- Email: [email protected]
Reducing homogenous solution oscillations of Lagrangian relaxation-based scheduling algorithms
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
Copyright (.h) 1999 IF A C ] 4th Triennial World Congress,
Beijing~ P.R.
14th World Congress ofIFAC
L-5a-04-5
China
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRANGIAN RELAXATJON-BASED SCHEDULING ALGORITHMS
Xiaohong Guan, Fei Lai Systef1ts Engineering Institute
Xian Jiaotong University Xian 710049, China Te!: +86 29 326-8673, FeLT: +86 2922 J -5248 Email: [email protected]@~ei.xjtu.edu.cn
Abstract: Solution oscillation often caused by discrete decision variables of homogeneous suhproblcms is a severe but inherent disadvantage in applying Lagrangian relaxation based methods for resource scheduling problems. In this paper~ the solution oscillation in Lagrangian relaxation framework is analyzed through an example with two identical subproblems. Based on this analysis, the key idea to alleviate the homogenous oscillation is to differentiate identical subproblems. A surrogate subgradient method is applied to update the multipliers at high level and all subproblenls are not solved simultaneously. The solutions to homogeneous subproblems can be different and the oscillation may be avoided or at least allevi.ated. Numerical testing fOT a short-teITIl generation scheduling problem v.rith hvo groups of identical units demonstrates that solution oscillation is greatly reduced and the feasible generation schedule is significantly improved. Copyright ©19991FAC
Key words: Scheduling algorithm, Resource allocation) Dynamic pro granuning, Minimization, Optimal search techniques, Power generation
I.
INTROD1JCTION
In many industrial production systems such as electric power generation and chemical batch proccss~ resource scheduling is indispensable. It is to allocate limited resources to meet customer demand, and possibly many discrete and continuous dynamic constraints of individual resources. The objects of the scheduling problems are usually to mininlize the operating costs, overdue penalties, etc., or to maximize the profit There would be very significant economic impact for scheduling activities. In short-tenn generation scheduling problem of an electric po,"ver system, a 0.5 to one percent of cost deduction can result in savings of more than ten million US dollars per year for a large utility company (Cohen, et a!', 1987). .However~ this type of scheduling problems generally belongs to NP-hard mixed integer progranuning problems, and it is extremely difficult to generate optimal schedules consistently for systems \vith practical sizes. Many methods have been developed for resource scheduling with continuous and discrete dynamic constraints. One of most successful methods is Lagrangian relaxation (Cohen, et aI., 1987; Sha\v, et aI., 1985~ Ferriera, et al.~ 1989; Guan, et or, 1992). The main idea of this method is to use Lagrange
multipliers to relax system-wide constraints such as demand and capacity so that the problem is converted into a t""ro-levcl optimization structure. The 10\\0' level consists of a number of subproblenls that are much simpler to be solved, one for each· individual resource. The Lagrange multipliers are updated at the high level usually by using a subgradient method. The two-level optimization is performed iteratively until the dual solution converges. A heuristic method is then used to modify a dual solution into a near-optimal feasible schedule. For discrete or mixed optimization problems with certain structures, Lagrangian relaxation is one of the most efficient approaches. The most visible advantage of this approach is its computational efficiency. In many cases, the computational times increase almost linearly with problem sizes (Guan, et 01., 1992). Solution oscillation IS a serious but inherent disadvantage in applying Lagrangian relaxation based methods. First, solution oscillation and singularity caused by hnf"'Br cost functions of so_me subproblcms is a \-\fell-recognized difficulty. The solutions to these subproblenls filay oscillate between the upper and lo\ver bounds of the admissible regions of decision variables with the slight change of the Lagrange multipliers, and nlUY become singular or undetetn1ined when the coefficients of their linear
5967
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
14th World Congress ofIFAC
can be illustrated by the follo"\ving sinlple exan1ple. Consider a single stage scheduling problem decomposable into hvo identical subproblems as follows:
cost functions are zeros or the coefficient matrix is singular. Detailed analysis is presented in (Guan, et al., 1992) and (Renaud~ 1993) to demonstrate the oscillation in primal and dual space. }\ugmented Lagrangian relaxation (Renaud, 1993; Cohen~ et af., 1988,~ \Vang~ et aI., 1995; Batut~ et aI., 1992) and nonlinear approximation (Guan, et al., 1994; Guan~ et al, 1995) ,vere developed to solve the linear osciIIation issues and impressive results have been obtained. Second, a more serious oscillation may be caused by discrete decision variables of homogeneous subproblems and the dual solution may be far away from the optimal schedule. If some subproblems are identical, no nlatter ho\v the multipliers are updated, the solutions to these subproblems are the same. For exa.mple~ suppose in hydrothcnnal generation scheduling problenls~ the optimal schedule is to have 5 of 10 identical tmits generate (up) and rest 5 units idle (do"vn) at a particular time. However since Lagrangian relaxation method is a price-based method~ the solutions to the subproblerns associated with the identical units are the same. These 10 units will be either scheduled up or down simultaneously, possibly causing serious updo\vn solution oscillation. Therefore it is difficult to obtain feasible schedule since there is not much infonnation on solution structure contained in a dual solution. This issue has long been recognjzed and considered as a major obstacle for applying Lagrangian approach especially to systems with significant amount of homogenous subproblems, and to our best knowledge, few good methods have been reported in the literature fOT resolving this issue. In this paper, the solution oscillation caused by homogenous subproblems in Lagrangian relaxation framework is thoroughly analyzed through an example with two identical subproblems. Based on this analysis) the key idea to allev iate the homogenous oscillation is to differentiate jdentical 5ubproblems. A surrogate subgradient method is used in this paper to realize this idea. The surrogate subgradient algoritlull is constructed in (Zhao, et aI., 1997) to update the Lagrange multipliers in order to reduce computational efforts. A search direction for the multiplieTs can he obtained without solving all the subproblems. In fact~ the solution to only one subprohIen) is necessary to obtain a proper surrogate subgradient directjon~ which saves efforts in obtaining a search direction and thus provides a different approach especially powerful for large size problems. Since the multipliers are updated at each high level iteration but all subproblems aTe not solved simultaneously, the dual cost functions of identical subproblems are not the same. The solutions to these subproblems can be different and the oscillation may be avoided or at least alleviated. More importantly, surrogate subgradient method can generate more dual solution patterns for identical subproblems than standard subgradient method so that it is easier to obtained good feasible schedules. Analysis on the above exalnple and nUlllerical testing for a system ,vith 2 groups of identical subproblems shows that surrogate subgradient method is a simple but efficient lllelhod. In comparison with standard subgradient method, surrogate subgradient tnethod achieves better feasible schedules. 11
HOMOGENOlJS SlJBPROBLE1\1 SOLlJTIOK OSCILLATION: AN EXA!\1PLE
L.
L C i (xi(l), zi (1)
min J ==
~
(1)
i=1
subject to 2
L di
(Xi
(1)~ zi (1)) == D(l) == 2,
(2)
i=J
\vhere 1 == ~i S -'-j (1) 5 if = 3 , i = 1) 2~ is continuous decision variable and can be considered as generation level in power generation scheduling problems; z i (1) E {O,l} is binary decision variables; d j (xi (1), zi (1)) is a mixed-integer contribution function with definition:
- rx (1)
dr(xi(l)~zi(l))==~
i
lO
C i (xi (1)
Zi
(1) > 0
(1) :s; 0 is cost function, defined as
C i (Xi (1), Zi (1))
~
(3)
Zi
= [100 + xi~ (1)]- Z i (I)
for
1~xi(1)~3,i=1,2.
