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IEEE Transactions on Power Systems, Vol. 8, No. 3, August
1993
1341
EFFECTS OF RAMP-R ATE LIMITS ON UN IT COMMITMENT
AN D ECONOMIC DISPATCH
C. Wang, Member
S. M. Shahidehpour, Senior Member
Department of Electrical and Computer Engineering
Illinois Institute of Technology
Chicago, Illinois 60616
ABSTRACT
This paper proposes an algorithm to consider the ramp
characteristics in starting up and shutting down the
generat-
ing units as well as increasing and decreasing power
generation.
In power system generation scheduling, a number of studies
have
focused upon the economical aspects of the problem under the
assumptions that the changes in the generating capacity
follow
a step function characteristic and unit generation can be
ad-
justed instantaneously. Even though these hypotheses greatly
simplify the problem, they do not reflect the actual
operating
processes of generating units. The use of ramp-ra te
constraints
to simulate the unit state and generation changes is an
effec-
tive and acceptable approach in the view of theoretical
devel-
opments of industrial processes. Since implementing ramprate
constraints is a dynamic process, dynamic programming (DP)
is a proper tool to treat this problem. In order to overcome
the
computat ional expense which is the main drawback of DP,
this
study initially employs artificial intelligence techniques to
pro-
duce a unit commitment schedule which satisfies all system
and
unit operation constraints except unit ramprate limits.
Then,
a dynamic procedure is used to consider the ramp properties
as units are started up and shut down. According to this ad-
justment, maximum generating capabilities of units will
change
with the unit operation status instead of following a step
func-
tion. Finally, a dynamic dispatch procedure is adopted to obta
in
a suitable power allocation which incorporates the unit
generat-
ing capability information given by unit commitment and unit
ramping constraints, as well as the economical
considerations.
Two examples are presented to demonstrate the efficiency
of
the
method.
Keywords- Ramp Rates, Unit Commitment, Dynamic Dis-
patch, Artificial Neural Networks, Heuristics.
1. INTRODUCTION
There are two tasks considered in power system generation
scheduling. One is the unit commitment which determines the
unit start up and shut down schedules in order to minimize
the
system fuel expenditure. The other is the economic dispatch
which assigns the system load demand to the committed gen-
erating units for minimizing the power generation cost. The
economic operation at tracts a great deal of attention as a
mod-
est reduction in percentage fuel cost leads to a large saving in
the
system operation cost. Many studies for power system genera-
tion scheduling have successfully applied various
mathematical
algorithms such as Lagrangian relaxation [1,2], dynamic pro-
gramming [3,4],and artificial intelligence techniques e.g.,
expert
systems
[5 , 6 ] ,
rtificial neural networks (ANN) [7,8], etc. The AI
techniques have incorporated the practical operational
policies
in the mathematical techniques to improve the system models
considerably. The mechanism of
A N N
simulates the learning
92 SM 413-5 PWRS
A
paper recommended and approved
by the IEE E Power System Engineering Committee of
the
IEEE
Power Engineering Society for presentation
at th e IEEE/PES 1992 Summer Meeting, Seattle,
WA,
July 12-16, 1992. Manuscript submitted Januar y
28,
1992; made available
for
printing May 13, 1992.
process of the human brain. One class of ANN learns the
knowl-
edge through examples,
or
training facts, composed of various
inputs and their corresponding outputs. The extent of the
in-
telligibility of ANN depends upon the diversity of the
training
facts. For an input which is not in the training facts, the
trained
A N N can estimate an output based on its previous knowledge
about the problem.
A number of studies
[l-81
dealing with the unit commitment
problem have held the assumption that the unit generating
ca-
pability follows a step change from zero to the rated
capacity
and vice versa. In fact, when a unit is in the star t up
process,
a
pre-warming process must be introduced in order to prevent
a brittl e failure. Therefore, because of the unit physical
limita-
tions, the unit generating capability increases as a ramp
func-
tion. In contrast , using a step function for representing
changes
in the generating capability, all processes are initiated as
the
unit achieves its rated capacity which represents an
unrealistic
treatment of the energy, especially when the unit start up is
a
long process. Similarly, when a unit is in the shut down
pro-
cess, it will take a while for the turb ine to cool down.
