Contingency Constrained Economic Dispatch problem and proposes a method for its solution. ... and software was implemented to support it. Since the Northern.
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IEEE Transactions on Power Systems, Vol. 12, No. 3, August 1997
Economic Dispatch with Generation Contingency Constraints B. Awobamise
M. Aganagic
G. Raina
Empros Power Systems Control Division Siemens Energy & Automation, Inc., Brooklyn Park, MN 55428
Abstract - This paper presents a new model of the Generation Contingency Constrained Economic Dispatch problem and proposes a method for its solution. The operating policy of the Northern Ireland Electricity was the basis for the formulation, and software was implemented to support it. Since the Northern Ireland Electricity system operates in relative isolation the operating security criteria are rather stringent. In particular, it is required that loss of generation of any unit in the system must be covered by fast, 3 or 30 second generation reserves on other units in the system. The fast response unit reserve capabilities are represented by concave curves. The solution method is based a nonlinear version of Danbig-Wolfe decomposition principle. Numerical results are presented.
-
Keywords: Economic
Dispatch, Contingency Constraints, Generation Reserves, Dantzig-Wolfe Decomposition 1. INTRODUCTION
The Economic Dispatch (ED) is a standard function of an Energy Management System (EMS). Its task is to allocate non-base load over the set of dispatchable units so that the required power is generated at the least cost, and that a minimum reserve capacity is provided. The minimum reserve capacity is usually based on an operating requirement that there should be a sufficient reserve capable to respond within a specified time to a generation contingency and/or to a sudden increase in the demand. While recognizing that this definition of the h c t i o n has served well for a large number of EMSs, its role in providing the required level of security should be examined with respect to the following contingencies: 0
0
Demand contingencies: unexpected increase in power demand Generation contingency: loss of a single generating unit.
Namely, it is certain, that the ED provides a definite reserve capacity for demand contingencies. On the other hand, if a 96 SM 509-0 PWRS A paper recommended and approved by the IEEE Power System Engineering Committee of the IEEE Power Engineering Society for presentation at the 1996 IEEUPES Summer Meeting, July 28 - August 1, 1996, in Denver, Colorado. Manuscript submitted December 28, 1995; made available for printing June 27, 1996.
A. I. McCartney Northern Ireland Electricity Belfast, Northern Ireland
generation contingency oc(:urs the reserve capacity that is actually available to remedy it depends on the contingency itself. Since the outaged unit may have contributed toward the reserve requirement, there is no certainty as how much reserve actually may be available. It seems that this apparent deficiency of the standard ED has not been of a particular concern for utilities that have a large number of units, so that reserve contribution of any single unit is small. However, there are utilities that have to address the problem of the generation contingencies in the ED function in a more detailed way. A case in point is provided by the Northern Ireland Electricity (NIE). The NIE system is a relatively small system with a peak of around 1500 MW. The power is provided locally from 23 thermal units with capacities in the range fiom 30 to 260 MW. Having no tie lines lo the main British Grid and only recently connected to the Grid of the Republic of Ireland, the system operates in relative isolation. This fact has some very significant implications on the operational policy which had to be reflected in the furictional specifications and design solutions for the new NIE: Energy Management System. In particular, it is required that the ED h c t i o n be executed in such a way that it is ensured that the following criteria be satisfied 0
0
Impact of an outage of any single generating unit be limited. Loss of generation of any single unit be covered by available reserves on other on-line units.
The first criterion is definted by so called impact factor that determines the maximum percentage of the total system load that can be assigned to any unit in the system. In other words, the impact factor limits the: maximum percentage of load that may have to be made up if the worst single generation contingency occurs. The second criterion is defined by the corresponding cover factor which is the least percentage of generation of any unit that is covered by the reserves on the other units in the system. ‘Thus, for example if a 100% cover factor is applied, the second criterion ensures that failure of any single on - line unit cim be hlly compensated within the reserve time by redistributing the system load among the available on - line units. We refer to an ED with the impact and reserve cover constraints as the Generation Contingency Constrained Economic Dispatch (GCCED). The second criterion is applied with respect to the fast 3 and 30 second genemtion reserves. The sequence of
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1996 IEEE
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point the reserve capability is limited by the excess capacity, i.e. the difference between the high operating limit and the current power output. If the time interval is longer than 5 to 15 minutes, the pick-up rate is usually assumed to be constant over the whole operating range of the unit. When shorter time intervals are considered, the pick-up rates have to be modeled as functions of the present unit power output. Figure 2. illustrates typical 3 and 30 second reserve capability curves drawn up following extensive tests on units at various load points. It is important to note that the reserve characteristics are piecewise linear and concave.