(4) Obviously, optimal solution of this problem is Zl (1) == 1 ,Z2 (1) == 0, xJ (1) == 2 ,x 2 (1) =arbitrary, or Zl (1) == 0 , Z2 (1) == 1, X 2 (1) :::: 2, Xl (1) =arbitrary, (5) and the minimum cost is $ 104,
Solution oscillation will occur if this problem is solved by using the Lagrangian relaxation approach and the dual solution would be far away from the primal optimum. In the Lagrangian relaxation framework, two identical dual subproblems are formulated as follows: min L i ,withL i == C i (Xi (1), zi (1)) (6) A(l)xi (1)· z i (l),i = 1,2 subject to 1 S; xi(l) ~ 3 _ (7) where ;l(1) is the Lagrangian multiplier relaxing system demand constraint (2). Give A(l) , the optimal solution to problem (6) is obtained as Xi (1) = min{max(A(l) / 2'!i)' xi Zj (1) :=:: 1 if Ci(Xi,l) <2(1)xi (8) xi (1) = arbitrary, zi (1) == 0
J,
C j (Xi J) ~ .l(l)xi (9) It is seen that no matter \",hat the multiplier is updated,
if
the discrete variables in optimal dual solution fonn only t\Vo patterns: z] (1) == z2(1) == 1 ~ (10) or z}(I)=zz(I)=O,
(11)
and (2) cannot be satisfied so that the solution is not feasible. The optimal primal solution zl (1) :::: 0 , z2(1)=lor zt(l)==l, z2(1)~O can never be obtained. In another ",/ord~ the dual solution does not give much structural information for obtaining a good feasible schedule.
Define high level dual function $(,1,(1») == min L J + min L 2 + A(l)D(l) \'1 '~1l("2 ,z:'
~ L; (A(l»)
The oscillation caused by homogeneous subproblems
+ L; (;[(1») + A(l)D(l)
(12)
5968
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
According to duality theory (Geoffrion, 1974; Fisher, 1981; Hiriart-"LTrruty, et al., 1993; Nemhauser, et al., 1998), the task at the high level is to maximize the above dual function. ""Then subgradient method is used at the high level (Hiriart-lJrruty, et aI., 1993; Nemhauser, et ai., 1998; Held~ et aI., 1974;Camerini, et al., 1975), the multipliers updated by j+1(1) == j + a g~ , g-i
14th World Congress ofIFAC
the n1ininluln up and down times required to keep resource i up or dO\\TI after it is started up or shutdown due to mechanical stress. The number of states required to describe the up and dovln states are u (V l and l,(}id so that the state transition in (19) is truncated. • Discrete decision 'Variable limits (Minirnum up/do",,'n titnes) (21) Vi (t) ::= 1 ) if 0 < :2 i (t) <
(13)
tor
where
v j (t) == -1 ,if
J
g
A
== D(l) -
t
Xj'
(1)· Zi (1) ,
- aJl~ <
Z j (t)
<0
(22)
(14)
i=='
IV
is the subgradient of the dual function (12) to A(t) and a is step size. The low level subproblem solution is shown in Fig. 1
SOLUTION METIIODOLOGy'"
Based 011 the Lagrangiall relaxation framev/ork (Cohel1, et aI., 1987; Sha",r, et al., 1985; Ferriera, et al., 1989; Guan, et a!., 1992; Geoffrion, 1974; Fisher:1981; Hiriart-1JD1lty, et a!., 1993~ Nemhauser, et al, 1998), the system-\vide constraint (17) is relaxed by L,agrange multiplier ).. (t) and the Lagrangian of the prob]em is formed as
6'
I)(1)
J
2~------"';---------
L
:=
J + A(t)[D(t) -
L d (xi Ct), j
Zi (t))]
.
(23)
i=-l
o
Since the decision variables Xi (t) and Zi (t) in (23) and individual constraints (18-22) are decDupled, the problem can be converted into a two-level optimization proble.m. The low level consists of a number of subproblems, one for each resource and the high level is to update the multiplier A = [A(l\ A(2)~ ...... , Itv(T)] ~ the lo\v level subproblems and high level problem are solved iteratively until the dual solution converges. 'The key idea of the surrogate subgradient method is that for given multiplier :\(t), not all but only a certain number of subproblems is solved to update the decision variables of these subproblems (Zhao~ et al.~ 1997).
/.(1)
Fig. 1 Subproblem solution versus the multiplier Solution oscillation is observed around A(l)
= C{(Xi,l)! Xi.
(15)
It v.'iIl be sho\vn in Section IV that the solution asci llation can be significantly reduced, and for the above example, the primal optinlal solution can be obtained by using surrogate subgradient method to update the multiplier.
Ill. PROBLEM FOR1V1ULA-rIONS IV I Solving a Suhproblem
Suppose there are I resources and the scheduling time horizon is T. The resource scheduling problem can be formulated as the following mixed integer programnling problem J
min J
T
ELL [C j=1
j
(Xi
(t),
Zi
By arranging the tenns in (23) accordi !lg to individual resources, a subproblem at the lo'W' level is \vritten as follows: P-i, ; = 1,2, ......... ,[
(t)) + Si (Zi (t)~ Vi (t))] ,( 16)
T
. Lj ~
J==)
ffitn
subject to I
L (li(xi(t), Z,.(t))
=:
D(l) t = 1, 2, ;0
,T,
(18)
•
Discrete state equation and constraints d Zi(t + 1) ::::= lllax {-a}i , min {mt , Zj(t) + Vj(t»)} } ) if Z i (t) . vi (t) > 0 , ( 19) Zi(l+l)==vi(t), if Zi(t)·vj(t) <0;0 (20) \vhere zi (t) i~ the discrete state variable representing the time that resource i has been in operation or shutdo\vn; the value of vi(t) is '"1" for operating (up), H_l" for shut-do""rn; aJr~ and
Zi
,(24)
(t))]
subject to (18-22). The discrete state equations dese-ribed by (18-20) can be represented by a state transition diagram shown in Fig. 2. The cost terms C i (xi (t)~ Zi (t) and - A(t)di(xi(t), zi(t)) in (24) involve discrete states and arc associated ~'ith nodes in Fig. 2. The cost tem1 Sf(zi(t), l\(t) involves state transitions and is associated with arcs. For given multiplier )\.(t), these cost terms can be easily calcula ted and the solutions to above individual subproblems can be obtained efficiently by using dynamic programming with only a few states and \veIl-structured transitions.
;=1
Continuous state constraints ~i S xi(t) S Xi;
(=1
- A(t)d i (Xi (t),
(17)
where C i ( ) is cost associated with states such as po","er generation and Si (-) is costs with changes of discrete states such as on/off of generators. The above optimization is also su~ject to the individual operating constraints described as follows.
•
L rei (Xi (t), Zi (t)) + Si (Z; (t), vi(t)
IV. 2 initializing
With given initial ;r which can be from the previous scheduling cycle, the initial schedule
x?
cuf are
5969
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
--------+.•
z,(t)= 4 Zj
(I)
~
•
1
luore dual solution patterns in comparison with standard subgradient method. This could provide a base to obtain better feasible schedule.
