Abstract: A linear programming model is built to examine generic direct
load control scheduling. Based upon the
average cost function, the approach aims to reduce the overall production
cost. Instead of determining the amount
of energy to be deferred or paybacked, the algorithm controls the number of
groups of customer/load type to minimize the system production cost. In addition to the advantage of better
physical feel on how the control devices should operate, the linear programming algorithm provides a
relatively inexpensive and powerful approach to tackle the scheduling problem.
Keywords: load management, direct load control, cost-based
operation, linear programming, integer solutions.
I INTRODUCTION
Load
management, introduced in the 70’s, is aimed to reduce the operating cost while
maintaining the reliability of the electric power network. Generally, load management can be
categorized into the following sections: direct load control (DLC) which allows
the utilities to shed remote customer loads unilaterally, indirect load control
which allows customers to control their loads independently according to the
price signals sent by the utilities, and storage capacity which allows both
utilities and customers to store energy during the off-peak/low cost session
and consume during the peak/high cost session.
This paper examines only the cost-based DLC algorithm using linear
programming.
Selectively grouping the customers’
load, the utilities are then capable of offering incentive to respective
customers for direct control over selected loads. Various algorithms, dynamic programming primarily [1-5], have
been developed to reduce the system peak, operating cost, or spinning
reserve. However, while one approach
[4] failed to recognize the fact that the maximum controllable load varies from
one to another period, another [2] ignores the load-dependent payback
pattern. Linear programming model, a
relatively inexpensive and powerful method is approached by C. N. Kurucs and et
al.[6] to reduce the system peak.
However, the algorithm failed to consider all possible control
duration. Thus, if the load/cost
zig-zagged over time, the resulting solution of the approach may not be
optimal. Furthermore, while Kurucs
tries to reduce system peak, K. D. Le and et al.[7] pointed out that resulting
system production cost of flattening the load may not be the lowest. Thus, the approach does not include all
appropriate costs.
This paper introduces the cost-based DLC scheduling using
linear programming algorithm. A
relatively inexpensive and powerful approach, the developed algorithm includes
various benefits of others [1-6].
First, rather than determining amount of energy to be controlled during
a certain period, which is a rather indirect approach, the new algorithm tries
to decide the number of groups to be controlled during a certain period. Second, instead of “two-phase” strategy [6],
the developed algorithm is capable of including all possible control
durations. Third, including the non-fixed
maximum controllable load and payback pattern over time, the algorithm resolves
the two problems through determining groups to be controlled. Introducing
paybacked energy which depends on the load during the controlled period
resolves the varied payback pattern.
Finally, by changing the constant parameters of the objective function
and constraints from period to period, the LP algorithm considers nonlinear
cost function and yet obtains integer solutions that are optimal.
II NOMENCLATURE
: system production cost
at period j
: system cost at
period j
: system load level at period j
: uncontrollable load
lelve at period j
: system controllable
load level at period j
: load available for DLC
at period j by customer/load type i
: rate charged on
controllable load of customer/load type i ($/MW)
: average cost function
at period j in $/MW, reflecting fixed, operating amd maintenance cost.
: number of groups of
customer/load type i that are controllable at period j
: number of groups of
customer/load type i that are undergoing load shedding at period j of length k
n: maximum number of customer/load
type
m: maximum period under study
: maximum length of
period for load shedding for customer/load type i
: payback ratio at
period (j+k+s-1) for load shedding at period j of length k on customer/load
type i
q(i,j,k):the maximum payback
period for load shedding at period j of length k on customer/load type i
u(*): unit step function where
u(*) = 1 if * 
0
and u(*)
= 0 if * < 0
c: per megawatt cost of
conducting load shedding
: maximum divisible
groups of customer/load type i
: maximum load that can be
increased at any period for the price/cost to remain in the feasible range
P: minimum load that can be decreased at any period for
the price/cost to remain in the feasible range
III. MODEL
Figure 1. Forecast of
Controllable and Non-controllable Load
Dividing
the load forecast into two section, as shown in Figure 1, the amount of power
available for control from one to another period is obtained. Distinguishing the controllable load of one
customer/load type to another is then categorized in more detail. A lower rate structure is offered to the
customers for controllability over their load, denoted 
. The system
production cost at any period is calculated as follows:
(1)
since the non-controllable
loads are fixed during any period, they may be deleted from the function. Let
(2)
be the average cost
function. Then, the simplified function
is:
where
(3)
Divide
the control choice of each customer/load type by 
, at any period:
(4)
Then, redefine:
(5)
the average cost function
is transforrmed into a function of the number of controllable groups, 
. Then, based on the
new system cost function, the minimization problem has become a problem of
finding the number of 
which will result in
the optimal minimization of system cost.