From REAL TIME:
- Load To Be Distributed Set - Generator Sustained Limits I...____.__._____ ~
- Reserve Requirements
iALARM:
:Insufficient : : Capacity On Line I . .:
.................,
I
I
'.
The GCCED definition presented in the next section includes new constraints in addition to the one on the sum of unit reserves that, at the best, the available formulations [l4,6-131 provide. The step fiom the standard ED formulation [SI to the GCCED mirrors the one from the standard ED to the Security Constrained ED (SCED) [ 6 ] .Security constraints in the SCED are formulated with respect to the unit power outputs whereas in the GCCED the arguments are the unit reserve contributions. It should be mentioned that the approach of this paper could be easily extended to include the constraints on both power outputs and reserve contributions.
Feasible Solution
&------
The plan of the paper is as follows. In Section 2 . , we introduce notation and develop the mathematical model of the problem. Solution algorithm is presented in Section 3. A discussion of some computational results is included in Section 4. The paper is concluded in Section 5.
Figure 1. Sequence of Execution of GCCED
executions of the EDs is shown in Figure 1. The sequence is executed cyclically every 3 minutes, and spontaneously whenever a trigger condition occurs (a change in unit availability, change of load, etc.). At every pass, we check (IMPACT TEST) if there is sufficient unit capacity on-line especially in the view of the impact factor requirements. The GCCED with the 3 - second reserve requirement is attempted next. If this dispatch is not feasible, the 30 - second reserve requirements are enforced instead. If neither GCCED is feasible, the reserve requirements are relinquished, an appropriate message is issued, and a reserve unrestricted ED is executed. Generating unit reserve capability is determined by sustained pick-up rate, i.e. unit capability to take on an additional load in a given time interval. At any operating
2. PROBLEM FORMULATION
Notation
- set of dispatchable generating units - set of base - loaded generating units
D B d 1
-total system demand - load to be dispatched ( 0 < 15 d ) - power generation (output) of unit i E B UD
PI Pi,
q.
- sustained lower and upper limits on generation of unit
gi (Pi 1
iEBuD - impact factor - reserve capability of unit i E B UD at operating point
SI
- reserve contribution 0 I S,
P
- reserve cover factor
Pk
- vector of k-th segment slopes of reserve curves
ak
- vector of k-th segment reserve curve constants
5
PI
I gl ( p ,) of unit
iEBuD
Cl (PI )
a 0
20
40
60
80
IM)
120
140
160
180
1W
Mw
Figure 2. Typical 3 and 30 - second reserve characteristics
- nonlinear cost function of generation ofunit i E B u D
- system lambda, i.e. dual price of system load (power balance constraint)
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-
dual price of total reserve requirement for dispatchable units (base-loaded units reserve constraint) - vector of dual variables associated with reserve cover
P Cl
constraint of unit i
EBv D
2. Reserve cover constraints reB
raD
I4
-the number of elements of D,i.e. its cardinality
T
- superscript denoting vector or matrix transpose
are specified for each dispatchable unit i, i E D . The sums on the left hand side represent reserve contributions of base loaded and dispatchable units.
e
- vector of ones, e = (1. e l ) T
3. Base-loaded units reserve cover constraint
csi
- identity matrix I c = e eT - I - binary complement of identity matrix - diagonal matrix with vector x on its diagonal diag(x) I
2max{s, + p . p ;
iaD
The objective of the GCCED is to determine unit generations p I so as to minimize the total generation cost: minimize
ci ( p ,
-
(1)
rcD
subject to a set of constraints. The objective function is commonly assumed to be convex (See [l, 2,4,6-131). In case that this assumption is not valid, only a local optimum is guaranteed. No additional assumption about its differentiability is made. As matter of fact, in the case of NIE and many other systems, the objective function is not differentiable. Figure 3 provides an illustration of a typical incremental cost rate (ICR) curve with discontinuities. The discontinuities usually correspond to the power output levels at which a fuel switch (fuel topping) or change in valve position takes place.
