Lagrangian Relaxation Neural Network for Unit Commitment Peter B. Luh, Yajun Wang and Xing Zhao Department of Electrical and Systems Engineering University of Connecticut Storrs, CT 06269-2157
[email protected],
[email protected],
[email protected] Abstract: This paper presents a novel method for unit commitment by synergistically combining Lagrangian relaxation for constraint handling with Hopfield-type recurrent neural networks for fast convergence to the minimum. The key idea is to set up a Hopfieldtype network using the negative dual as its energy function. This network is connected to “neuron-based dynamic programming modules” that make full use of the DP structure to solve individual unit subproblems. The overall network is proved to be stable, and the difficulties in handling integer variables, subproblem constraints, and subproblem local minima plaguing current neural network methods are avoided. Unit commitment solutions are thus natural results of network convergence. Software simulation using data sets from Northeast Utilities demonstrates that the results are much better than what has been reported in the neural network literature, and the method can provide near-optimal solutions for practical problems. Furthermore, the method has the potential to be implemented in hardware with much improved quality and speed.
Hopfield-type neural networks for constrained optimization have been based on the well known “penalty method” that convert a constrained problem to an unconstrained one by having penalty terms on constraint violations. The unconstrained problem is then solved by Hopfield-type networks as mentioned above [11]. Generally, a solution to the converted problem is a solution to the original one only when penalty coefficients approach infinity. As coefficients become large, however, the converted problem is ill conditioned. To obtain a solution without letting coefficients go to infinity, tradeoff between objective optimality and constraint satisfaction must be made by fine tuning algorithm parameters. The tradeoff, however, is generally difficult to make. For problems with integer variables, Hopfield networks approximate these variables by continuous ones, and “high gain activation functions” or additional constraints are used to induce integrality. These approaches, however, cause convergence difficulties and impede solution quality. In addition, these networks may possess local minima, and the solution quality depends highly on initial conditions.
Keywords: Unit commitment, neural networks, optimization, Lagrangian relaxation, dynamic programming.
I. INTRODUCTION Unit commitment is to determine the commitment and generation levels of generators to meet the system demand and reserve requirements. It has played an important role over the past few decades in reducing utilities’ generation costs. It will continue to be important in the deregulated power market for utilities to support their bidding process, and for Independent System Operators (ISO) to select bids submitted by utilities and Independent Power Producers. A challenge is to further improve solution quality and computation speed to respond to the rapidly changing and competitive market. Conventional methods seem to have reached a plateau on this regard, and neural networks hold much promise for the next breakthrough.
Hopfield-type networks and its extended versions were developed to solve unit commitment problems [5, 7, 8, 9, and 12]. Since minimum up/down time constraints formulated as “IF-THEN” statements are difficult to handle by the penalty method, they were either omitted or simplified. As a result, individual unit constraints may be violated. “Noise methods” were introduced to randomly change neuron states to help jump out of local minima [12], though global minimum cannot be guaranteed. Although these methods demonstrated the possibility of using neural networks for unit commitment, they suffer from the above-mentioned difficulties. It is also not easy to scale up the methods for practical problems, and the largest problem reported is a system of 30 units with a time horizon of 24 hours [9, 12].
Neural networks for optimization. Neural networks for unconstrained optimization are based on the “Lyapunov stability theory” of dynamic systems – if a network is “stable,” its “energy” will decrease to a minimum as the system approaches its “equilibrium state.” If one can map the objective of an optimization problem to an “energy function” of a properly set up network, then the solution is a natural result of network convergence, and can be obtained at a very fast speed [3].
