Abstract-- We study the problem of dynamically scheduling a set of period stage control tasks controlling a set of large air conditioner loads (ACLs). To be able to solve the scheduling problem for realistic on-line cases, we utilize the technique of relaxed dynamic programming (RDP) algorithm to generate an optimal or near optimal daily control scheduling for ACLs with relaxing bounds. Field tests of controlling the ACLs located in the campus are tested on-site to demonstrate the effectiveness of the proposed load control strategy.
Index Terms—relaxed dynamic programming, optimization, load control scheduling.
I. INTRODUCTION
NERGY saving efficiency is an ongoing problem, especially for industries with high power consumption, such as the iron and steel, the petrochemistry, the cement, and the paper-making industries, which need to continually consider electricity saving schemes to increase their competitiveness [1-3]. The conventional control mode for ACL supports three types of control, demand control, cycling control and timer control, to assist customers in saving electricity costs. The proposed optimum loads control scheduling (LCS) scheme supports any combinations of these three control types to optimally save costs during the dispatch period [4, 5] by using RDP algorithm to adopt optimal loads scheduling. To be able to solve the scheduling problem for realistic on-line cases, we utilize the technique of RDP algorithm to generate an optimal or near optimal daily control scheduling for ACLs with relaxed bounds under constraints. To carry out the proposed strategy, the techniques of microprocessor hardware are applied with Visual C++ language.
II. PROBLEM FORMULATION
There are two objectives to be implemented for the LCS strategy [6]. The first is to maximize the customer electricity
T.-F. Lee and F.-M. Fang are with the Department of Radiation Oncology, Chang Gung Memorial Hospital Kaohsiung Medical Center, Kaohsiung, Taiwan. (E-mail: [email protected], [email protected]).
Y.-C. Hsiao is with the Department of Electrical Engineering, Fortune Institute of Technology, Kaohsiung County, Taiwan, ROC. (E-mail: [email protected]).
P.-J. Chao is with the Department of Radiation Oncology, Kaohsiung Yuan’s General Hospital, Taiwan. (E-mail: [email protected]).
M.-Y. Cho and H.-Y. Wu are with the Department of Electrical Engineering National Kaohsiung University of Applied Sciences. (*Corresponding author, phone: 886-7-3814526*5530; Fax: 886-7-392-3532, E-mail:[email protected], [email protected]).
saving benefits and the second is to minimize the perturbation of load interruption to the customer simultaneously. The proposed model can be built to achieve customer satisfaction and to achieve the electricity saving requirement, the mathematical model of the problem can be formulated as follows [7, 8].
A. Objective function
For minimizing the uncomfortable situation and disturbance to the customers, the objective function B considered in this problem comprises two terms. The first term is to minimize inconvenient and disturbance to customers during the daily dispatch period, and the second term considers the willingness of customers to be charged an incentive rate for accepting interrupted control schemes, which can be expressed as follows:
( )
(
( )
)
, 0 1. 4 1 S 1 M 1 N 1 ≤ ≤ × × − × =∑∑
= = δ δ g i g g i i CL Min B (1) Where M: number of ACLsN: number of control periods (daily)
CLg(i): capacity of the gth ACL group in period i (kW)
Sg(i): State of the gth ACL growth in period i
=1 if the gth ACL group is connected to the system in period i; =0 if the gth ACL group is disconnected from the system in period i δ: saving weighting percentage factor of the gth ACL group; its value is
between 0 and 1
B: objective weighting capacity; it is the shared by demand control and cycling control strategies
We adopt the knapsack technique based on a RDP algorithm to find a global optimal or near optimal solution but less complexity in dimensionality. In the first, assume Sm is the
solution set of objective function from 1 to the mth number of the ACL, and then to define benefit function bi and weighting
function wi at each period. The objective function solution is to