I
EB
-cq} ;€B
Note that the right hand side of (2.3) is a constant, which we denote by r B , and that it determines the minimum total reserves that has to be provided by the dispatchable units in order to provide the required reserve cover for each base loaded unit i E B.
4. Impact factor and generation capability constraints
E, I p I I min{z, 6.d )
(2.4)
are included for each dispatchable unit i, of unit i E D . 5. Reserve capability constraints 0 5 SI 5 g , ( P r )
(2.5
for units i E D . The reserve capability curves g,()are assumed to be concave functions. Using the well known fact that any concave function can be represented as a point-bypoint infmum of a set of affine functions (a functionfis said to be aflne if f-f(0)is linear), the reserve capability curves can be represented as (2.6 g,(Pr) = + f i k *PI)
ysk
As in the case of NIE, it is often sufficient to use only a finite number of affine hctions to describe the reserve capability curves. For notational convenience, and without any loss of generality, we will assume that for every unit the same number of affine functions is required to determine its reserve capability curve. In that case, the contributions of the units can be written in a compact vector form
P Figure 3. Typical Unit Incremental Cost Rate Curve
The minimum solution is required to satisfy the following constraints:
1. Power balance constraint i ED
If the assumption that system losses are constant that is implied by (2.1) is not acceptable, then the constraint can be replaced by
CPI=l i d Kt
where
K,
are the loss penalty factors.
and the problem constraint matrix is as shown in Figure 4. Figure 4 shows only the equality and non-trivial inequality constraints. Lower and upper bounds on the problem variables are not shown as they are treated implicitly. The matrix has dense rows corresponding the power balance and the cover constraints. In contrast to these constraints that couple units, the remaining constraints corresponding to the reserve capability curves are dense and separable. The latter constraints dominantly determine the row size of the matrix. These facts suggest that the method [l], that was originally developed for the SCED with the idea to take advantage of the block angular matrix structure resulting fiom the reserve constraints, could be adalpted to the GCCED problem. The method is based on a two level decomposition using a
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nonlinear version of the Dantzig-Wolfe Decomposition Principle [ 5 ] . W
S
=I 2 rB
2-(Csi)-e IGB
I a0
Figure 4. Problem Constraint Matrix
where p k and sk denote vectors of power outputs and reserve contributions obtained as solutions of the subproblems described below, It should be noted that the master problem is a linear programming problem whose columns have practically the same density structure as the constraints (2.12.3). The number of columns is variable depending on the number of columns (proposals) being generated. However, we know that no more I q+ 3 will be needed to span the optimal solution. The master problem is solved using the revised simplex method ( [ 5 ] ) with an implicit treatment of lower and upper bounds. Let A , p and o be the optimal dual variables of the master problem (3) corresponding to the constraints (3.2), (3.3) and (3.4), respectively. These dual variables are used to determine the objective fimctions of the subproblems.