• • •
= 3
z.,(t) =2 z . (r)
14th World Congress ofIFAC
The surrogate subgradient method is applied to solve the simple example with nvo identical subproblems in Section [I. The dual solutions versus iterations are S!lOW'l1 in Fig. 3. It can be seen that all possible pattenlS {O,O;I,O;O,l;1,l} for the binary variables Zl (1) and z ~ (1) can be obtained in dual solutions. The feasible primal solution for this example can be obtained with z 1 (1) =: 1 , Z 2 (1) == 0 or z. (1) =: 0 , Z2 (1) ::::; 1 and adjustment of xl (1) or x 2 (1) .
Zi(r) =-1 Zi(r.)
=-2
zi(t)"""
-t1J1 t+l
Fig 2. Discrete state transition diagram
ZI~1
and z? can be obtained by solving all individual subproblems of (24). The initial surrogate subgradient is calculated as gO = [go(l), gO(2),.~... _, gO(T)] , (25)
r
where
zi~J
I
gO (t)
==:
D(l) -
Ld
j
(x; (t),
Zi (t))
(26)
,
i==1
~-ariQbles
(27)
j-=o1
and si is step size at iteration t. Not all decision variables will not be updated at each iteration~ The most commonly used surrogate subgradient method is interleaved surrogate subgradient method, where only one subproblem is solved and the decision variables associated \vith that subproblem are updated for each high level iteration. Assume subproblemj is selected to be solved at iteration I, the decision variables are determined by solving P-j in (24), that is [X~+l (I), z.~+l (t)] ;;::: arg minL j lj(t),Zj{t)
r
L [e
j
(x j (t),
Zj
"
~
(t)) + S j (z j (t), v j (t))
f=l
- A' + j (t) d.i (x j (t), z.i (t))], (28) For all other subproblems, the decision variables are kept the same: 1 Xf-ll (t) = x;' (t),zf+ Ct) =: zf (t),Vi 7 j . (29) lV. 4 Consfnlcting Feasible Schedules
The dual solution is generally associated \vith an infeasible schedule. TThat is, the once relaxed system-wise constraint (17) are not satJsficd. A heuristic method is usually applied in every dual iteration or in a few heuristic iterations after the dual solution converges to modify the dual solution into a feasib le schedule as in (Cohen? et al., 1987; Shaw, eta!., 1985; Ferricra, eta/., 1989; Guan, eta!., 1992). Since surrogate subgradient method is used~ there are
~~MERICAL
TESTING
The numerical testing is perfonned for short-term po\ver generation scheduling problem with 10 generating units among which two groups of units (lJnit 1-2 and Unit 3-8) are identical respectively. The scheduling horizon is 24 hours. The generation cost function Ci (Xi (t), Zi(t)) is assumed as a quadratic function
I
g~(t)==D(t)- Ldj(xf(t),zf(t));
=:
:'
-.. -V
Given the current schedule and multipliers at iteration 1, the nlultipliers for the next iteration are updated as .J!+1 == i + si g~ , \vhere surrogate subgradient is
vtlithL j
;:
Fig.3 Discrete decision variables in dual solution
the same as the subgradicnt. 1V, 3 Updating J.\1ultipliers and Decision
·
01---------------------·...,.---------.-.~--.:. .-----:----l;---t-.;..-te-:n-at-j-on-.-.~
C;. (x{(t),zi (I» = {f4Xf(t) +b;X; +c" 0,
ifz;(t) > 0 I fzj(t) ~ 0
(30)
The system parameters are summarized in Table 1 and Table 2. The algorithm in implemented in MATLAfJ. A principle to construct feasible schedule is to keep binary decision variables unchanged if possible. In another word, the changes of Z i (t) in constructing feasible schedules are kept minimum. The schedules obtained by standard subgradient method (STS) and surrogate 5ubgradient method (SUS) are shown in Table 3 and Table 4 respectively. Important observations can be noted: • Unit 1 and IJnit 2 are identical and cheap units. 1'hey are scheduled running through by STS and by SUS. This makes sense since cheap units should be utilized as much as possible. • For identical Unit 3-8, their generation scheduled by STS is the same; while the commitment status and generation scheduled by SUS are different. The costs are sununarized in Table 5, where the duality gap, defined as the relative difference between the costs of feasible and dual solution~ is a mea~UTe on solution quality since dual cost is lower bound on costs of all feasible schedules. Due to the reduction of homogeneou~l C!scillation_~ the cost of the feasible schedule obtained by SUS is about 3. 7CjiJ lower in comparison \vith STS, and for the same dual cost~ the duality gap is reduced by 3.8 % • Since the cost base of power generation problems is usually huge, the improvement is very significant.
5970
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
14th World Congress of IFAC
--fable 1 Parameters of Cost Function and Constraints Unit
1
Xi
455
2 455
150
150
:!i aj
bi Cl m~1
r
(1)~ l
Unit xj ~i ll;
bi cj
ll/!I
(l}£j
3 162 25
4 162
~~~~1 t
5
I
8000'
162 25
25
0.00031 0.00031 0.003980.003980.00398 19.70 19.70 19.70 17.26 17.26 970 450 450 970 450 8 8 6 6 6
8
8
6 162
6
7
6
8 162
162
6
6
6
4000 3000
2000
o
'Table 3 Generation Schedule by STS (in
C'Z ]
1
750
2
800
3 4 5 6
900 1000 1050 1150 1200 1250
7
8
Hour 9 10 11 12 13 14 15 16
1)(/)
1350
1450 1500 1550 1450 1350 1250 1100
l-Iour 17 18 19 20 21 22 23
8 9 10
Dft) 1050
1150 1250 1450
2 3 4 5
6 7 8 9
T
Ilg ~ IL = 2: 19 A (1)( ~ t--::]
which represents the degree of the violation of system-\vide constraint (17), against dual iterations in Fig. 4. Based on our experience in constructing feasible schedules (Guan~ et aI., 1992;Guan, et a!., 1994;Guan, eta!., 1995), the smaller the norm the fewer changes of discrete or integer variables and the better feasible schedules usually obtained, It can be seen in Fig. 4 that A rf of the SUS is much 1 smaller than that of STS.
ffg)!fl'
JO
iS2 ]
7
8
9
10 11
12
455 455 455 455 455 455 455 25 40 48.356.773.3 90 98.3 25 40 48,356.773,3 90 98.3 25 40 48.356.773.3 90 98.3 25 40 48.356.773.3 90 98.3 25 40 48.356.773.3 90 98.3
455 107
107 107
107 ] 07
0 0 25 25 25 40 48.356.773.3 90 98.3 107 000000000000 0 0 0 0 0 0 0 0 0 0 0 0
455 455 455 455 455 455 455 455 455 455 400 350 90 73.356.731.7 25 40 56.7 90 73.3 40 25 25 90 73.356.731.7 25 40 56.7 90 73.3 40 25 2S 90 73.356.731.7 25 40 56.7 90 73.3 40 25 9a 73.356.73 L 7 25 40 56.7 90 73.3 40 25 90 73.356.731. 7 25 40 56.7 90 73.3 40 25 90 73.356.731,7 25 40 56.7 90 73.3 40 25 00000000000 0 0 0 0 0 0 0 0 0 0 0
25 25 25 25 () 0
1 2 3 4 5 6 7 8 9 10 11 12 375 400 425 455 455 455 455 455 455 455 455 455
2 375 400 425 455 455 455 455 455 455 455 455 455 3 4 5 6 7 8 9
000000000000 0 0 0 0 0 0 0 0 o a t 18 128 0 0 0 0 U 0 0 85 110 108 118 ] 28 0 0 0 0 08096.785110108]18128 0 02545708096.785110108]18128 0 0 25 45 70 80 96.7 85 110} 08 118 128 000000000000
lO
0
0
0
0
0
0
1"(
13
14
15
16
17
18
CO~CLUSIOKS
Lagrangian relaxation-based resource scheduling algorithms, solution oscillation caused by homogenous subproblems ,"vith discrete decision variables is very difficult to resolve. Based on the analysis for a simple example, a surrogate subgradient method is applied to dlfferenti ate identical subproblelus in order to alleviate the honlogenous oscillation. Numerical testing for a short-tenn generation scheduling problem with t\.vo groups of identical units sho\vs that solution oscillation is greatly reduced and the improvement on the quality of feasible generation sc.hedule is significant. \~.rork is being carried out to develop a unified method to resolve the inherent solution oscillations caused by linear and homogeneous subproblems to obtain stable dual solution and to accelerate converge.