Recent development in neural networks for constrained optimization includes combining Lagrangian Relaxation (LR) or Augmented Lagrangian Relaxation with Hopfield-type networks showing significant improvement on solution quality [4]. Unfortunately, with “traveling salesman problems” as the reference model by most researchers, method development has been problem specific with many important issues overlooked and great opportunity missed. Scope of This Paper. This paper presents a novel method for unit commitment by synergistically combining Lagrangian relaxation for constraint handling with Hopfield-type recurrent neural networks for fast convergence to minimum. Problem formulation is provided in Section 2, and for simplicity of presentation, only thermal units without ramp rates are 1
considered. The key ideas as presented in Section 3 are to set up a Hopfield-type network using the negative dual as the energy function. This network is connected to “neuron-based dynamic programming modules” (NBDP) developed in Section 4 to solve individual unit subproblems. The resulting Lagrangian relaxation neural network (LRNN) is proved to be stable, therefore unit commitment solutions are natural results of network convergence. As the dynamic programming structure is fully exploited by NBDP to handle integer variables and unit-wise constraints, LRNN transcends the majority of difficulties associated with conventional neural network methods such as local minima, slow convergence, and violations of local constraints. Software simulation using data sets from Northeast Utilities demonstrates in Section 5 that the method is able to provide near optimal solutions for practical problems, and the results are much better than what has been reported in the neural network literature. Furthermore, the method has the potential to be implemented in hardware with much improved quality and speed.
and the 10-minute spinning reserve requirements: I ∑ ri ( p i ( t )) ≥ Pr ( t ). i =1
Individual unit constraints may include:
The above problem is “separable” since the objective function is the sum of generation costs of individual units, and the system-wide demand and reserve requirements are also additive. Fully exploiting the separable structure can reduce algorithm complexity and improve solution quality. III. LAGRANGIAN RELAXATION NEURAL NETWORKS FOR UNIT COMMITMENT
Unit commitment is to determine the commitment and generation levels of I thermal units over a specified time period T. The time unit is one hour and the planning horizon may vary from one week to ten days. To mathematically formulate the problem, the following symbols are introduced:
I: i: Pd ( t ) :
Pr ( t ) : pi (t) : ri (p i ( t )) : x i (t) : Si ( x i ( t )) :
T: t:
Lagrangian Relaxation. Playing a fundamental role in constrained optimization over the decades, Lagrangian relaxation is particularly powerful for separable problems. The idea is to relax system-wide constraints on demand and reserve (2.2) and (2.3) by using multipliers, and to form a twolevel optimization. The “Lagrangian” is first formulated as
Fule cost of thermal unit i for generating power pi(t) at time t, a piece-wise linear function of pi(t), in dollars; Number of thermal units; Index of thermal units; System demand at time t, in MW;
T
t =1 i =1
i =1
i =1
min L i , with L i (λ, µ) ≡
T ∑ {[C i ( p i ( t )) + S i ( x i ( t ))] t =1
−λ( t )p i ( t ) − µ( t )ri ( t )]},
(3.2)
subject to individual unit constraints. Given a set of multipliers, (3.2) can be solved by dynamic programming. Let L*i (λ, µ) denote the optimal sub-Lagrangian for subproblem i. The high level dual problem is then given by I
* max L(λ, µ), with L(λ, µ) ≡ ∑ L i (λ, µ) λ, µ
(2.1)
i =1
+
subject to system-wide constraints including system demand:
= Pd ( t ),
I
where λ(t) and µ(t) are Lagrange multipliers associated with demand and reserve requirements at time t, respectively. After re-grouping the terms in (3.1), the low level consists of individual unit subproblems:
Based on the rules of New England Power Pool (NEPOOL), we have the following separable mixed-integer optimization problem:
I ∑ pi ( t ) i =1
I
+ λ( t )[Pd ( t ) − ∑ p i ( t )] + µ( t )[Pr ( t ) − ∑ ri (p i ( t ))]} , (3.1)
10-minute spinning reserve contribution of thermal unit i at hour t; State of unit i at time t, denoting the number of hours that unit i has been on or off; Start-up cost of thermal unit i, a linear funciton of time since last shut down, in dollars; Time horizon, in hours; Hour index, t = 1, …, T.
min J, with J ≡
I
L(λ, µ) ≡ min ∑ { ∑ [Ci (pi ( t )) + Si ( x i ( t ))]
10-minute spinning reserve requirement at time t, in MW; Power generated by unit i at hour t, in MW;
T I ∑ { ∑ [C i ( p i ( t )) + S i ( x i ( t ))]}, t =1 i =1
Capacity Minimum up/down time Must-run or must-not-run Minimum generation for the first and last hour (this is required by NEPOOL for steam units)
Detailed descriptions can be found in [2].