Subproblems There are as many subproblems as there are units in the system. They have the following form
3. SOLUTION ALGORITHM
min c J ( p J ) - [ A - P . p l ’ P J-[Pu((OT.‘c)Jl’sJ (4) subject to (2.4-2.5). The subproblem objective is a function in two variables: p J ands,. However, since ,u and are
The steps of the algorithm are the same as in [l]. It proceeds in two phases. The objective of the first phase is to nonnegative, it follows that (4) is equivalent to the following obtain a primal feasible solution by minimizing the sum of one - dimensional minimization: infeasibilities, whereas the problem objective (1) is being min ( P I ) -in -P’Pl ‘ P J - bU+ (Or J 1’ gJ ( P J ) (5) minimized in the second phase. At each iteration of the over the interval , ’Z I p J 5 PJ . Since cJ(.) is convex and algorithm a sequence of subproblems and a master problem gJ (.) concave, the subproblem objective function is convex. are solved. As the result of the new problem formulation, we have different defmitions of the master problem, subproblems Thus, the subproblem can be solved by funding a minimum of and the stopping criterion. convex one-dimensional function which is usually an easy task. Master Problem The master problem has the following form
Stopping Criterion
n
Suppose that all the subproblems (5) are solved. For each j E D let p , be the solution of the subproblem and let subject to
- ( P J )=
‘ J ( p J ) - [ A - P ’ p l ‘ P J- [ p + “)J 1. g J ( P J ) be the corresponding minimum. Then the algorithm terminates if the following condition: ‘J
n
J=1
is satisfied where 77 is the optimal value of the master problem dual variable corresponding to the constraint (3.5). If the condition is not satisfied then a new column 1s appended to the master problem, and a new iteration of the algorithm is initiated. In the NIE implementation, the relation implied by ( 6 ) is tested with an accuracy of The proof of the stopping criterion is given in [I].
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4. A DISCUSSION OF SOME COMPUTATIONAL
RESULTS A C code implementation of a real time version of the GCCED was developed and fully integrated into the NIE EMS. The function resides on an IBM RS 6000 workstation running under the AIX operating system.
cover factor point is emphasized because it corresponds to the standard operating requirement of NIE. As it can be seen, the cost increase of 2.23%is inciurred. The available on-line
e+
6-
I%]
4b
5 -
6
4-
4)
3 2 23%
2-
...................................... 41
i
1-
: : : +e
.o
20
0
40
e I 60
80
100
120
Cover Factor rh] Figure 5. Cost of Security
units can provide the maximum cover factor of around 108%. Beyond this point the problelm becomes infeasible. The influence of cover factors on the system lambda is shown in Figure 6.
120 1
13 14 15
16
100 105 108 110
6
9 10
i
29 41 45
Infeasible
105 100
I 0
20
40
60 68
80
IO0
120
Cover Factor ph] Figure 6. System Lambda
It is interesting to point out that the system lambda doesn’t exhibit a monotone behavior with respect to the cover factor. In a part, this is a consequence of the facts that the minimum of a convex function subjiect to linear constraints is not necessarily a convex function of the constraint matrix elements, and that the cover factor p appears there (See Figure 4.). However, the arbitrary straight - line approximations between the points in the plot is used to suggest only the continuity of the relationship. 5. CONCLUSIONS Figure 5 shows the cost implications of the cover factor constraints. It can be seen that the cover factors up to around 45% come at a zero cost. From there on, the cost of generation increases monotonically. This is to be expected since the set of feasible solutions of any GCCED with an increased cover factor is a subset of the original set. The 68%
The paper has presented ix new formulation of the problem of load allocation to the generation units which models in detail an “n-1” security criterion. The problem is named the Generation Contingency Constrained Economic Dispatch. The security criterion is specified by a parameter called the
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reserve cover factor which is new in the literature. The cover factors are in principle different fiom unit to unit and can be derived using a probabilistic analysis. In general, one would expect that a less reliable unit be assigned a higher cover factor. The formulation provides an optimal solution while ensuring that a preventive reserve for any single generation contingency is ensured. For any contingency, the guaranteed amount of the reserve is determined as the minimum percentage, corresponding to the cover factor, of the lost generation. The formulation was developed on the basis of the operating policy requirements of the Northern Ireland Electricity (NIE). A solution method based on a nonlinear version of Dantzig-Wolfe decomposition is proposed. The solution is based on an adaptation of the method [l] which has been proven to be very efficient for solving the traditional security constraint dispatch. The method was implemented and hlly integrated into the new NIE EMS system. A discussion of some computational results based on an NIE case is presented. In particular, the costs of security criterion are examined. 6 . BIBLIOGRAPHY S. Mokhtari, "Security Constrained Economic Dispatch Using Nonlinear Dantzig-Wolfe Decomposition", IEEE PES Winter Meeting, Paper No. 96 WM 544-PWRS, Baltimore, MD, Jan. 1996
[l] M. Aganagic and
[2] K. Aoki and T. Satoh, "Economic Dispatch with Network Security Constraints Using Parametric Quadratic Programming", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-101 (12), pp. 4548-4556, Dec. 1982. [3] B. H. Chowdhury and S. Rahman, "A Review of Recent Advances in Economic Dispatch", IEEE Trans. on Power Apparatus andsystems, Vol. PAS-101 (5), Nov. 1990. [4] A. J. Elacqua and S. L. Corey, "Security Constrained Dispatch at the New York Power Pool", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-101 (S), pp. 2876-2883, Aug. 1982.