obtain the maximum corresponding B under constraints which will discuss next had to be satisfied. Thus, the objective function and constraint function can be expressed as follows:
, maximize : Objective b B T i i =
∑
∈∑
∈ ≤ T i i W w . : Constraint (2) (3) Namely, the optimal result is decided by the benefitRelaxed Dynamic Programming for Constrained
Economic Direct Loads Control Scheduling
Tsair-Fwu Lee, Member, IEEE, Horng-Yuan Wu, Ying-Chang Hsiao, Pei-Ju Chao, Fu-Min Fang, and Ming-Yuan Cho*, Member, IEEE
function B, for which the mathematical equation takes the following form:
(
)
(
[
(
)
) (
)
]
+ − − − > − = . otherwise , , 1 , , 1 max , if , , 1 m m m b w w m B w m B w w w m B m,w B (4) WhereSm: set of items numbered 1 to m
B(m,w): best solution in Sm with weight exactly equal to w
B. Load Constraints
Apart from fitting the above-mentioned formulation rules, the proposed algorithm must also satisfy the loads constraints. Due to the characteristics of ACLs, the load demand of ACL will be increased due to the energy payback phenomenon. In this paper, the energy payback is expressed as follows [9]:
( )
i =0.6×P( )
i−1 +0.3×P (i−2)+0.1×P(
i−3)
.PPB LC LC LC (5)
Once the load recovery demand increase caused by the energy payback is calculated, the load demand in period i after control must be modified as follows:
( )
( )
( )
. ' ) (i P i P i P i P = − LC + PB (6) WhereP(i): modified load demand in period i P’(i): forecasted load demand in period i
) (i
PLC : amount of load reduced by control in period i The load demand control can be stated as follows:
( )
∑
(
( )
( )
( )
)
= + − = N i PB LC i P i P i P Min i P 1 , ' Minimize (7)subject to 0≤PLC
( )
i ≤Pmax, for 1≤i≤96.III. RELAXED DYNAMIC PROGRAMMING
The idea of RDP was first proposed by B. Lincoln and A. Rantzer of LTH, Lund University, Sweden in 2003 [10-13]. The principle is shortly described in the following.
The optimal value function is characterized by the “Bellman equation” as follows [14]:
( )
min{
*(
( )
,) ( )
,}
,( )
0 0. * x = B f x u +l xu B = B u (8)A common method to find the optimal value function is value iteration, i.e., to start at some initialB0(x), for exampleB0 ≡0, and update iteratively
( )
min{
(
( )
,) ( )
,}
.1 x B f x u l x u
B k
u
k+ = + (9)
Where l(x,u) is a given cost function. In RDP method, we have to find a B(x) with relaxed bounds which fulfills B(0)=0 and
( )
(
) ( )
{
}
( )
( )
(
) ( )
{
, ,}
. min , , min u x l u x f B x B u x l u x f B k u k k u + ≤ ≤ + (10)In particular, there is a lower bound in (10) implies that B is a Lyapunov function for the closed-loop system [9]. Usually l and l are chosen to satisfyl
(
x,m) (
≤l x,m) (
≤l x,m)
, for example(
x,m)
=λl(
x,m)
, λ ≥1, l(
x,m)
=λl(
x,m)
, λ≤1. l (11) (12)With this relaxation of Bellman’s equation, we can search for a solution B(x) which is more easily parameterized than optimal B*(x). From this, a simplified B (x) which satisfies k
( )
x B( )
x B (x)Bk ≤ k ≤ k (13)
is calculated. This satisfies
( ) {min} 0 0
( )
,( )
{( )min} 0 0( )
, ,∑
∑
= = = = ≤ ≤ k m m u k k m m u u x l x B u x l k m k m (14)The λ’s (and the l’s) are chosen as a trade off between complexity (time and memory) and accuracy. If λ and λ are close to 1, then the iterative condition (13) becomes close to ordinary value iteration (9), which gives high accuracy and high complexity. On the other hand, if the fraction λ/λ is very big, then the accuracy drops, but (13) can be satisfied with less complex computations. Note that if l is chosen as in (11) and (12), then the relative error in the value function defined by λ and λ is independent of the number of iterations [10-13].
A. RDP solution methodology
The LCS period is first divided into a number of intervals, each of which is defined as a stage in the RDP. In each stage, a number of state-sets are given. All the states contained in a state-set are faced with the same load levels, but those in a different state-set have distinct load levels, ranging from the criteria λl (upper bound) to lλ (lower bound) around the predicted load level to suit for the constrained uncertainties and to reduce the dimension curse. Fig.1 displays the search structure of the RDP with lower dimensions than the DP structure. Each state of a state-set in the current stage
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