II. PROBLEM FORMULATION
C i (p i ( t )) :
(2.3)
T ∑ [λ ( t ) Pd ( t ) + µ ( t ) Pr ( t )]. t =1
(3.3)
Maximizing the dual (3.3) without its explicit representation can be done by several methods, including the most widely used subgradient method described by
(2.2) 2
λk+1 = λk + αk ∇λL(λk, µk), and
(3.4)
µk+1 = µk + αk ∇µL(λk, µk),
(3.5)
the negative dual function that is always convex without local minima, a subnetwork may have many local minima, and the solution quality depends highly on initial conditions. To overcome these difficulties, an innovative way to construct subnetworks, the neuron-based dynamic programming, is presented next.
where k is the iteration index, λk and µk are the multiplier vectors at iteration k, αk the step size, and ∇λL and ∇µL the subgradient of L with respect to λk and µk, respectively. The dual function is always concave, and provides a lower bound to the optimal primal cost [1].
IV. NEURON-BASED DYNAMIC PROGRAMMING The dynamic programming structure is fully utilized by neuron dynamic programming to handle integer variables and individual unit constraints. To better understand NBDP, the standard DP is presented first.
Iterative updating of multipliers, repeated resolution of subproblems, and the heuristic adjustment of subproblem solutions lead to a near-optimal solution of the original unit commitment problem. This approach can be naturally mapped onto a neural network to be presented next.
Dynamic Programming for Thermal Subproblems. Each subproblem can be viewed as a multistage optimization problem and solved by DP. A simplified DP structure is shown in Fig. 4.1, with stages corresponding to hours. The number of “up states” is equal to the minimum up time plus one, where the extra one is needed to consider “last hour generation” required by the NEPOOL rules for steam units. Similarly, the number of “down states” is equal to the “cold start-up time” because the start-up costs remain constant after the cold start-up time. Minimum up/down time constraints are handled through the proper delineation of allowable transition patterns as presented in [2], with start-up costs associated with certain state transitions.
Lagrangian Relaxation Neural Network. The key idea of LRNN is to set up a network using the negative dual -L(λ, µ) as the energy function. In this network, multipliers are represented by “Lagrangian neurons,” and values of the energy function as well as the subgradients of the function with respect to multipliers are provided by the “neuron-based dynamic programming modules” (NBDP) to be presented in the next section. As LRNN evolves, multipliers and NBDP are simultaneously updated, and the energy decreases to its minimum (or the dual L(λ, µ) increases to its maximum). To construct the network, multiplier updating formula (3.4) and (3.5) are viewed as a set of differential equations [5]. The dynamics of “Lagrangian neurons” can thus be described by the following differential equations:
∂L(λ(t ), µ(t ), x i (t ), p i (t )) dλ (t ) = α(t) · , and ∂λ(t ) dt
(3.6)
∂L(λ(t ), µ(t ), x i (t ), p i (t )) dµ(t ) = α (t) · . ∂µ(t ) dt
(3.7)
Up States
Down States t
t+1
Figure 4.1 A Simplified State Transition Diagram
In the above, λ, µ, and L are continuous-time functions with time argument t which is different from the hourly index t used in (3.1). The function L(t) is also slightly different from L(λ, µ) where the primal decision variables xi(t) and pi(t) have been optimized according to (3.1). For L(t), however, this is not the case since the primal variables xi(t) and pi(t) are evolving at the same time with λ and µ, and generally have not reached their optimal values for the multipliers λ(t) and µ(t) at that time instant except at convergence.
The backward DP algorithm starts with the last stage, and computes the terminal cost. As the algorithm moves backwards, the cost-to-go of a state belonging to a particular stage is the sum of the stage-wise cost, the transition cost, and the optimal cost-to-go of the succeeding stage. Minimizing the sum of the transition cost and the optimal cost-to-go of the succeeding stage is performed over all allowable transitions to obtain the state’s optimal cost-to-go. The optimal subproblem cost is then obtained as the minimum of the optimal costs-togo at the first stage, and the commitment status and generation level at each hour for the given set of multipliers can then be obtained by forward tracing the stages.
