproblems forthermal units

Subproblems forhydro units

Subproblems forother special units

Obtain feasible schedule

Multipliers Generation levels

19

Solve Thermal Subproblems

• Objective function

• Individual constraints of thermal units

– Discrete dynamic state transactions

– Minimum down/up times

– Discontinuous operating regions

• Major method: Dynamic Programming

t

tititiitititii tprttpttxStpcL ))(()()()())(()(Min Li with

• Optimal generation at a particular hour (ignoring ramping constraints)

Stage-wise cost

pti(t)p*ti(t)total stage-wise cost

generation cost cti(pti(t))

shadow cost-(t) pti(t)-(t) pti(t)

Obtain optimal stage-wise cost and generation by optimizing a single variable function !

• Typical start up cost functions

Start up cost Si(xti(t))

xti(t)

exponential

Saturate linear

Minimum down time Cold start time

• Optimal states across time obtained with efficiently with only a few states and transitions

States t t+1

Up min time

Up one hourUp two hour

Down one hour

Down two hour

Down min down time

Down cold start time

generationcosts cti(pti(t))

Start up costsSti(xti(t))

23

Dealing with ramping constraints

Difficulties of ramping constraintsRamping couples generation levels across time

Continuous dynamics The optimal generation of an “Up State” can no

longer be a single pointDynamic programming on discrete states can

no longer be applied straightforwardly

Approach to resolve the issue Heuristics Discretizing generation levels greatly

increasing the number of states and computational efforts

Relaxation of ramping constraints three level optimization structure

Constructive dynamic programming for continuous optimal generation level and regular dynamic programming for optimal discrete states

Double dynamic programming method for solving subproblems with ramping constraints best algorithm so farbest algorithm so far

Ideas of constructive dynamic programmingOptimal generation level can only be at the

corner points of the cost function or the active points of the ramping constraints

The optimal generation levels of the previous or next stage w.r.t the above points can be mapped across time systematically

The possible optimal generation levels are constructed backwardly without discretization

t t+1mapping of optimal generation levels

The number of states would increase but not significantly

The method is efficient

Redefine the discrete state as an “up” or “running” cycle Apply constructive dynamic programming to obtain

optimal general levels and cost for all running cycles Apply dynamic programming to obtain the optimal cycle

1T

1T 1T

T T T

1T 1T 1T

1T

1T

T

2

1 1

2

0

1

1U

1D

2U

Ui: the number of hours before the unit committed (up) for the ith time,

Di: as the number of hours before the unit decommitted (down) for the ith time.

29

Solve Hydro Subproblems (including Pumped Storage)

• Objective function

• Individual constraints of hydro units– Water balance– Reservoir levels– Discontinuous operating regions – Discrete operating constraints such as minimum down

times• Major difficulties

– Hydraulic coupling integrated with discontinuous operating regions and discrete dynamic constraints

min Lj, with j

jhjjhjhjj twrttwpttSL ))(()())(()()(

Method 1: Network flow optimizationIgnore discontinuous operating regions and

discrete operating constraintsApply minimum cost flow optimization to

schedule generation levels with water balance and reservoir level constraints

Meet other constraint by heuristicsTwo reservoir example:

w1(t)

w2(t)

v1(t)

v2(t)

)(1 t

)(2 t

t t+1

v1(·)

v2(·)

-w1(t)

w1(t)

-w2(t)

)(1 t

)(2 t

)1(1 t

)1(2 t

Method 2: Relaxation of reservoir level constraintsSubstitute continuous hydro dynamics and relax

limits on reservoir levels Solve subproblems w.r.t. individual hydro units

using dynamic programming similar to thermal subproblems

Apply minimum cost flow optimization with fixed discrete states to schedule generation levels to meet water balance and reservoir level constraints

Method 3: General mixed integer programming Solve the hydro subproblems as a mixed integer

programming problem using solver such as CPLEX

Efficiency closely related to the problem formulation

33

Solve High Level Dual Problem

Subgradient

),,(max,

with ((t), (t)) =

)()()()()(),()(),( ** tPttPtttLttL rdj

hji

ti

jhj

itid tptptPtg )()()()(

jhjhj

ititir tprtprtPtg ))(())(()()(

34

Updating Multipliers

• The multipliers are updated using an efficient subgradient algorithm– Adaptive step sizing– Good initial multipliers using priority list scheduling

35

Over generationPrice down

Time

Total generation

Under generation price up

System demand

Updating Multipliers

36

Obtaining Feasible Schedules

• Goal: to satisfy once relaxed system demand and reserve requirement constraints

• Heuristics should be applied• If possible, satisfy these constraints by adjusting generation

levels only economic dispatch– For piece-wise linear cost function, sorting all power

blocks of all scheduled “up” units and piling these blocks up till the the system wide constraints satisfied

• Adjust discrete operating (commitment) states. Calculate the “opportunity cost” of state change based on

the state transition and cost-to-go information in the dual solution for all units

Adjust the commitment state of a unit with the smallest cost increase

Repeat if sufficient and necessary feasibility conditions not satisfied

38

In this talk



Resolution

Numerical resultsBased on Northeast Utilities system with 70 thermal

units, 7 hydro units and 1 large pumped storage unit

Data set Feb W2, 1991 Feb W3, 1991

CPU time (s) 408 408

# of iterations 50 40

Best feasible cost ($)

