Power Generation Loading Optimization using a Multi-Objective ConstraintHandling Method via PSO Algorithm Lily D LI1, Xiaodong Li2, Xinghuo Yu2
-----------------------------------------------------------------------------------------------------------------
1
Central Queensland University, Australia 2 RMIT University, Australia
IEEE INDIN 2008
1
Outline
Background Problem formulation The proposed approach PSO algorithm Constraint handling Simulation Results Discussion Conclusion IEEE INDIN 2008
2
Background A typical coal-fired power generation unit – principle: consume coal, produce electricity and emission
(image from: http://www.tva.gov/power/images/coalart.gif) IEEE INDIN 2008
3
Background (cont.)
Environmental Regulation Rising fuel costs Green house gas emission
Require Power plants generators to be more efficient (i.e. higher performance) IEEE INDIN 2008
4
Background (cont.)
A generator unit’s performance is determined by its thermal efficiency ( or heat rate) A unit’s thermal efficiency changes all the time (depends on design, plant life, operation & maintenance practice etc.) Unit loading distribution should consider each unit’s performance and its capacity IEEE INDIN 2008
5
Background (cont.)
Ideally: economically distributed higher performance unit lower performance unit
IEEE INDIN 2008
Higher workload
Lower workload
6
Problem formulation Modelling
IEEE INDIN 2008
7
Problem formulation (cont.)
Objective function
Heat consumption should be minimized F ( X ) =
n
∑
i = 1
n: xi: hi : fi(xi):
h
i
=
n
∑
x
i
fi ( xi )
i = 1
number of unit workload to unit i heat consumption to unit i unit i’s heat rate IEEE INDIN 2008
8
Problem formulation (cont.)
Unit i’s heat rate f i ( x i ) = a i k x i + a i ( k −1 ) x i k
( k −1 )
+ ... + a i 1 x i + a i 0
Unit i’s NOx emission level q i ( x i ) = bi 1 x i + b i 0
Above two functions for each unit are obtained from field testing or unit modelling IEEE INDIN 2008
9
Problem formulation (cont.)
Constraints
Total output load must be maintained and adjustable n
∑
xi = M
to ta l
i =1
Each unit’s NOx gas emission has to be restricted within a license limit P qi (xi ) − P ≤ 0
(i = 1,2,... n )
Unit capacity constraints M i min ≤ xi ≤ M i max (i = 1, 2,...n ) IEEE INDIN 2008
10
Problem formulation (cont.)
Optimization statement minimize F ( X ) = ∑ n
x
i
fi ( xi )
i =1
subject to
g1 ( X ) =
n
|
∑
xi − M
to ta l
|− ε ≤ 0
i =1
ri (X) = qi (xi ) −P ≤ 0 (i = 1,2,... n ) M
i m in
≤ xi ≤ M
i m ax
( i = 1 , 2 , . . .n )
ε : a minimum error criterion for equality constraint IEEE INDIN 2008
11
PSO algorithm
A stochastic optimization method Mimic social behaviour of flocks of birds and schools of fish Population based Evolve from generation to generation
Picture from: http://www.ieeeswarm.org/
IEEE INDIN 2008
12
PSO algorithm (cont.)
Each particle represents a solution Particles fly in the search space During flying, particles adjust direction by using each individual’s best past experience (individual behavior) and its neighbors’ best experience (social behavior) Eventually, fly to the optimal (or close to optimal) solution IEEE INDIN 2008
13
PSO algorithm (cont.)
Formula v id
(t +1)
v id
(t +1)
v id
(t +1)
xid
(t +1)
= χ [ w v id
)⎫ ⎪ + c 2 r 2 i d ( t ) ( l B e s t i d ( t ) − x i d ( t ) ) ] ⎪⎪ ⎬ i f v id ( t +1) > V m ax = V m ax , ⎪ ⎪ i f v id ( t +1) < − V m ax = − V m ax , ⎪⎭
= xid + vid (t )
(t )
+ c 1 r1id
(t )
( p B e s t id
(t +1)
(t )
− x id
(t )
(a) (b)
vidt+1: velocity for particle i, dimension d, generation t+1 xidt+1: position for particle i, dimension d, generation t+1 IEEE INDIN 2008
14
Constraint-handling
Multi-objective constraint-handling method Treat constraints as objectives One is to transform single objective constrained problems into bi-objective unconstrained problems m in im ize w here
F( X )= ( f ( X ), Φ ( X )) Φ (X ) =
m
∑
i =1
⎫ ⎪ ⎬ m ax(0, g i ( X )) ⎪ ⎭
Φ: is a total amount of constraint violations m: is number of constraints gi: is the number i constraint function IEEE INDIN 2008
15
Constraint-handling (cont.)