[5] 6. B. Dantzig, "Linear Programming and Extensions", Princeton University Press, N. J, 1963. 161 R. Lugtu, "Security Constrained Economic Dispatch", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-98 (l), pp. 270-274, Jan./Feb. 1979. [7] M. V. F. Pereira and L. M. V. G. Pinto, "A Decomposition Approach to the Economic Dispatch of Hydrothermal Systems", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-IO1 (lo), pp. 3851-3860, Dec. 1982.
[S] R. Romano, V. H. Quintana, R. Lopez and V. Valadez, "Constrained Economic Dispatch of Multi -area Systems Using the Dantzig-Wolfe Decomposition Principle", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-100 (4), pp. 21272137, Apr. 1981. [ 91 W. 0. Stadlin, "Economic Allocation of Regulating Margin", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-90 (4), pp. 381-388, JulylAUg. 1971.
101 B. Stott and J. L. Marinho, "Linear Programming for Power System Network Security Applications", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-98 (3), pp. 837-848, MayIJune 1979. 111 D. Streiffert, "Multi - Area Economic Dispatch with Tie Line Constraints", IEEE PES Winter Meeting, Paper No. 95 WM 179-2-PWRS, New York, Jan. 1995. [12] L. S. Vargas, V. H. Quintana and A. Vannelli, "A Tutorial Description of an Interior Point method and its Applications to Security - Constrained Economic Dispatch", IEEE Trans. on Power Apparatus and Systems, Vol. 8, pp. 1315-1323, Aug. 1993. [13] J. G. Waight, A. Bose, and G. B. Sheble, "Generation Dispatch Reserve Margin Constraints Using Linear Programming", IEEE Trans. on Power Apparatus and Systems, Vol. PAS-100 (l), pp. 252-258, Jan. 1981.
7. BIOGRAPHICAL NOTES
Muhamed Aganagic holds a Diploma Engineer degree in Electrical Engineering and a Ph. D. in Operations Research Erom Stanford University, Stanford, CA. Dr. Aganagic is the lead for Midterm/ Long Term Operations Planning applications in the Operational Planning Group at Siemens Empros. Bola Awobamise holds a Ph. D. degree in Electrical Engineering fiom the UMIST, Manchester, Great Britain. Dr. Awobamise is the Manager of Power Applications Group at Siemens Empros.
Gokal Raina holds a Ph. D. degree in Electrical Engineering fiom the University of Calgary, Alberta. He has been working in the Power and Scheduling Applications section at Siemens Empros since 1988. He is a member of Power Engineering Society. Allen I. McCartney holds an Honors B. Sc. degree in Electrical and Electronic Engineering from Queen's University, Belfast. He is System Operational Planning Engineer at Northern Ireland Electricity and Project Manager for the replacement Energy Management System being provided by Siemens Empros.