Before
the unit generating capability decreases to its lower limit,
the
residual energy is to be used to meet the load demand, which
is contrary to the case where the changes in unit generating
ca-
pacity are modelled as
a
step function. In the past, ramping
process was considered in the economic dispatch denoted as a
dynamic dispatch
[9-111.
In order to satisfy the ramping con-
straints, a dynamic process was performed in conjunction
with
the economic dispatch. Reference [9] has proposed
a
practical
and efficient method to calculate a suboptimal generation
sched-
ule for a system with ramping constraints.
This paper considers the inclusion
of
ramping constraints
in both unit commitment and economic dispatch. Since DP is
a time consuming algorithm, this study avoids the use of DP
to compute the generation schedule. Three steps are used to
complete the task of generation scheduling. First, the
ramping
constraints in the unit commitment are relaxed, so that the
unit
commitment problem would merely employ step functions for
representing the generating capability. In order to expedite
the
execution, an ANN is used to generate a possible unit
commit-
ment schedule and a heuristic procedure is employed to
modify
the unit commitment to achieve a feasible and near optimal
so-
lution. Then, a dynamic adjusting process is adopted for the
resulting unit commitment schedule in order to incorporate
the
ramping constraints. Finally, a dynamic dispatch is
performed
to obtain a suitable unit generation schedule.
2. MATHEMATICAL MODEL
The objective of the generation scheduling problem is to
minimize the system operation cost. This cost includes the
fuel
cost for generating power and the start up cost over the en-
tire study time span, while satisfying the system operating
con-
strain ts, e.g., power balance, spinning reserve requirements,
unit
generation limits, min up/down times, ramp-rate limits, etc.
The list of symbols used in this paper is as follows:
F:
otal operation cost of the entire system
Pi@):
power generat ion of unit at hour t
N,:
total study time span in hours
N :
total number of units
I; t):commitment sta te of unit at hour t (1
or
0)
F, P; t ) ) : uel cost of unit when generat ing power is
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SI(X*?fft - 1)):
D
) :
Pi :
i:
R t ) :
T0 l ff:
I
T :
C : t ) :
U R i :
D R , :
UH,:
DH,:
The objective of
N , N
equal to P, t)
time duration for which unit has been on/off
at hour t
start up cost of unit i after Xloff(t- 1) hours
Off
system load demand at hour t
rated upper generation limit of unit
i
rated lower generation limit
of
unit i
system spinning reserve requirement at hour t
minimum up/down time of unit
i
time constant in the start up cost function for
unit
i
constrained generating capability of unit at
hour t
ramp-up rate limit of unit (MW/ h)
ramp-down rate limit of unit (MW/ h)
ramp-u p time of unit (h)
ramp-down time of unit (h)
the problem is to minimize,
F
=
C [ I i t ) F i P i t ) ) Ii ( t )(l- l t - ) ) s l (xpf f (
t
-
))]
t=l i = l
1)
The constraints for the problem are:
1) System power balance
C p i ( t ) ~ t > t = 1,. . . ,N ,
(2)
i=l
2) System spinning reserve requirements
N
PiIi t) D t )+
R(t)
t
= 1, .
.
,Nt
(3)
i 1
3) Unit generation limits
pi
Pi(t) 5 C i t )
i
=
1, .