6
rrable 4 Generation Schedule by SUS (in M~...J
Ilg
VI
5
14 15 16 l7 18 19 20 2] 22 23 24 1 455 455 455 455 455 455 455 455 455 455 400 350
To further illustrate the reduction of solution oscillation, \-'le plot the norm of the subgradient of the
dual function to the multiplier)
4
~ J3
1350 1150 950 850
24
3
375 400 375 425 0 0 25 25 4 0 0 2S 25 5 0 0 25 25 6 0 0 25 25 7 0 0 25 25
Table 2 System Demand D(t)
2
3
l
Hour
1
M\~l)
375 400 375 425 455 455 455 455 455 455 455 455
2
I
3
15
Fig.4 Violation of systclll-\vide constraint
10 55
25 10 25 20 25 0.003980.003980.003980.007120.00222 19.70 19.70 19.70 22.26 27.27 450 370 665 450 450 6 1 6 3 6
I
5000
6
9 80
I
7000 6000
FOT
0
0
0
0
19 20 21
22
0
0
23 24
1 455 455 455 455 455 455 455 455 455 455 450 400 2 455 455 455 455 455 455 455 455 455 455 450 400 3 000000000000 4 108 88 68 38 28 48 68 108 0 0 0 0 5108886838284868 )0811080 0 0 6
108 88
68
38
28
A~
68
lOB 110 gO
25
25
7 108 88 68 38 28 48 68 J 08 110 80 25 25 8 108 88 68 38 28 48 68 108 110 0 0 0 9 000000000000 10
0
0
0
0
0
0
0
0
0
0
0
0
5971
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
Table 5 Dual and Feasible Costs (in US $) Dual cost Cost of feasjble schedule Duality gap (%)
STU
sus
595,790
595,750 605,100
1.5
5.3
VII ACKNOWL,EDGMENT The research presented in this paper is part by National Outstanding Y oung
Grant
#6970025,
National
suppo~ed in Investl~ator
Natural
SCIence
Foundation of China, 863 Project of China, Li Heritage Prize for Excellence. in Creative Activities, Li Foundation, San FranCISCO, USA and Key Research Project of Ministry of Education of China.
14th World Congress ofIFAC
Fisher, M. L. (1981). ~'The I~agrangian Relaxat~on J'vfethod for Solvi.ng Integer ProgTamnllng Problems,~~ Management Science~ Vo!. 27, No.I, pp.1-18. . . . Geoffrion, A. M. (1974). '"Lagranglan Relaxation for Integer Programming,'"' Math. Progrolnnling Stud. 2, North Holland: Amsterdam, pp. 82-114. Guan, X.~ P. Luh, H. Yan~ and J. A. Amalfi. (1992). ~~A Optiulization-Based Method for UnIt Con1l11itment, ~, Electric Po"ver & Energy s..vstems, V 01. 14, No. 1, pp. 9-1 7 ~ Guan, X., P. B. Luh, H. Van, P. N1. Rogan.(1994). '''Optimization-Based Scheduling of Hydrothermal Power Systems \vith Pumped-Storage Units," IEEE Tra~sactions on POlrver S}'stems, Vol. 9, No. 2,~ay
pp~
1023-1031.
P. B. Luh and L. Zhang. (1995). HNonlu{ear Approximation Method in Lagrangian Relaxation-Based Algorithms for Hydrothermal Scheduling," IEEE Transactions on Power Svstems, Vot JO, 1\0. 2, May, pp. 772-778. I-Ield M., P. Wolf and I-I. P. Crovlder. (1974). "Validation of Subgradient Optimization,H Mathematical Prograuuuing, Vol. 6, pp.66-28. Hiriart-Urruty, J. B., and C. Lemarechal. (1993). Convex Analysis and Alinimization Algorithll1S /&/1) Springer-Verlag, Berlin. ~emhauser, G., and L. Wolsey. (1988)~ integer and Combinatorial Optimization, John Wiley & Sons, Guan
VIII REFERENCES Batut, J. P., A. Renaud. (1992). "'Daily Genc~at~on Scheduling Optimization with TransmISSion Constraints: A New Class of Algorithms," IEEE Transaction on POl-ver System, Vo1.7, No,3, pp.982-989. . Camerini, P., 1... Fratta; and F~ Maffioh. (1975).
Inc.~
"On Improving Relaxation Methods by Modified Gradient Techniques," l\1athematical Progralnming Study 3, Amsterdam, pp. 26-34.
Cohen A. and V. Sherkat. (1987). "OptimlzationBas~d Methods for Operations Scheduling," Proceedings v/IEEE, Vol. 75, No. 12, pp. 15741591. Cohen~ G. and D. L. Zhu. (1988). '~Decomposition Coordination Methods in Large Scale Optimization Problems: The Non-differentiable Case and the Use of Augmented Lagrangian," Advances in Large Scale Systems: Theory and Applications, Vo!. 27, JAI Press Inc. eT, USA, pp. 203-266. Fcrrcira, L. A. F. M., T. Anderson, C. F. Imparato. (1989). T. E. Miller, C. K. Pang, A. Svobod~, A. ~. \'ojdani, tlShort-Term Resource Scheduhng I~ Multi-Area Hydrothermal Po\ver SysteulS, Electric POlver & Energ)-' Systems, Vol. 11, No. 3, pp. 200-212.
X.
0Jew York_
Rcnaud, A.( 1993). HDaily Generation Management at Electricite de France: Form Planning Towards Real l'ime/' IEEE Tran.saction on Automatic Control, \roI. 38, :No. 7, pp. 1080-1093. Shaw, J. 1., and D. P. Bertsekas4 (1985). " Optimal Scheduling of Large Hydrothermal Power Systems," IEEE Transactions 011 POlver Apparatus and Systems, \roL PAS-I04, pp. 286-293. vVang, S. J., S. M. Shahidehpour, D. S. KiIschcn~ S~ M~khtari and G~ D. Irisarri. (1995). " Short-Term GcneTati~n Scheduling with Transmission Constraints Using Augmented Lagrangian Relaxation, ," IEEE Transactions on Power Syste,nJ, \1 0 1. 10, No. 3, Aug~ pp~ 1294-1301. Zhao, X., P. B. Luh, J. Vlang. (1997). ~'The Surrog~te Gradient Algorithm for Lagranglan RelaxatIon Method,'" Proceedings of the 36th DeCl~{jion and Control Conference, San Diego, CA, USA, Dec. 1997, V 0 l. 1, pp. 30 5-3 10.
5972
Copyright 1999 IFAC
ISBN: 0 08 043248 4
Copyright (.h) 1999 IF A C ] 4th Triennial World Congress,
Beijing~ P.R.