A key of LRNN is to develop efficient subnetworks for subproblems, which should be solved simultaneously as multipliers are updated. Mapping subproblems onto subnetworks, however, is not as easy as setting up Lagrangian neurons. Difficulty comes from minimum up/down time constraints that are formulated by “IF-THEN” statements. These constraints must be converted to equality or inequality constraints before they can be handled by either the penalty or relaxation methods. This, however, is not easy. In addition, both discrete and continuous variables exist. High gain activation functions and other methods could be used to induce integrality. These approaches, however, cause convergence difficulties and impede solution quality. Furthermore, unlike
Neuron-Based Dynamic Programming. To solve a subproblem by a sub-network, one approach is to let neurons represent DP states, and introduce constraints among neurons to delineate allowable state transitions. Constraints among neurons can then be handled by the penalty or relaxation methods. This, however, is not a good approach since if the penalty method is used, subproblem solutions may not be feasible, and the local minima difficulty cannot be avoided. If
3
the relaxation method is used, excessive number of multipliers will be introduced, and this will lead to slow convergence.
L* = L(λ*,µ *, x *i , p *i ) ≤ L(λ*,µ *, x i (t), p i (t)).
To overcome these difficulties, neuron-based dynamic programming (NBDP) is developed. The key idea is to make the best use of the DP structure already existed, and implement the DP functions by neurons. In doing this, the DP structure as illustrated in Fig. 4.1 is maintained, and each state is represented by a “state neuron.” The function of a state neuron is to obtain the optimal cost-to-go by adding up two values, the stage-wise cost derived from multipliers, and the minimum of the sum of the transition cost and optimal cost-togo of the succeeding stage. This minimization is carried out through the introduction of another layer of “comparison neurons.” The connections of state neurons and comparison neurons are subject to allowable state transitions as shown in Fig. 4.2, where comparison neurons are represented by gray circles. To obtain stage-wise costs, “stage-wise cost neurons” are introduced to calculate the stage-wise costs given the multipliers as input. The two stage-wise cost neurons in Fig. 4.2 correspond to the two cases of optimal generation and the first/last hour minimum generation. The standard DP is thus mapped onto a subnetwork with simple topology and elementary functional requirements that can be implemented in hardware. The number of neurons required for subproblem i is (2×Ji+2)×T, where Ji is the number of states for unit i, and T the time horizon.
(4.3)
Lagrangian Neurons λ(t)
µ(t) Stage-wise Cost Neurons State Neurons
S. G. (*)
t
Comparison t+1 Neurons Neuron Dynamic Programming *
S.G. denotes the Subgradient Generator Fig. 4.2 LRNN Structure
Therefore,
The commitment status can be obtained by tracing forward the stages. The generation level at each state is feed to the Subgradient Generator to calculate the subgradient, which is then feed to the multiplier neurons to update the multipliers as described by (3.6) and (3.7). The multiplier neurons and NBDP are thus integrated and evolve simultaneously.
T I T ∂L( t ) T ∂L( t ) L* ≤ ∑ ∑ [C i (p i ( t )) + Si ( x i ( t ))] + λ* + µ* ∂λ( t ) ∂µ( t ) t =1i =1
Since the DP structure is fully exploited, a sub-network does not have local minima, and an optimal solution can be obtained at the convergence of LRNN. In addition, the solution satisfies all unit constraints that can be handled by the standard DP. Difficulties such as infeasibility and local minima of subproblem solutions encountered by using the penalty or relaxation method are completely avoided.
Given (4.1), this yields
= L(t) + (λ* − λ (t ))T
(4.4) 0 < L* - L(t) ≤ (λ* − λ (t ))T
From (3.6) and (3.7), we then have 0 < α(t)[ (λ* − λ (t ))T < (λ* − λ(t )) T
Proposition 1: Given the current multipliers (λ(t), µ(t)), if
0 < (λ − λ(t ))
T
dµ(t ) dλ(t ) . + (µ * − µ(t )) T dt dt
∂L(t ) ∂L(t ) + (µ * − µ(t )) T ] ∂µ(t ) ∂λ(t )
dλ(t ) dµ(t ) + (µ * − µ(t )) T , dt dt
therefore (4.2) is proved.
(4.1)
(4.6) Q.E.D.