,
N
(4)
t
=1,
. . ,
Nt
4) Unit minimum up/ down times
[XPn(t- 1)- Ttn] [I - 1)- i(t)]2 0
[XIoff t-
1)
-
Tff f ]
* [ I i t )- , t -
)]
2
0
5 )
(6)
5 )
Ramp rate limits
for
unit constrained generating capability
as unit i starts up (7)
as unit i shuts down
(8)
C,(t)
-
Ci(t
-
) 5 U R ;
Ci(t-
)
- Ci(t)5 D R ,
6)
Ram pra te limits for unit generation changes
Pi t)
-
Pi t
-
1) 5 U R , as generation increases (9)
Pi t - 1) - Pi t)5 DRi as generation decreases (10)
Since eqns.
(7-8) and
(9-10)
are dynamic constraints
for unit commitment and economic dispatch, respectively, it
is
necessary to formulate algorithms which consider these
charac-
teristics properly.
3. S O L U T I O N
METHOD
The generation scheduling problem considered in this paper
is solved by two separate procedures which deal with unit
com-
mitment and economic dispatch, respectively. For the unit
com-
mitment, we train an ANN according to t he available
knowledge
for the o ptimal unit commitment schedules which correspond
to typical system load curves. Then, for
a
given load profile,
the ANN will generate a unit commitment schedule which may
represent an infeasible but close to the optimal solution.
In
or-
der to generate a feasible solution, a heuristic method is
used
to adjust the ANN unit commitment schedule according to the
system operat ing experience[l2]. After obtaining an
economical
unit commitment schedule, which satisfies the system operat-
ing Constraints except the ramping limits,
a
dynamic adjusting
procedure is adopted to incorporate the ramp-rate
constraints.
There are four possible unit st ate trajectories between two
adja-
cent hours which are staying decommitted
(0 -
0 ,
starting up
(0
-
l ) , hutting down 1 0) and staying committed
1
1).
Because of ramping, th e last three cases may require
additional
adjustments. In unit commitment, ramping up a unit at ith
hour may affect the unit combinations at hours
i
+
1, . . . Nt.
Since we do not include the effect of ramping up a unit at a
cer-
tain hour on the unit combinations of the later hours, the
unit
commitment schedule will be a suboptimal solution.
The economic dispatch is based on th e outcome of t he unit
commitment. The fundamental idea for considering ramping
limits in adjusting the unit power generation is similar to th
at of
the unit commitment. For the last hour of the study time
span,
the
X
method is used to calculate the power generation of every
committed unit. Based upon the power generation schedule at
i t h hour, the X method is applied to dispatch the load
demand
at i - ) t h hour. If some of the unit generation changes at (i
-
1)th
hour exceed their ramp-rates as compared with the results
of the
i t h
hour, we fix those changes at their reachable limits
and dispatch the remaining required power generation
optimally
among the available units. Again, since we
do
not coordinate the
power generation over the entire study time span, the
generation
schedule is suboptimal. However, the adopted approach is
much
faster than those with optimal solutions [12-141.
The reason for adjusting the unit commitment and genera-
tion schedule from t.he last hour to the first hour of the study
is
to generate a feasible solution. If we modify the solution
from
the first hour to the last hour in the course of
incorporating
the ramping limits, the best unit combination or load
dispatch
may result in an infeasible solution for the following hours,
as
we cannot look into the load demand for the future hours
[ll] .
3.1
Unit Commitment
by
Relaxing
Ramping Constra ints
If the ramping constraints of the unit commitment prob-
lem, that is, eqns.
(7)
and (8 described in Section
2
are re-
In this regard, when a unit starts up, its generating
capability
is assumed to increase immediately from zero to
PI.
Likewise,
when a unit shuts down, its generating capability jumps from
PI to zero spontaneously. Therefore,
C, t)
in eqn. 4) s always
equal to PIwhen unit i is on. There are several available meth
-
ods which have implemented this type of constraints,
including
the Lagrangian relaxation, dynamic programming, AI methods,
etc. From our experience, those approaches which exploit ar-
tificial intelligence techniques usually require shorter
execution
time and can provide satisfactory results as long as the trend
in
the behavior of the system is well documented. In this
paper,
we adopt the
ANN
approach enhanced by heuristics t o generate
the unit commitment schedule for a given system.
laxed, then eqns.