Adopt domination concept Definition: A solution X(1) is said to dominate the other solution X(2), if both conditions 1 and 2 are true: The solution X(1) is no worse than X(2) in all objectives, (for all j =1,2,…,m). The solution X(1) is strictly better than X(2) in at least one objective, for at least one
Second objective Φ has high priority
Final results must be selected from Φ = 0 or Φ ≤ δ
IEEE INDIN 2008
16
Selection rules
Selection rules
Non-dominated particles are better than dominated ones Particles with lower Φ is better than those with higher Φ (means closer to the feasible area)
IEEE INDIN 2008
17
Modified PSO algorithm 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.
GlobalF = POSITIVE_INFINITY; P0 = URand (Li, Ui) V0 = 0 F0 = Fitness_F (P0) Φ0 = Fitness_Φ(P0) pBest0 = P0 For i = 0 To N lBesti = LocalBest (Pi-1, Pi, Pi+1) End for Do For i = 0 To N Vi+1 = Speed ( Pi,,Vi, pBesti, lBesti) Pi+1 = Pi+ Vi+1 IEEE INDIN 2008
(Selection rules)
(Equation (a)) (Equation (b)) 18
Modified PSO algorithm (cont.) 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25.
Fi+1 = Fitness_F(Pi+1) Φi+1 = Fitness_Φ(Pi+1) If (Pi+1 isBetterThan pBesti ) pBesti = Pi+1 If (Φi+1 ≤ δ) If (Fi+1 < GlobalF) GlobalF = Fi+1 End For For i = 0 To N lBesti = LocalBest (Pi-1, Pi, Pi+1) End for While (iteration < max_iteration)
(Selection rules)
(Selection rules)
IEEE INDIN 2008
19
Features of the modified PSO algorithm
When calculating fitness, both objectives F and Φ need to be evaluated;
If a particle’s new location is better than its best past location, the pBest is updated (decided by selection rules);
IEEE INDIN 2008
20
Features of the modified PSO algorithm (cont.)
A particle’s best neighboring particle is determined by the two steps:
Find all the non-dominated particles in the neighborhood (by comparing two objective functions); If there is only one non-dominated particle in the neighborhood, select it as lBest; otherwise, select one with the lowest Φ as lBest (the lower Φ means closer to the feasible region).
IEEE INDIN 2008
21
Simulation
Unit heat rate and emission functions (from a local Australia power plant)
IEEE INDIN 2008
22
Parameters
Neighbourhood topology: ring (circular) w = 0 (Empirically derived) c1=c2=2 χ=0.63 Vmax= 0.5*(decision variable range) number of particles is 40 the maximum iteration is set to 10,000 ε = 1.0E-3 (minimum error criterion for equality constraint) δ = 1.0E-8 (feasibility tolerance) 10 independent runs
IEEE INDIN 2008
23
Results Table II is the optimized loading distribution for whole range of the generation capacity
IEEE INDIN 2008
24
Results (cont.)
• Higher performance unit receives higher workload (unit 1)
• Lower performance unit receives lower workload (unit 3) IEEE INDIN 2008
25
Constraint-handling methods comparison
Compared two constraint handling methods (result in Table III)
9 Preserving Feasibility constraint-handling method 9 Multi-objective constrainthandling method
The multi-objective constraint-handling method performs better
IEEE INDIN 2008
26
Discussion
Preserving feasibility constraint-handling method only consider feasible solutions initialization takes long time may be impractically too long or impossible for large search spaces and small feasible spaces Penalty function constraint-handling method penalty factor is difficult to choose problem-dependent IEEE INDIN 2008
27
Discussion (cont.)
Multi-objective constraint-handling method
no problem dependent parameters applied to a wider applications faster initialization infeasible solution are not ignored which contribute to the diversity and convergence (recent research shown)
IEEE INDIN 2008
28
Discussion (cont.)
Multi-objective constraint-handling method
performance depends on feasibility tolerance δ i,e., the larger δ, the easier to search
IEEE INDIN 2008
29
Conclusion
A multi-objective constraint handling method for power generation loading optimization problem is proposed Domination concept and selection rules are adopted No problem-dependent parameters Simulation demonstrated: capability, efficiency Methodology can be applied to wider applications IEEE INDIN 2008
30
Thank you!!
IEEE INDIN 2008
31