Proposition 2. If the initial point satisfies
then the subgradient direction of L(t) with respect to (λ(t), µ(t)) is always in an acute angle with the direction towards the optimal (λ*, µ *), i.e., *
∂L(t ) ∂L(t ) + (µ * − µ(t )) T . ∂µ(t ) ∂λ(t ) (4.5)
Convergence of LRNN. Let L(t) denote the continuous-time Lagrangian function, and L* the optimal dual solution. The following steps establish the convergence of LRNN for the unit commitment problem. L(t) < L*,
∂L(t ) ∂L(t ) + (µ * − µ(t )) T . ∂µ(t ) ∂λ(t )
L ( 0) < L*
(4.7)
and the dynamics of Lagrangian neurons satisfies
(4.2)
α(t) ≤
*
Proof. Since minimization is performed in defining L , the optimal dual always satisfies
then
4
L* − L( t ) ∂L( t ) ∂λ ( t )
2
+ ∂L( t ) ∂µ ( t )
2
,
(4.8)
the results of Sasaki et al. 1992 [9] and Walsh et al. 1997 [12], are also presented in the table. Since LRNN does not have the local minimum difficulty, the problem was solved only once whereas the problems presented in [9] and [12] were solved 100 times for different initial conditions, with best and worst results presented. In addition, since [9] and [12] are based on the penalty method, the parameters must be tuned to trade off between optimality and feasibility. This is not easy, however, as stated in [12] that “The problem of accurately tuning the parameters remains,” and “no rigorous method for selecting these parameters exists.” It is difficult to scale up these methods for practical applications since the larger the problem, the more difficult to tune parameters.
L( t ) < L*. Proof: Please refer to [6]. Theorem: (λ(t), µ(t)) in LRNN described by (3.6) and (3.7) will converge to an optimal point (λ*, µ *) as long as
L ( 0) < L* and α(t) ≤
L* − L( t ) ∂L( t ) ∂λ ( t )
2
+ ∂L( t ) ∂µ ( t )
2
.
Proof: From the above proposition, the gradient direction of (λ(t), µ(t)) is always in an acute angle with the direction pointing to (λ*, µ *). From (4.6), we have
d λ* − λ(t )
TABLE 5.1 COMPARISON OF LRNN VS. OTHERS Methods
2 + µ * − µ (t ) dλ(t ) = − 2(λ* − λ(t )) T dt dt
No. of Units
2
− 2(µ * − µ(t )) T
d µ (t ) < 0. dt
(4.9)
CPU (min)
[9]
30
24
*
[12]
17
24
**
LRNN
60
168
*
Therefore (λ(t), µ(t)) gets closer and closer to (λ*, µ *). It can also be shown that (λ(t), µ(t)) converges to an optimal solution of the dual problem [6]. Q.E.D.
Time Horizon
Cost Higher than LR best
Worst
Average
5
0.51%
2.79%
1.50%
15 3.1
-0.27%
7.58%
N/A
0.57%
Obtained on NEC PC9801RA21 with an INMOS T80020 Transputer board as the accelerator [9]. Obtained on PC 486 [10].
**
The solution quality in the above table is obtained by comparing neural network results versus those obtained by the LR method following what was done in [9]. The closer the costs are to what were obtained by LR, the better the method is. In [9], the average cost is 1.5% higher than that of LR. The result of LRNN is 0.57% higher than what was obtained by our LR method [2] using a fine tuned subgradient method to update multipliers. For [12], the average cost is not available. The best cost is slightly better than our results while the worst is much worse, and the difference between the best and the worst is quite large. Compared with [9] and [12], LRNN is able to solve much larger problem with satisfactory results.
A Word on Hardware Implementation. Our ultimate goal is to implement LRNN in hardware. Hardware implementation can drastically improve the computation speed, and this in turn allows more “iterations” to improve solution quality. Since the multiplier neurons can be readily implemented [10], the major challenge is to implement NBDP. This is not totally out of reach since a simpler but similar algorithm, the Viterbi algorithm, has already been implemented in hardware and has found wide applications in telecommunications. In view of LRNN’s simple topology and elementary functional requirements, it has a strong potential to be implemented in hardware with much improved quality and speed over the standard LR or other methods.