14th World Congress ofIFAC
L-5a-04-5
China
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRANGIAN RELAXATJON-BASED SCHEDULING ALGORITHMS
Xiaohong Guan, Fei Lai Systef1ts Engineering Institute
Xian Jiaotong University Xian 710049, China Te!: +86 29 326-8673, FeLT: +86 2922 J -5248 Email: [email protected]@~ei.xjtu.edu.cn
Abstract: Solution oscillation often caused by discrete decision variables of homogeneous suhproblcms is a severe but inherent disadvantage in applying Lagrangian relaxation based methods for resource scheduling problems. In this paper~ the solution oscillation in Lagrangian relaxation framework is analyzed through an example with two identical subproblems. Based on this analysis, the key idea to alleviate the homogenous oscillation is to differentiate identical subproblems. A surrogate subgradient method is applied to update the multipliers at high level and all subproblenls are not solved simultaneously. The solutions to homogeneous subproblems can be different and the oscillation may be avoided or at least allevi.ated. Numerical testing fOT a short-teITIl generation scheduling problem v.rith hvo groups of identical units demonstrates that solution oscillation is greatly reduced and the feasible generation schedule is significantly improved. Copyright ©19991FAC
Key words: Scheduling algorithm, Resource allocation) Dynamic pro granuning, Minimization, Optimal search techniques, Power generation
I.
INTROD1JCTION
In many industrial production systems such as electric power generation and chemical batch proccss~ resource scheduling is indispensable. It is to allocate limited resources to meet customer demand, and possibly many discrete and continuous dynamic constraints of individual resources. The objects of the scheduling problems are usually to mininlize the operating costs, overdue penalties, etc., or to maximize the profit There would be very significant economic impact for scheduling activities. In short-tenn generation scheduling problem of an electric po,"ver system, a 0.5 to one percent of cost deduction can result in savings of more than ten million US dollars per year for a large utility company (Cohen, et a!', 1987). .However~ this type of scheduling problems generally belongs to NP-hard mixed integer progranuning problems, and it is extremely difficult to generate optimal schedules consistently for systems \vith practical sizes. Many methods have been developed for resource scheduling with continuous and discrete dynamic constraints. One of most successful methods is Lagrangian relaxation (Cohen, et aI., 1987; Sha\v, et aI., 1985~ Ferriera, et al.~ 1989; Guan, et or, 1992). The main idea of this method is to use Lagrange
multipliers to relax system-wide constraints such as demand and capacity so that the problem is converted into a t""ro-levcl optimization structure. The 10\\0' level consists of a number of subproblenls that are much simpler to be solved, one for each· individual resource. The Lagrange multipliers are updated at the high level usually by using a subgradient method. The two-level optimization is performed iteratively until the dual solution converges. A heuristic method is then used to modify a dual solution into a near-optimal feasible schedule. For discrete or mixed optimization problems with certain structures, Lagrangian relaxation is one of the most efficient approaches. The most visible advantage of this approach is its computational efficiency. In many cases, the computational times increase almost linearly with problem sizes (Guan, et 01., 1992). Solution oscillation IS a serious but inherent disadvantage in applying Lagrangian relaxation based methods. First, solution oscillation and singularity caused by hnf"'Br cost functions of so_me subproblcms is a \-\fell-recognized difficulty. The solutions to these subproblenls filay oscillate between the upper and lo\ver bounds of the admissible regions of decision variables with the slight change of the Lagrange multipliers, and nlUY become singular or undetetn1ined when the coefficients of their linear
5967
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
14th World Congress ofIFAC
can be illustrated by the follo"\ving sinlple exan1ple. Consider a single stage scheduling problem decomposable into hvo identical subproblems as follows:
cost functions are zeros or the coefficient matrix is singular. Detailed analysis is presented in (Guan, et al., 1992) and (Renaud~ 1993) to demonstrate the oscillation in primal and dual space. }\ugmented Lagrangian relaxation (Renaud, 1993; Cohen~ et af., 1988,~ \Vang~ et aI., 1995; Batut~ et aI., 1992) and nonlinear approximation (Guan, et al., 1994; Guan~ et al, 1995) ,vere developed to solve the linear osciIIation issues and impressive results have been obtained. Second, a more serious oscillation may be caused by discrete decision variables of homogeneous subproblems and the dual solution may be far away from the optimal schedule. If some subproblems are identical, no nlatter ho\v the multipliers are updated, the solutions to these subproblems are the same. For exa.mple~ suppose in hydrothcnnal generation scheduling problenls~ the optimal schedule is to have 5 of 10 identical tmits generate (up) and rest 5 units idle (do"vn) at a particular time. However since Lagrangian relaxation method is a price-based method~ the solutions to the subproblerns associated with the identical units are the same. These 10 units will be either scheduled up or down simultaneously, possibly causing serious updo\vn solution oscillation. Therefore it is difficult to obtain feasible schedule since there is not much infonnation on solution structure contained in a dual solution. This issue has long been recognjzed and considered as a major obstacle for applying Lagrangian approach especially to systems with significant amount of homogenous subproblems, and to our best knowledge, few good methods have been reported in the literature fOT resolving this issue. In this paper, the solution oscillation caused by homogenous subproblems in Lagrangian relaxation framework is thoroughly analyzed through an example with two identical subproblems. Based on this analysis) the key idea to allev iate the homogenous oscillation is to differentiate jdentical 5ubproblems. A surrogate subgradient method is used in this paper to realize this idea. The surrogate subgradient algoritlull is constructed in (Zhao, et aI., 1997) to update the Lagrange multipliers in order to reduce computational efforts. A search direction for the multiplieTs can he obtained without solving all the subproblems. In fact~ the solution to only one subprohIen) is necessary to obtain a proper surrogate subgradient directjon~ which saves efforts in obtaining a search direction and thus provides a different approach especially powerful for large size problems. Since the multipliers are updated at each high level iteration but all subproblems aTe not solved simultaneously, the dual cost functions of identical subproblems are not the same. The solutions to these subproblems can be different and the oscillation may be avoided or at least alleviated. More importantly, surrogate subgradient method can generate more dual solution patterns for identical subproblems than standard subgradient method so that it is easier to obtained good feasible schedules. Analysis on the above exalnple and nUlllerical testing for a system ,vith 2 groups of identical subproblems shows that surrogate subgradient method is a simple but efficient lllelhod. In comparison with standard subgradient method, surrogate subgradient tnethod achieves better feasible schedules. 11
HOMOGENOlJS SlJBPROBLE1\1 SOLlJTIOK OSCILLATION: AN EXA!\1PLE
L.
L C i (xi(l), zi (1)
min J ==
~
(1)
i=1
subject to 2
L di
(Xi
(1)~ zi (1)) == D(l) == 2,
(2)
i=J
\vhere 1 == ~i S -'-j (1) 5 if = 3 , i = 1) 2~ is continuous decision variable and can be considered as generation level in power generation scheduling problems; z i (1) E {O,l} is binary decision variables; d j (xi (1), zi (1)) is a mixed-integer contribution function with definition:
- rx (1)
dr(xi(l)~zi(l))==~
i
lO
C i (xi (1)
Zi
(1) > 0
(1) :s; 0 is cost function, defined as
C i (Xi (1), Zi (1))
~
(3)
Zi
= [100 + xi~ (1)]- Z i (I)
for
1~xi(1)~3,i=1,2.