(1-6)
form a sic unit commitment problem.
In implementing the
A N N
technique, the daily load curves
of a system generally are classified into several categories,
e.g.,
load curves for weekdays, weekends, holidays, Christmas,
etc.
Within each category, available load curves are slightly
differ-
ent and generally correspond to similar unit commitment
sched-
ules[7]. The t raining facts are a set of load curve - unit
commit-
ment pairs which include several cases in every category.
The
unit commitment schedules used as training facts are
obtained
by rigorous methods, e.g., Lagrangian relaxation. The input t
o
the trained ANN is a daily load profile and the outpu t of
the
ANN is the corresponding unit commitment schedule.
So, if
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the given load curve is one of th e training facts, th e ANN
can
quickly generate an optimal unit commitment schedule. If the
given load curve belongs to one of the categories but is
slightly
different from the tr aining facts, the ANN generates a fair
esti-
mate of the unit commitment schedule. If the given load
curve
differs significantly from the training load curves in the
avail-
able categories, the ANN output may be a sub-optimal unit
commitment schedule. As time goes on, we can improve the
ANN
outcome by carefully analyzing the system and classifying
the categories to cover a wide range of system loads.
Since there are limited trainin g facts representing the
char-
acteristics of load curves in each category, a given load curve
may
be close but different from the trainingfacts , so the unit
commit-
ment schedule identified by ANN may be an infeasible
solution.
In this regard, a heuristic approach is adopted for adjusting
the
ANN outcome to achieve a feasible and optimal solution. The
system deficiency and surplus capacities are defined as
follows:
H1:
H2:
H3:
H4:
H5:
H6:
H7:
H8:
N
N
PiI i t ) < C P i t ) I i t )
R(t)
11)
(12)
i=l
i = l
N N
C F i I i t )
-
C P i t ) I i t ) R(t)
2 mini[Fi]
i= l i= l
The heuristics used in this s tudy a re listed below:
If deficiency exists at hour
t ,
list all the shut down units
in an ascending average incremental cost order denoted by
Oplist.
Omit the units from the Oplist which cannot be committed
at hour t because of their minimum down time require-
ments.
Commit units on the Oplist sequentially until deficiency
either becomes zero or reaches its minimum negative value.
If surplus exists at hour t , list
all
the committed units in
a descending average incremental cost order denoted by
Lplist.
Omit those unit s from the Lplist which cannot be shut down
at hour t because of their minimum up time constraints.
Shut down units given in the Lplist sequentially until
surplus either becomes zero or reaches its minimum non-
negative value.
If a unit is on for certain hours, then off and on again, we
may compare its operation cost with that of maintaining
this unit in operation continuously, in order to save the
start up cost.
In a period T, compare the total operation cost
of
commit-
ted small-capacity units with that of a large-capacity unit
which is not in operation. If we can preserve the spinning
reserve requirements, the replacement may be cost efficient.
An oDtimal
or
near oDtimal unit commitment schedule is
now obtafned which satisgees all system constraints except
the
ramp-rate limits. By implementing the following procedures,
the
unit commitment schedule will be adjusted t o meet the
ramping
requirements.
3.2 Dynamic Adjustment for Incorporat ing
Ramping Cons tra ints into Uni t Commitment
The unit
i
constrained generating capability,
Ci(t),
which
changes abruptly as the unit starts up or shuts down, was
used
in Section 3.1 and is shown in Fig.
1.
Because of the ramp-
rate limits, Ci t ) must be modified as shown in Fig.
2.
It is
necessary, at this time, to explain the adopted ramping
policy
as a
unit starts up/shuts down. In Fig.