Even though the average cost function is nonlinear, it
can be perceived as constant slope for a rnage of change in load levels,
denoted by the limits of the range 
. Thus, the minimization
of nonlinear system cost function is reduced into a linearized problem.
IV FORMULATION
The reduced cost of deferring the energy is:
(6)
The increased in cost for paying back the energy is:
(7)
If the summation of payback ratio of any customer/load
type at any period j, 
, dos not equal 1, there is a loss or increase in revenue.
Thus, the possible revenue loss must be
included as a penalty.
(8)
Then, the objective of controlling the load is to
minimize the overall cost, which is
minimize - DF + PB + PT (9)
subject to:
(10)
(11)
where at any period, for any
customer/load type i, the controlled (deferred or paybacked) groups can never
exceed the preset maximum value, 
.
Since the nonlinear objective is approximated by
piece-wise linear segments, extra constraints, equation (12-13), should be
included to ensure that the change in load at any period will not exceed the
maximum allowable increase or decrease.
(12)
and
(13)
A cautious step should be noted, where, for any period,
the largest deferred and paybacked load, denoted by 
, and 
respectively, should
at least smaller than or equal to 
or 
.
Graphical Representation:
To visualize equations (6 - 13), a graphical
representation is shown in Figure 2.
There are n pieces laying on top of each other to represent the
individual customer/load type. For each
individual category,the studied m periods are listed from left to right. From top to bottom of each layer are the
load shifting choices and how they interact throughout the time frame. The boxes (undashed) on each layer represent
the periods when the load is deferred while the dashed boxes represent the
period of paybacked load. For
simplicity, the load shifting choices of each customer/load type is set to be
three, i.e., 
, and 
s of all customer/load types are equal.
Figure 2.
Graphical Representation of Control Pattern
In accordance with equation (6), the DF coefficient of
each control choice, 
, is represented by the un-dashed boxes. For instance, when k = 3, the DP coefficient
is found as follow:
From equation (7), the PB coefficient of each control
choice, 
, is represented by the dashed boxes. The PB coefficient of k = 3, for instance,
is determined as:
where 
is determined by
statistical measurement of past data and present control load pattern.
The PT function, equation (8), takes action only when the
payback ratio of 
does not sum up to
1. For instance, The PT coefficient of
k = 3 is calculated as follow:
If the payback back
ratio sum up to 1, then the expression, 
, will become zero, and the penalty will not take any
action. If the sum of payback ratio is
greater than 1, the PT coefficient becomes negative and is considered as bonus
rather than the penalty.
By equation (10), each control choice, 
, has to be greater or equal to zero, i.e., 
, 
, 
and etc.
From equation (11), the maximum control choices of any
period, 
, of individual customer/load type should be smaller than or
equal to the maximum allowable divisible groups. At period 3, for instance, the limitation on control choices are
found to be:
In short, at any period, for any customer/load type, if
the box (dashed or un-dashed) corresponding to 
is extended to the
observed period, 
is then considered as
one of the elements that will confine the maximum controllable groups during
the period.
According to equations (12-13), at any period, the
difference between the energy deferred and paybacked should be less than the
predetermined values 
and P. At period 3, for instance, equation (12-13)
are expressed as
and
Iterative Procedure:
To obtain optimal, integer solutions, the objective
function and constraints have to be updated to reflect the change in the
average cost and in available controlling load group. To achieve the mentioned task, an iterative process is employed.