(4) Obviously, optimal solution of this problem is Zl (1) == 1 ,Z2 (1) == 0, xJ (1) == 2 ,x 2 (1) =arbitrary, or Zl (1) == 0 , Z2 (1) == 1, X 2 (1) :::: 2, Xl (1) =arbitrary, (5) and the minimum cost is $ 104,
Solution oscillation will occur if this problem is solved by using the Lagrangian relaxation approach and the dual solution would be far away from the primal optimum. In the Lagrangian relaxation framework, two identical dual subproblems are formulated as follows: min L i ,withL i == C i (Xi (1), zi (1)) (6) A(l)xi (1)· z i (l),i = 1,2 subject to 1 S; xi(l) ~ 3 _ (7) where ;l(1) is the Lagrangian multiplier relaxing system demand constraint (2). Give A(l) , the optimal solution to problem (6) is obtained as Xi (1) = min{max(A(l) / 2'!i)' xi Zj (1) :=:: 1 if Ci(Xi,l) <2(1)xi (8) xi (1) = arbitrary, zi (1) == 0
J,
C j (Xi J) ~ .l(l)xi (9) It is seen that no matter \",hat the multiplier is updated,
if
the discrete variables in optimal dual solution fonn only t\Vo patterns: z] (1) == z2(1) == 1 ~ (10) or z}(I)=zz(I)=O,
(11)
and (2) cannot be satisfied so that the solution is not feasible. The optimal primal solution zl (1) :::: 0 , z2(1)=lor zt(l)==l, z2(1)~O can never be obtained. In another ",/ord~ the dual solution does not give much structural information for obtaining a good feasible schedule.
Define high level dual function $(,1,(1») == min L J + min L 2 + A(l)D(l) \'1 '~1l("2 ,z:'
~ L; (A(l»)
The oscillation caused by homogeneous subproblems
+ L; (;[(1») + A(l)D(l)
(12)
5968
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
According to duality theory (Geoffrion, 1974; Fisher, 1981; Hiriart-"LTrruty, et al., 1993; Nemhauser, et al., 1998), the task at the high level is to maximize the above dual function. ""Then subgradient method is used at the high level (Hiriart-lJrruty, et aI., 1993; Nemhauser, et ai., 1998; Held~ et aI., 1974;Camerini, et al., 1975), the multipliers updated by j+1(1) == j + a g~ , g-i
14th World Congress ofIFAC
the n1ininluln up and down times required to keep resource i up or dO\\TI after it is started up or shutdown due to mechanical stress. The number of states required to describe the up and dovln states are u (V l and l,(}id so that the state transition in (19) is truncated. • Discrete decision 'Variable limits (Minirnum up/do",,'n titnes) (21) Vi (t) ::= 1 ) if 0 < :2 i (t) <
(13)
tor
where
v j (t) == -1 ,if
J
g
A
== D(l) -
t
Xj'
(1)· Zi (1) ,
- aJl~ <
Z j (t)
<0
(22)
(14)
i=='
IV
is the subgradient of the dual function (12) to A(t) and a is step size. The low level subproblem solution is shown in Fig. 1
SOLUTION METIIODOLOGy'"
Based 011 the Lagrangiall relaxation framev/ork (Cohel1, et aI., 1987; Sha",r, et al., 1985; Ferriera, et al., 1989; Guan, et a!., 1992; Geoffrion, 1974; Fisher:1981; Hiriart-1JD1lty, et a!., 1993~ Nemhauser, et al, 1998), the system-\vide constraint (17) is relaxed by L,agrange multiplier ).. (t) and the Lagrangian of the prob]em is formed as
6'
I)(1)
J
2~------"';---------
L
:=
J + A(t)[D(t) -
L d (xi Ct), j
Zi (t))]
.
(23)
i=-l
o
Since the decision variables Xi (t) and Zi (t) in (23) and individual constraints (18-22) are decDupled, the problem can be converted into a two-level optimization proble.m. The low level consists of a number of subproblems, one for each resource and the high level is to update the multiplier A = [A(l\ A(2)~ ...... , Itv(T)] ~ the lo\v level subproblems and high level problem are solved iteratively until the dual solution converges. 'The key idea of the surrogate subgradient method is that for given multiplier :\(t), not all but only a certain number of subproblems is solved to update the decision variables of these subproblems (Zhao~ et al.~ 1997).
/.(1)
Fig. 1 Subproblem solution versus the multiplier Solution oscillation is observed around A(l)
= C{(Xi,l)! Xi.
(15)
It v.'iIl be sho\vn in Section IV that the solution asci llation can be significantly reduced, and for the above example, the primal optinlal solution can be obtained by using surrogate subgradient method to update the multiplier.
Ill. PROBLEM FOR1V1ULA-rIONS IV I Solving a Suhproblem
Suppose there are I resources and the scheduling time horizon is T. The resource scheduling problem can be formulated as the following mixed integer programnling problem J
min J
T
ELL [C j=1
j
(Xi
(t),
Zi
By arranging the tenns in (23) accordi !lg to individual resources, a subproblem at the lo'W' level is \vritten as follows: P-i, ; = 1,2, ......... ,[
(t)) + Si (Zi (t)~ Vi (t))] ,( 16)
T
. Lj ~
J==)
ffitn
subject to I
L (li(xi(t), Z,.(t))
=:
D(l) t = 1, 2, ;0
,T,
(18)
•
Discrete state equation and constraints d Zi(t + 1) ::::= lllax {-a}i , min {mt , Zj(t) + Vj(t»)} } ) if Z i (t) . vi (t) > 0 , ( 19) Zi(l+l)==vi(t), if Zi(t)·vj(t) <0;0 (20) \vhere zi (t) i~ the discrete state variable representing the time that resource i has been in operation or shutdo\vn; the value of vi(t) is '"1" for operating (up), H_l" for shut-do""rn; aJr~ and
Zi
,(24)
(t))]
subject to (18-22). The discrete state equations dese-ribed by (18-20) can be represented by a state transition diagram shown in Fig. 2. The cost terms C i (xi (t)~ Zi (t) and - A(t)di(xi(t), zi(t)) in (24) involve discrete states and arc associated ~'ith nodes in Fig. 2. The cost tem1 Sf(zi(t), l\(t) involves state transitions and is associated with arcs. For given multiplier )\.(t), these cost terms can be easily calcula ted and the solutions to above individual subproblems can be obtained efficiently by using dynamic programming with only a few states and \veIl-structured transitions.
;=1
Continuous state constraints ~i S xi(t) S Xi;
(=1
- A(t)d i (Xi (t),
(17)
where C i ( ) is cost associated with states such as po","er generation and Si (-) is costs with changes of discrete states such as on/off of generators. The above optimization is also su~ject to the individual operating constraints described as follows.
•
L rei (Xi (t), Zi (t)) + Si (Z; (t), vi(t)
IV. 2 initializing
With given initial ;r which can be from the previous scheduling cycle, the initial schedule
x?
cuf are
5969
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
--------+.•
z,(t)= 4 Zj
(I)
~
•
1
luore dual solution patterns in comparison with standard subgradient method. This could provide a base to obtain better feasible schedule.
• • •
= 3
z.,(t) =2 z . (r)
14th World Congress ofIFAC
The surrogate subgradient method is applied to solve the simple example with nvo identical subproblems in Section [I. The dual solutions versus iterations are S!lOW'l1 in Fig. 3. It can be seen that all possible pattenlS {O,O;I,O;O,l;1,l} for the binary variables Zl (1) and z ~ (1) can be obtained in dual solutions. The feasible primal solution for this example can be obtained with z 1 (1) =: 1 , Z 2 (1) == 0 or z. (1) =: 0 , Z2 (1) ::::; 1 and adjustment of xl (1) or x 2 (1) .