2 ,
as an example, unit
i is shut down at hour
m
and started up at hour n. After hour
rn it is unnecessary for unit
i
to contribute to the capacity of
the system, so Cj t ) , t > m should be less than
pi
n order
it
rn
hour
Fig.
1
Unit maximum generating capability
modeled as a step function
C
i
+
hour
Fig. 2
Unit maximum generating capability
modeled as a ramp function
to minimize the operating cost
of
unit
i.
In this regard, the
required time for ramping down unit at hour m is equal to
t l . Since the study time interval is one hour in this paper,
the
constrained generating capability of unit
i
at hour
m
is equal to,
At hour
n,
unit has started up. In order to preserve the security
of th e system, a conservative policy is employed which ramps
up
this unit U H ; hours earlier and assumes C;(n)= pi.
It ha.s been shown in Fig.
2
that unit states a.t later hours
will affect the decision made
for
previous hours and sometimes,
in order
to
let unit generating capacity reach a certain point at
specific hours, it is necessary to s tart up units at earlier
hours[9].
In this regard, we adjus t the unit commitment schedule from
the
last hour to the first hour, while the unit combination at the
last
hour is the same as that obtained in Section 3.1.
We now proceed to determine the unit states I ; t ) and unit
constrained genera.ting capability
C ; t )
t hour
t , t < Nt
under
the assumption that
I i t + 1)
and Ci(t
+ 1)
are known. The re
are only four possible cases for a unit stat e changing from
hour
t to hour t + 1.
Case 1: I , t ) = Ii(t
+ 1 =
0
any changes to unit
i
at hour
t .
Case 2: I ; t )= 0,
are two situations which are to be considered.
In this ca.se,C,(t)
= C , t + l )
= 0, it is unnecessary to make
I ; t + 1) = 1 and U H ;
2
This means the unit is to start up at hour t + 1. There
a.
As
shown in Fig. 3, the period between last shut down to
the present s tart up,
K
hours, is longer than the sum of the
unit minimum down time and ramp-up time. That is,
Hence, we can apply the adopted rampin up policy directly
to this situation. For the hour
h , h
E
fn + 11, where n
is as shown in Fig.
3 ,
the cons trained generating capability
becomes,
C , h )= min{P,,
UR;
* ( h n ) }
h E [n, + 11
(15)
and if
C , h )
E , ,
the commitment sta te changes to on,
where I , h ) = 1.
b. As
shown in Fig. 4 , the period between last shut down to
the present start up, K hours, is shorter than the sum of
the unit minimum down time and ramp-up time. T hat is,
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C
excess energy Ei t)
Fig.
3
Ramping up unit i as the minimum shut
down constraint is satisfied
C h excess energy
Ei( t)
Fig.
4
Running unit i continuously in order to
preserve minimum shut
down
constraints
hou r
In this situation, it is impossible to let unit shut down
at hour m and start up again a t hour
t
+ 1, since it would
violate the minimum down time constraint. So, unit will
run continuously from hour m to hour t+ by changing
the constrained generating capability to Pi as illustrated
in
Fig.
4.
Therefore, Ii (h), E
[m,
]
changes to
1
and the
constrained generating capability is adjusted to,
C, h)= P , h E [m , l
(17)
In both situations, comparing with the step function oper-
ation, unit generates excessive energy during hour
t ,
shown as
shaded areas inFigs. 3and 4 , which is equal to,
where E,( t ) s the excessive energy generated by unit
during
ramping up at hour
t .
Case
3:
I l ( t )= 1, I z t
+
1) = 0 and D H ,
If unit is asked to shut down at hour t
+ 1,
based on the
adopted ramping down policy after hour t
+
1, the constrained
generating capability of unit should be less than
E, ,
that is,
Ci(h)= Pi
-
D R , * h
-
I
h
E
[n ,
]
(19)
C i t
+
1)
= Pi
DRi
t
+ 1- .)
5
Pi
where R. is as shown in Fig.
5 .