4,617,464 7,883,044

Max dual cost ($)

4,595,055 7,831,425

Duality gap 0.47% 0.66 %

Consistent convergence and near optimal Consistent convergence and near optimal schedules obtainedschedules obtained

Only a few seconds on P-IV computerOnly a few seconds on P-IV computerSignificant cost saving in comparison with the Significant cost saving in comparison with the

schedules by NU engineersschedules by NU engineersProduction use for many yearsProduction use for many years

41

In this talk



Resolution

42

Inherent Issues of Lagrangian Relaxation Based Scheduling Algorithms

Homogeneous subproblem solutions to the subproblems of identical unitsMay deviate far away from the optimal scheduleDifficult to obtain feasible solution since not much

information on solution structureLong been recognized and considered as a major

obstacle for applying Lagrangian approach

Existing approaches to solving these issue parameter perturbation (heuristics)

43

Homogenous solution: 2-Unit Example

2

1

))1(),1((mini

iii xpCJ

Subject to

2

1

2)1()1(i

i Dp 3)1(1 ip 0)1(or ip

Optimal solutions: 2)1(1 p 0)1(2 p

2)1(2 p 0)1(1 p

)1()]1(100[))1(),1(( 2iiiii xpxpC with

44

Dual solution with Lagrangian Relaxation

2,1),1()1())1((with,min ippCLL iiiii

Subject to 3)1(1 ip

Optimal dual solution:

,3),1,2/max(min)1( ip 3)1,3(iCif

3)1,3(iCif0)1( ip

45


Dual solution patterns: 0)1()1( 21 pp 1)1()1( 21 ppor

Primal optimal solution:

2)1(,0)1( 21 pp 0)1(,2)1( 21 ppor

0

2

6

3/)3(iC )1(

)1()1( 21 pp

D( 1)

46


Solutions oscillation around

3/)1,3(iC

as the multiplier being updated Subproblem solutions far away from primal optimum Primal optimal solution never obtained

47

Key Idea of the New Algorithm: Differentiate homogenous subproblems

Add quadratic or piece-wise linear penalty terms to Larangian

Solve individual subproblem successively with each high level iteration to keep decomposability

Update Lagrange multipliers using surrogate subgradient at the high level

48

New Algorithm: Successive Subproblem Solving Method (SSS)

InitializationLet w = 0, solve standard LR

Update multipliers

),(1 jjjj xgs

)(1 jjjj xgs

Step 0 Step 1

Solve only one subproblems

Step 2

CheckConvergence

Step 3

49

Features of the New Algorithm

Surrogate subgradient = proper search direction Dual cost still the lower bound of the primal cost Larger penalty weight smaller constraint violation

in the dual problem Rigorous convergence proof

50

Numerical Testing for SSS Algorithm

Testing results of the simple problem with two identical units

Testing results of generation scheduling problem of 10-units with two groups of identical units

Excellent results observed

51

Numerical Testing

Testing results of the simple problem

Iteration l

1 0 0 0 100.0000

2 0 0 0.4750 100.9500

3 0 0 0.9545 101.9090

4 0 0 1.4386 102.8772

5 0 0 1.9272 103.8545

6 2 0 2.4206 104.0000

)1(2lp)1(1

lp l lL

SLR SSS

Hour Unit 3~8 Unit 3 Unit 4 Unit 5 Unit 61 0 0 0 0 02 0 0 0 0 03 0 0 0 0 04 0 0 0 0 05 0 0 0 0 06 0 0 0 0 257 0 0 0 0 107.28 25.0 0 0 25 114.059 120.9 0 25 120.9 120.9

10 127.75 0 127.75 127.75 127.7511 134.6 0 121.6 134.6 134.612 162.0 0 25 144.0 162.013 127.75 0 127.75 127.75 127.7514 120.9 0 25 120.9 120.915 107.2 0 0 25.0 107.216 52.4 0 0 52.4 52.417 38.7 0 0 25.0 38.718 66.1 0 0 66.1 66.119 107.2 0 0 25.0 107.220 162.0 0 0 162.0 162.021 120.9 0 0 120.9 120.922 100.35 0 0 100.35 100.3523 25.0 0 0 25.0 25.024 0 0 0 25.0 0

Testing results of the 10-units generation scheduling

53

Degree of Constraint Violation

0

0.51

1.52

2.53

3.5

0 20 40 60 80 100

Degree of constraint violation (in 10, 000 MWHr)

Iterations

54

Conclusions

Lagrangian relaxation is an efficient and effective approach for solving generation scheduling problems, and other similar resource scheduling problems.

Double dynamic programming is an efficient and effective method to solve subproblems with hybrid dynamic constraints.

The new SSS algorithm is effective to resolve the issues of homogenous subproblems and quantitative measure of the solution quality can still be obtained.

Numerical testing results demonstrated the efficiency and effectiveness of the above method.

The methods have been applied to the scheduling problems of manufacturing, optical switching networks, etc.

55

On-going and Future Work

Forecasting, simulation, bidding strategy, game theoretic analysis and mechanism design for electric power markets

Modeling and analysis of networked systems: power system collapse

Computer network securitySensor networks

Optimization Based Power Generation Scheduling

Documents

Transcript of Optimization Based Power Generation Scheduling