Zi(r) =-1 Zi(r.)
=-2
zi(t)"""
-t1J1 t+l
Fig 2. Discrete state transition diagram
ZI~1
and z? can be obtained by solving all individual subproblems of (24). The initial surrogate subgradient is calculated as gO = [go(l), gO(2),.~... _, gO(T)] , (25)
r
where
zi~J
I
gO (t)
==:
D(l) -
Ld
j
(x; (t),
Zi (t))
(26)
,
i==1
~-ariQbles
(27)
j-=o1
and si is step size at iteration t. Not all decision variables will not be updated at each iteration~ The most commonly used surrogate subgradient method is interleaved surrogate subgradient method, where only one subproblem is solved and the decision variables associated \vith that subproblem are updated for each high level iteration. Assume subproblemj is selected to be solved at iteration I, the decision variables are determined by solving P-j in (24), that is [X~+l (I), z.~+l (t)] ;;::: arg minL j lj(t),Zj{t)
r
L [e
j
(x j (t),
Zj
"
~
(t)) + S j (z j (t), v j (t))
f=l
- A' + j (t) d.i (x j (t), z.i (t))], (28) For all other subproblems, the decision variables are kept the same: 1 Xf-ll (t) = x;' (t),zf+ Ct) =: zf (t),Vi 7 j . (29) lV. 4 Consfnlcting Feasible Schedules
The dual solution is generally associated \vith an infeasible schedule. TThat is, the once relaxed system-wise constraint (17) are not satJsficd. A heuristic method is usually applied in every dual iteration or in a few heuristic iterations after the dual solution converges to modify the dual solution into a feasib le schedule as in (Cohen? et al., 1987; Shaw, eta!., 1985; Ferricra, eta/., 1989; Guan, eta!., 1992). Since surrogate subgradient method is used~ there are
~~MERICAL
TESTING
The numerical testing is perfonned for short-term po\ver generation scheduling problem with 10 generating units among which two groups of units (lJnit 1-2 and Unit 3-8) are identical respectively. The scheduling horizon is 24 hours. The generation cost function Ci (Xi (t), Zi(t)) is assumed as a quadratic function
I
g~(t)==D(t)- Ldj(xf(t),zf(t));
=:
:'
-.. -V
Given the current schedule and multipliers at iteration 1, the nlultipliers for the next iteration are updated as .J!+1 == i + si g~ , \vhere surrogate subgradient is
vtlithL j
;:
Fig.3 Discrete decision variables in dual solution
the same as the subgradicnt. 1V, 3 Updating J.\1ultipliers and Decision
·
01---------------------·...,.---------.-.~--.:. .-----:----l;---t-.;..-te-:n-at-j-on-.-.~
C;. (x{(t),zi (I» = {f4Xf(t) +b;X; +c" 0,
ifz;(t) > 0 I fzj(t) ~ 0
(30)
The system parameters are summarized in Table 1 and Table 2. The algorithm in implemented in MATLAfJ. A principle to construct feasible schedule is to keep binary decision variables unchanged if possible. In another word, the changes of Z i (t) in constructing feasible schedules are kept minimum. The schedules obtained by standard subgradient method (STS) and surrogate 5ubgradient method (SUS) are shown in Table 3 and Table 4 respectively. Important observations can be noted: • Unit 1 and IJnit 2 are identical and cheap units. 1'hey are scheduled running through by STS and by SUS. This makes sense since cheap units should be utilized as much as possible. • For identical Unit 3-8, their generation scheduled by STS is the same; while the commitment status and generation scheduled by SUS are different. The costs are sununarized in Table 5, where the duality gap, defined as the relative difference between the costs of feasible and dual solution~ is a mea~UTe on solution quality since dual cost is lower bound on costs of all feasible schedules. Due to the reduction of homogeneou~l C!scillation_~ the cost of the feasible schedule obtained by SUS is about 3. 7CjiJ lower in comparison \vith STS, and for the same dual cost~ the duality gap is reduced by 3.8 % • Since the cost base of power generation problems is usually huge, the improvement is very significant.
5970
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
14th World Congress of IFAC
--fable 1 Parameters of Cost Function and Constraints Unit
1
Xi
455
2 455
150
150
:!i aj
bi Cl m~1
r
(1)~ l
Unit xj ~i ll;
bi cj
ll/!I
(l}£j
3 162 25
4 162
~~~~1 t
5
I
8000'
162 25
25
0.00031 0.00031 0.003980.003980.00398 19.70 19.70 19.70 17.26 17.26 970 450 450 970 450 8 8 6 6 6
8
8
6 162
6
7
6
8 162
162
6
6
6
4000 3000
2000
o
'Table 3 Generation Schedule by STS (in
C'Z ]
1
750
2
800
3 4 5 6
900 1000 1050 1150 1200 1250
7
8
Hour 9 10 11 12 13 14 15 16
1)(/)
1350
1450 1500 1550 1450 1350 1250 1100
l-Iour 17 18 19 20 21 22 23
8 9 10
Dft) 1050
1150 1250 1450
2 3 4 5
6 7 8 9
T
Ilg ~ IL = 2: 19 A (1)( ~ t--::]
which represents the degree of the violation of system-\vide constraint (17), against dual iterations in Fig. 4. Based on our experience in constructing feasible schedules (Guan~ et aI., 1992;Guan, et a!., 1994;Guan, eta!., 1995), the smaller the norm the fewer changes of discrete or integer variables and the better feasible schedules usually obtained, It can be seen in Fig. 4 that A rf of the SUS is much 1 smaller than that of STS.
ffg)!fl'
JO
iS2 ]
7
8
9
10 11
12
455 455 455 455 455 455 455 25 40 48.356.773.3 90 98.3 25 40 48,356.773,3 90 98.3 25 40 48.356.773.3 90 98.3 25 40 48.356.773.3 90 98.3 25 40 48.356.773.3 90 98.3
455 107
107 107
107 ] 07
0 0 25 25 25 40 48.356.773.3 90 98.3 107 000000000000 0 0 0 0 0 0 0 0 0 0 0 0
455 455 455 455 455 455 455 455 455 455 400 350 90 73.356.731.7 25 40 56.7 90 73.3 40 25 25 90 73.356.731.7 25 40 56.7 90 73.3 40 25 2S 90 73.356.731.7 25 40 56.7 90 73.3 40 25 9a 73.356.73 L 7 25 40 56.7 90 73.3 40 25 90 73.356.731. 7 25 40 56.7 90 73.3 40 25 90 73.356.731,7 25 40 56.7 90 73.3 40 25 00000000000 0 0 0 0 0 0 0 0 0 0 0
25 25 25 25 () 0
1 2 3 4 5 6 7 8 9 10 11 12 375 400 425 455 455 455 455 455 455 455 455 455
2 375 400 425 455 455 455 455 455 455 455 455 455 3 4 5 6 7 8 9
000000000000 0 0 0 0 0 0 0 0 o a t 18 128 0 0 0 0 U 0 0 85 110 108 118 ] 28 0 0 0 0 08096.785110108]18128 0 02545708096.785110108]18128 0 0 25 45 70 80 96.7 85 110} 08 118 128 000000000000
lO
0
0
0
0
0
0
1"(
13
14
15
16
17
18
CO~CLUSIOKS
Lagrangian relaxation-based resource scheduling algorithms, solution oscillation caused by homogenous subproblems ,"vith discrete decision variables is very difficult to resolve. Based on the analysis for a simple example, a surrogate subgradient method is applied to dlfferenti ate identical subproblelus in order to alleviate the honlogenous oscillation. Numerical testing for a short-tenn generation scheduling problem with t\.vo groups of identical units sho\vs that solution oscillation is greatly reduced and the improvement on the quality of feasible generation sc.hedule is significant. \~.rork is being carried out to develop a unified method to resolve the inherent solution oscillations caused by linear and homogeneous subproblems to obtain stable dual solution and to accelerate converge.