As represented by the shaded
area in Fig.
5 ,
unit will generate less energy than that of step
functions during hour t , which is equal to,
where L;
t ) is
the energy tha t unit does not supply during
ramping own at hour
t .
y
T*,
~ r
pi
. . . . . . . , . . . , . . . _ . . . . _ . . .
b-
DHi
Fig. 5
Ramping down unit i to be shut down
at hour t+l
Case 4:
I z ( t )
= I t t
+ 1
=
1
(1) If
C, t)
=
C, t
+ 1) = P , , which means tha t unit is in
the steady operating st atus (not in the process of ramping
up/down), then there is no change in the generating energy
of unit during hour
t ,
as shown in Fig. 6(a).
(2 ) If
C, t )
< C, t + l ) , hen unit is in the process of ra m p
ing up, as in Fig. 6(b). Therefore, the excessive energy
generated by unit can be calculated by using eqn. (18).
(3 )
If
C, t)
>
C,(t
+
l ) ,
hen unit is in the process of ramping
down,
as
in Fig. 6(c) . Therefore, unit generates less en-
ergy tha n tha t of the ste p function operation, and the
lower
energy can be calculated by eqn.
(19).
( 4 b) ( c )
Fig. 6
Three possible situations when unit i
is on during hour
t
At hour
t ,
after we analyze all the units according to the
above rules, we calculate the net gcnerating energy change
as,
N N
A E ( t )
=
E,( t ) L , t )
2 =
i= 1
and perform the following adjustments accordingly:
1. If
1
I z t ) C , t ) D t ) + R( t ) or A E ( t )< 0
2= 1
then there exists a capacity or energy deficiency a t hour t
.
List all the peaking units, which satisfy
C, t)
= 0 and the
minimum down time requirements, in an ascending ramp-
up time U H , ) order denoted by Uplist. Commit the first
unit in the Uplist according to the rules given in Case 2
and
goto eqn. (21).
2. If
N
I z t ) C z t ) D t )
R(t)> z G o n z t 8 { ~ l
t = 1
and
A E ( t ) >
min
{ P }
i c o n u n i i s
then there exist capacity and energy surplus at hour t .
ist
.
all the peaking units, which satisfy
C,(t)
= P and minimum
up t ime requirement, in a n ascending ramp-down time order
denoted by Dnlist.
Decommit the first unit in the Dnlist
according to the rule given in Case 3 and goto eqn.
(21).
3.
If neither of the above situations occurs after ramping mod-
ifications a t hour t , hen we have obtained the unit
commit-
ment schedule at hour
t .
The same procedure would apply
to hours
t
-
1
2 ..., until the first hour is reached.
It is necessary to point out that peaking units, rather than
the more economical units which have longer ramping up/down
times, are used in compensating for the deficiency
or
surplus
caused by the unit ramping characteristics.
A s
we would require
a short period of compensation, units with lower operating
costs
and longer ramping up/down times are not efficient and may
not
be regarded as economical for this purpose.
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3.3 Dynamic Dispatch
If we do not consider the unit ramping in the economic
dispatch of a system which consists of thermal units, the
eco-
nomic dispatch can be implemented by the
X
method at each
hour. In reality, a turbine with a high temperature and
pres-
sure state would require additional time to increase
or
decrease
its power generation. A dynamic dispatch considers
additional
constraints, eqns.
(9)
and
( l o ) ,
for economic dispatch, similar to
that of implementing ramping limits in unit commitment.
First,
the
X
method is used to dispatch the power generation among
the committed units at the last
hour.
It should be emphasized
that the upper generating limit of a unit at a certain hour
is
equal to the constrained generating capability of the unit at
this
time, C, t),which is obtained by ramping the unit, instead
of
the rated capacity p, . Generally, after calculating the
genera-
tion schedule at hour t + l , he economic dispatch at hour t
can
be considered
as
follows:
Step 1:Use the X method to dispatch the load demand among
the committed generating units by neglecting the ra m p
ing properties.