Table 5.2 COMPARISON OF LRNN WITH LR LRNN
V. TESTING RESULTS Simulation was performed on a Sun Ultra 1 Model 170 workstation using data from Northeast Utilities billing data files. The contributions of hydro and pumped-storage units, together with power provided by contracts, were deducted from the system demand and reserve requirements. The number of thermal units was about 60, the commitment horizon was one week, i.e., 168 hours, and the reserve requirements considered were 10-minute spinning reserve. To simulate LRNN by software, the Lagrangian neuron dynamics described by (3.6) and (3.7) are converted to difference equations similar to (3.4) and (3.5). The resulting solutions are then fed into a heuristics as described in [2] to obtain feasible schedules. Results are summarized below.
LR
Cost
A
7,955,163
0.74
187
7,910,256
0.53
112
B
4,792,862
0.98
206
4,750,111
0.79
116
0.90
C
6,928,102
0.67
221
6,996,911
0.77
123
-0.98
($)
*
Gap
CPU
(%) (sec.)
Cost
Cost
Cases
($)
Gap CPU
Difference (%) (sec.) (%)
*
0.57
It is calculated by (LRNN-LR)/LR*100%.
Case 2: Beyond the data set of June Week 1, 1995 (A), two more NU data sets were tested: July Week 3, 1995 (B), and April Week 2, 1997 (C). The LRNN results are summarized in Table 5.2. For comparison purpose, results obtained by using our LR method [2] for the same data sets are also presented in the table. It can be seen that LRNN results are very close to what were obtained by our LR, and the differences are small. The small gap of LRNN indicated
Case 1: The LRNN result for NU’s data set of June Week 1, 1995 is summarized in Table 5.1. For comparison purpose, 5
consistent convergence, and implies that near optimal solutions are obtained. [9] VI. CONCLUSION In this paper, LRNN is presented to solve thermal unit commitment problems by combining Lagrangian relaxation with Hopfield-type neural networks. The difficulties of integer variables, and subproblem local minima and solution infeasibility are avoided. Numerical testing demonstrated that the method is able to obtain near optimal solution for practical problems, and is much better than what has been reported in the neural network literature. In addition, the method has the potential to be implemented in hardware with much improved quality and speed, where applications are not limited to unit commitment problems only. Rather, it can be applied to separable integer or mixed-integer optimization problems with stage-wise additive subproblem cost functions, a generic class of problems with many important applications.
[10]
[11]
[12]
ACKNOWLEDGMENT
IX. BIOGRAPHIES
This work was supported in part by the National Science Foundation grants ECS 97 26577. The authors would like to thank Prof. L. S. Thakur and Dr. Daoyuan Zhang of UConn and Mr. Krishnan Kasiviswanathan of NU for valuable suggestions.
Peter B. Luh (S'77-M'80-SM'91-F’95) received his B. S. degree in Electrical Engineering from National Taiwan University, Taipei, Taiwan, Republic of China, in 1973, the M. S. degree in Aeronautics and Astronautics from M.I.T., Cambridge, Massachusetts, in 1977, and the Ph.D. degree in Applied Mathematics from Harvard University, Cambridge, Massachusetts, in 1980. Since 1980, he has been with the University of Connecticut, and currently is the Director for the Taylor L. Booth Center for Computer Applications and Research, and a Professor in the Department of Electrical and Systems Engineering. His major interests include schedule generation and reconfiguration for manufacturing and power systems. Dr. Luh is a Fellow of IEEE, and an Editor of the IEEE Transaction on Robotics and Automation.
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Yajun Wang was born in Shenyang, P. R. China on Dec. 26, 1971. He received his B.S. degree in Electrical Engineering from Tsinghua University, Beijing, P. R. China in 1995. Currently he is a Ph. D. candidate in the Department of Electrical and System Engineering, University of Connecticut, Storrs, CT. Xing Zhao received the B.S. degree in Electrical Engineering from Harbin Institute of Technology, Harbin, P. R. China in 1993, and the M.S. degree in Electrical Engineering from Tsinghua University, Beijing, P. R. China in 1996. Currently he is a Ph.D. candidate in the Department of Electrical and System Engineering, University of Connecticut, Storrs, CT.
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