6
rrable 4 Generation Schedule by SUS (in M~...J
Ilg
VI
5
14 15 16 l7 18 19 20 2] 22 23 24 1 455 455 455 455 455 455 455 455 455 455 400 350
To further illustrate the reduction of solution oscillation, \-'le plot the norm of the subgradient of the
dual function to the multiplier)
4
~ J3
1350 1150 950 850
24
3
375 400 375 425 0 0 25 25 4 0 0 2S 25 5 0 0 25 25 6 0 0 25 25 7 0 0 25 25
Table 2 System Demand D(t)
2
3
l
Hour
1
M\~l)
375 400 375 425 455 455 455 455 455 455 455 455
2
I
3
15
Fig.4 Violation of systclll-\vide constraint
10 55
25 10 25 20 25 0.003980.003980.003980.007120.00222 19.70 19.70 19.70 22.26 27.27 450 370 665 450 450 6 1 6 3 6
I
5000
6
9 80
I
7000 6000
FOT
0
0
0
0
19 20 21
22
0
0
23 24
1 455 455 455 455 455 455 455 455 455 455 450 400 2 455 455 455 455 455 455 455 455 455 455 450 400 3 000000000000 4 108 88 68 38 28 48 68 108 0 0 0 0 5108886838284868 )0811080 0 0 6
108 88
68
38
28
A~
68
lOB 110 gO
25
25
7 108 88 68 38 28 48 68 J 08 110 80 25 25 8 108 88 68 38 28 48 68 108 110 0 0 0 9 000000000000 10
0
0
0
0
0
0
0
0
0
0
0
0
5971
Copyright 1999 IFAC
ISBN: 0 08 043248 4
REDUCING HOMOGENOUS SOLUTION OSCILLATIONS OF LAGRA...
Table 5 Dual and Feasible Costs (in US $) Dual cost Cost of feasjble schedule Duality gap (%)
STU
sus
595,790
595,750 605,100
1.5
5.3
VII ACKNOWL,EDGMENT The research presented in this paper is part by National Outstanding Y oung
Grant
#6970025,
National
suppo~ed in Investl~ator
Natural
SCIence
Foundation of China, 863 Project of China, Li Heritage Prize for Excellence. in Creative Activities, Li Foundation, San FranCISCO, USA and Key Research Project of Ministry of Education of China.
14th World Congress ofIFAC
Fisher, M. L. (1981). ~'The I~agrangian Relaxat~on J'vfethod for Solvi.ng Integer ProgTamnllng Problems,~~ Management Science~ Vo!. 27, No.I, pp.1-18. . . . Geoffrion, A. M. (1974). '"Lagranglan Relaxation for Integer Programming,'"' Math. Progrolnnling Stud. 2, North Holland: Amsterdam, pp. 82-114. Guan, X.~ P. Luh, H. Yan~ and J. A. Amalfi. (1992). ~~A Optiulization-Based Method for UnIt Con1l11itment, ~, Electric Po"ver & Energy s..vstems, V 01. 14, No. 1, pp. 9-1 7 ~ Guan, X., P. B. Luh, H. Van, P. N1. Rogan.(1994). '''Optimization-Based Scheduling of Hydrothermal Power Systems \vith Pumped-Storage Units," IEEE Tra~sactions on POlrver S}'stems, Vol. 9, No. 2,~ay
pp~
1023-1031.
P. B. Luh and L. Zhang. (1995). HNonlu{ear Approximation Method in Lagrangian Relaxation-Based Algorithms for Hydrothermal Scheduling," IEEE Transactions on Power Svstems, Vot JO, 1\0. 2, May, pp. 772-778. I-Ield M., P. Wolf and I-I. P. Crovlder. (1974). "Validation of Subgradient Optimization,H Mathematical Prograuuuing, Vol. 6, pp.66-28. Hiriart-Urruty, J. B., and C. Lemarechal. (1993). Convex Analysis and Alinimization Algorithll1S /&/1) Springer-Verlag, Berlin. ~emhauser, G., and L. Wolsey. (1988)~ integer and Combinatorial Optimization, John Wiley & Sons, Guan
VIII REFERENCES Batut, J. P., A. Renaud. (1992). "'Daily Genc~at~on Scheduling Optimization with TransmISSion Constraints: A New Class of Algorithms," IEEE Transaction on POl-ver System, Vo1.7, No,3, pp.982-989. . Camerini, P., 1... Fratta; and F~ Maffioh. (1975).
Inc.~
"On Improving Relaxation Methods by Modified Gradient Techniques," l\1athematical Progralnming Study 3, Amsterdam, pp. 26-34.
Cohen A. and V. Sherkat. (1987). "OptimlzationBas~d Methods for Operations Scheduling," Proceedings v/IEEE, Vol. 75, No. 12, pp. 15741591. Cohen~ G. and D. L. Zhu. (1988). '~Decomposition Coordination Methods in Large Scale Optimization Problems: The Non-differentiable Case and the Use of Augmented Lagrangian," Advances in Large Scale Systems: Theory and Applications, Vo!. 27, JAI Press Inc. eT, USA, pp. 203-266. Fcrrcira, L. A. F. M., T. Anderson, C. F. Imparato. (1989). T. E. Miller, C. K. Pang, A. Svobod~, A. ~. \'ojdani, tlShort-Term Resource Scheduhng I~ Multi-Area Hydrothermal Po\ver SysteulS, Electric POlver & Energ)-' Systems, Vol. 11, No. 3, pp. 200-212.
X.
0Jew York_
Rcnaud, A.( 1993). HDaily Generation Management at Electricite de France: Form Planning Towards Real l'ime/' IEEE Tran.saction on Automatic Control, \roI. 38, :No. 7, pp. 1080-1093. Shaw, J. 1., and D. P. Bertsekas4 (1985). " Optimal Scheduling of Large Hydrothermal Power Systems," IEEE Transactions 011 POlver Apparatus and Systems, \roL PAS-I04, pp. 286-293. vVang, S. J., S. M. Shahidehpour, D. S. KiIschcn~ S~ M~khtari and G~ D. Irisarri. (1995). " Short-Term GcneTati~n Scheduling with Transmission Constraints Using Augmented Lagrangian Relaxation, ," IEEE Transactions on Power Syste,nJ, \1 0 1. 10, No. 3, Aug~ pp~ 1294-1301. Zhao, X., P. B. Luh, J. Vlang. (1997). ~'The Surrog~te Gradient Algorithm for Lagranglan RelaxatIon Method,'" Proceedings of the 36th DeCl~{jion and Control Conference, San Diego, CA, USA, Dec. 1997, V 0 l. 1, pp. 30 5-3 10.
5972
Copyright 1999 IFAC
ISBN: 0 08 043248 4