Step 2:
For
every committed unit, check the following condi-
tions:
a. If P, t ) > P,(t + 1) and P,(t)
- P, t
+ 1) > DR,, then
the required reduction in the power generation of unit
is beyond its ramp-rate limit. Fix the power generation
of unit z within its limit,
and let t his unit out of coordination. GO to Step
1.
b. If
P, t) YR,
then the
required additional power generat ion of unit 1s beyond
its ramp-rate limit. Fix the power generation of unit 2
within its limit,
and let this unit out of coordination. Go to Step 1.
c. If -uR, Pt( t ) -Pl ( t+l ) < DR,, thenunit
igeneration
is at its optimal operating status. Proceed to check the
next committed unit.
Step 3: Once all unit generations are checked and adjusted
to
meet the system constraints, the generation schedule at
hour
t
will be formed. Carry out the same procedure to
the previous hour t
-
1.
P, t ) = P, t + 1) +DR,
(2 2 )
P,(t)= Pz(t
+
1) - UR:
(2 3 )
Fig. 7 presents the outline of the proposed method for
the power system generation scheduling problem. It should be
emphasized that the heuristic techniques emulate the process
followed by th e mathematical techniques which are enhanced
by
the human operators intuition for a least cost operation of
a
large scale power system. The reason various rule-based and
heuristic methods are introduced by different investigators
for
studying unit commitment is that the rigorous mathematical
techniques require a significant amount of computation time.
We can implement the rigorous techniques off-line and use
its
output as training facts for ANN or improvising rule-based
ap-
proaches. In this respect, t he proposed heuristic techniques
can
provide a satisfactory and economically viable unit
commitment
schedule (7,121 for a certain load curve, and the
corresponding
final solution will be either optimal or quite close to
optimal.
Since the ramping limits are incorporated by dynamic adjust-
ment instead of global optimization, the final solution is
subop-
timal. The sta rt u p cost is considered when a unit is required
to
shut down and start up again. As shown in Fig. 4, if the
initial
unit commitment indicates that the unit will be shu t down
for
a
short period, the ramping characteristics mandate the
continu-
ous operation of the unit to save the st art
up
cost. On the other
hand, if a unit is to be shut down for a long period of time
as
shown in Fig. 3, even when the ramping limits are considered
it is more economical
to
shut the unit down for a while to save
the operation cost, which may be more expensive than the
start
up cost. In this regard, the solution obtained is
satisfactory.
/Read in data]
Give the load curve
to
ANN and
generate a unit Commitment
schedule
se the heuristics
to
modify the unit
It
=
Nt
-11
.
i = i
+
1 1
A E t ) = z E i t ) - Z L i t )
I
spinning reserve, or
Start up some peaking units to
com pensate the deficiency and AE t) have surplus ?
Do committed unit capaci
A
k=k+l and let unit i
out
of dispatch
r v
Fig.
7
The outline
of
the
proposed
method
-
8/10/2019 ramp up
6/10
1346
16 25.0 100.0 I 0.00598 I 18.2000 I 218.7752
42.,348
17
I
54.25
I
155.0
I
0.00463
I
4 COMPUTATION RESULTS
A
system with 26 thermal unit s is used to test the efficiency
and reliability of the proposed method.
The relationship of a
unit fuel cost with the unit power generation is described by
a
quadratic function and the unit start up cost is an
exponential
function of the time that the unit has been shut down, that
is,
(2 4 )
; P ; t ) ) a ; P ; t ) 2+
b;P;(t)
+ c ;
=
1,.
. N
t = 1 , . . . ,
Nt
I
The unit characteristics are given in Tables l( a) an d l( b).
The
program is written in C which
runs
on an IBM PC/386. Two
examples are discussed below.
The load demand in the first study case is given in Table
2.