To facilitate the explanation of the iterative process,
the following representations are defined:
From equation (10):
, 
(14)
, 
(15)
From equation (11):
(15)
(16)
With these defined representation,
an iterative process searches for the optimal solution as shown in Figure 3.
Evaluation:
Upon completion of iterative
procedure where 
and 
are found, the final
step is needed to exclude the operating and maintenance cost, c, when
evaluating the total savings of achieved from shifting the load.
VI RESULTS
To show how the algorithm achieves the cost saving, a
program written in MATLAB code is used to examine a system consisting of two
customer/load type for a 12 hours duration.
Five percent of each customer/load type is assumed directly
controllable. At individual hour, 3 control
choices are available, i.e., 
. Also, for simplicity,
the payback pattern where assumed equal for both customer/load type at any
hour. For k = 1, a 100% payback occurs
at the next hour. For k = 2, a 100%
payback occurs at the third hour. For k
= 3, a 60% payback occurs at the fourth hour and a 40% payback occurs at the
fifth hour. For simplicity, the effect
of penalty function is not examined.
Figure 4 shows the load pattern for both customer/load types.
Figure 3 LP
Iterative Procedure
Before any load shifting, the system cost is $
201,760. Upon completion of the load
shifting, the system cost is $199,860, while the operating and maintenace cost
of DLC is $28.82. This is a total
savings of $1871.18.
Figure 5 shows the change in demand before and after
optimal load shifting. Figure 6 shows
how the average cost changes after load is shifted to reduce the system cost.
VI CONCLUSION
A cost-based load management solution using linear
programming is presented. This new
approach includes the benefits of other approaches [1-6]. The algorithm relates the control devices to
the DLC. Furthermore, the integer
solutions achieved by the approach are more practical representation of the
control system.
Figure 4. Demand
Before Load Shifting
Figure 5. Demand
Before and After Load Shifting
Figure 6.
Average Cost Curve Before and After Load Shifting
VII REFERENCES
1. A. I. Cohen, C. C. Wang, “An
optimization Method for Load Management Scheduling”, IEEE Trans. Power Systems PWRS, Vol. 3, No. 2, 1988.
2. Y. Hsu, C. Su, “Dispatch of
Direct Load Control Using Dynamic Programming”, IEEE Trans. PWRS, Vol. 6, No. 3, 1991.
3. A. I. Cohen, J. W. Patmore, D.
H. Oglevee, R. W. Berman, L. H. Ayers and J. F. Howard, “An Integrated System
for Load Control”, IEEE Trans. Power
Systems, Vol. PWRS-2, No. 3, August 1987.
4. J. Chen, F. N. Lee, A. M.
Breipohl, R. Adapa, “Scheduling Direct Load Control to Minimize System
Operational Cost”, IEEE Trans. Power Systems,
Vol. 10 November 1995.
5. F. N. Lee, A. M. Breipohl,
“Operational Cost Savings of Direct Load Control”, IEEE Trans. Power Apparatus and Systems, Vol. PAS-103, No. 5, May
1984.
6. C. N. Kurucz, D. Brandt, S.
Sim, “A Linear Programming Model for Reducing System Peak Through Customer Load
Control Programs”, presented at the IEEE PES winter meeting, 96 WM 239-9 PWRS,
Baltimore, Maryland, 1996.
7. K. D. Le, R. F. Boyle, M. D.
Hunter, K. D. Jones, “A Procedure for Coordinating Direct-Load-Control
Strategies to Minimize System Production Cost”, IEEE Trans. Power Apparatus and Systems, Vol. PAS-102, No. 6, June
1983.
Kah-Hoe Ng
received his BSEE from Iowa State University.
He is currently working towards a MS degree.
Gerald B. Sheblé
(M 71, SM 85) is a Professor of Electrical Engineering, Iowa State University,
Ames, Iowa. Dr. Sheblé received his
B.S. and M.S. degrees in Electrical Engineering from Purdue University and his
Ph.D. in Electrical Engineering from Virginia Tech. His industrial experience includes over fifteen years with a
public utility, a research and development firm, a computer vendor, and a
consulting firm. His research interests
include power system optimization, scheduling and control.