The system spinning reserve
is
based on the capacity of the
largest online unit. Table
3
is the optimal unit commitment
schedule without considering the ramping limits. Table
4
is the
final unit commitment schedule which satisfies all the
system
operating constraints. The asterisk indicates that the unit is
in
the process of ramping
up
or down, and the unit constrained
generating capability is not equal to the unit rated
capacity.
According to Table
3,
at hour
23,
unit 16 is
off
but needs to
Table l( a) Generating units capacity and coefficients
68.95
140.0
100.0
Table l( b) Generating units operating and ramp limits
min init.
down cond.
UH DH, UR, DR,
(h) (h) (h)
(h)
(MW/h) (MW/h)
0
-1
0 0
48.0
60.0
n
-1 1
n
w 5 70 0
min init.
(h) (h) (h)
(h)
(MW/h) (MW/h
0 -1
0
0 48.0
60.0
down cond. I
DH, UR,
n I
-1
I
1
n I ~5
1 70n
II
10--13
3 -2
3
I
2
I
1
I
38.5
14--16 I 4 I -2
I -3 I
2 I 2
I 51.0 74.0
t
II
24 10
start up at hour
24.
Since unit 16 is shut down for only two
hours and its minimum down time and ramp-up time are two
hours each, it will be impossible to shut down this unit at
hour
22.
Accordingly, unit 16 will run continuously and generate
excess energy at hour 23, as shown in Table 4.
Unit
22
is on at hour
23,
but
is
asked to shut down at hour
24. Since unit 22 needs two hours to ramp down, we should
let
unit
22
to ram p down at hour
22.
So, the constrained generatin
capability of unit
22
at hour
23
is equal to 197MW- 99MW/h
l h
=
98MW, and hence the energy generated by unit 22 during
f
Table
2
Load demand
in
case study
1
Table 3 Unit commitment schedule without ramping
limits in case study 1
our
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
0 0 0 0 0 0 0
1 1 1 0 0 0 0
1 1 1 1 0 1 1
1 1 1 0 0 0 0
1 1 1 0 0 0 0
0 0 0 0 0 0 0
unit
1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
1 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
0 0 1 1 1 1
--- 26 )
1 0 0 1 1 1 1 0 0 0 1 1 1
1 0 0 1 1 1 1 0 0 0 1 1 1
1 0 0 1 1 1 1 0 0 0 1 1 1
1 0 0 1 1 1 1 0 0 0 1 1 1
1 0 0 1 1 1 1 0 0 0 1 1 1
1 1 0 1 1 1 1 0 0 0 1 1 1
1 1 0 1 1 1 1 1 0 0 1 1 1
1 1 0 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1
o n o o o o o o o i i o o i o o i i i i i i i i i i
0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1
Table
4
Final unit commitment in case study 1
our
1
2
3
4
5
6
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
unit ( 1 - - - 26 )
0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 1 1 1 1 0 0 0 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 * 0 1 l 1 1 * 0 0 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 * * * 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 0 1 1 1 1 1 * * 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 * 1 1 1 1 l l l l l l
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 1 0 1 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
1 1 1 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
~ 1 0 0 0 0 0 0 0 1 1 1 1 1 * ~ 1 1 1 1 l l l l l l
1 1 1 1 1 1 1 1 1 1 1 / 0 1 0 / 1 1 1 1 1 * * 1 1 1
0 0 0 0 0 0 0 0 0 1 1 0 0 1 0 1 1 1 1 1 1 0 0 1 1 1
1. * indicates the unit is in the process of ram in upldown
2.
underline indicates the unit state is modifie l &er the
inclusion of the ramping limits
-
8/10/2019 ramp up
7/10
1347
Table
6
Unit commitment schedule without ramping
our 23 will be less than that of the system modeled as a
step
function. The same situation occurs to unit 23.
After ramping up/down un its at hour 23 as necessary, the
on units are not able to supply the required energy, that is
A E t )