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Discussion R. BILLINTON, M. FOTUHI-FIRUZABAD AND S. ABORESHAID (Power Systems Research Group, University of Saskatchewan): The authors are to be congratulated for presenting an interesting paper on economic load dispatch. The problem is formulated by including a constraint designated as the Generation Contingency Constraint Economic Dispatch (GCCED). It is concerned with the economic allocation of generation or proper division of the load among the committed units such that the system should have sufficient response to tolerate the outage of any single on-line unit. This is an important feature in operating reserve studies and is of interest to electric utility operators. The authors' comments on the following questions will be appreciated. There are two tasks considered in power system generation scheduling. The first is unit commitment followed by the dispatch of the committed units. This paper deals with the second aspect. If the units are committed to the system based on the largest unit contingency, the committed units may or may not be able to respond to the single unit outages within the margin time. The authors comments regarding the unit commitment criterion is appreciated. In order to formulate the operating criteria, two factors namely as impact and cover factors are introduced. Based on the definitions each unit has its own impact and cover factors. They, however, were expressed in Section 2, i.e. in notations and equations, as single values. This can also be noticed in the results presented in Figures 5 and 6. The question i s , do all the units have the same cover factors?. If not, then which cover factor is used in Figures 5 and 6. If yes, then how does the size and ramp rate of different units affect the cover factor?. The test system used in this paper has 23 thermal units ranging from 30 to 260 MW. Typical thermal units can pick up 1-3% of their full output capacity in one minute. If the largest unit in the system is fully loaded, the system should have a response of at least 260 MW in 30 second in case of the loss of the largest unit. This means that each unit should pick up a portion of about 10 to 15 MW of the load within the margin time. In practice, a thermal unit at this range cannot pick up load at a rate of 30 MW per minute. The authors' comments on the possible availability of this amount of response for the system is appreciated. Hydro units play a dominant role in responding to load increase and or sudden generation loss where hydro generation constitutes a significant portion of the overall generation, as in some systems in Canada. Rapid start gas turbine units can also pick up load in a relatively short time. This units are not on-line and usually called stand-by units. The system can operate with lower operating cost and higher response capability by including stand-by units. Would the authors please indicate that how the proposed technique can be extended to include stand-by units. Figures 5 and 6 present the impact of the cover factor constraints on the operating cost and system lambda. The cover factor of 45% is associated with the standard economic dispatch which has the lowest operating cost. There are several points before the 45% level where the cost is identical to that of the basic economic load dispatch. What is the benefit of operating at lower cover factors and why the system cost does not vary at these point? As the authors point out in the conclusion, the cover factors are derived using a probabilistic analysis. This is an interesting point and we would like the authors comments on how these factors can be derived based on the available outage data of different units. Does this indicate that the technique considers probabilistic concepts in addition to the deterministic criteria? Are these factors and the presented results sensitive to the accuracy of the units data? The authors are again to be congratulated on an interesting paper. We look forward to reading their comments. Manuscript received August 27, 1996.
M. Aganagic, Gokal Raha (Power Systems Control Division, Siemens Energy & Automation, Brooklyn Park, MN 55428): We want to thank the discussers for their interest in our work. Their insightfbl questions and comments give us an opportunity to expand on our paper. Here are our responses to the discussion: 1. The unit commitment function, which was implemented as a part of the project included the same generation contingency constraints as the ones described in the paper. Moreover, the GCCED code was reused in the economic dispatch part of the unit commitment function.
2. In the project in question ihe cover and impact factors were specified on the system wide basis, i.e. they were the same for all the units. As we chose to introduce the concept of the GCCED on a real life case, we derived the economic dispatch model with these assumptions in mind. However, it is easy to see that the model be extencled to include unit specific cover and impact factors. From an operational point of view this may be a useful extension that could provide for a variable hedging against unit outages. In other words, we may want to specify lower impact factors and am higher cover factors for less reliable units. Regarding the second part of the question, it does not appear that either the size or ramp rate of a unit directly affect its cover factor. However, it is true that cover factor is usually of no concern for smaller units as they will normally be covered by already available reserves. It should be noted also that unit ramp rate affect their sustained limits imposed in the economic dispatch. 3. We were informed that the reserve curves like the one presented in Figure 2 are based on extensive studies and measurements. They are obtained after a significant work on unit governing systems and upgrading event recording systems. 4. We believe that an extensiion of the GCCED to systems with hydro generation capacities is possible and straight forward. In an unlikely case that the start-up costs can be neglected, the units can be made subject to the dispatch with respect to both power and reserve generation as any other units. For hydro units, we of course have to assume that the water values are available. Otherwise, the problem of unit start-ups including hydro units and gas turbines we usually delegate to a unit commitment (hydro-thermal icoordination or hydro scheduling) function. In the later case, the presence of these stand-by units will affect reserve constraints;by decreasing the amounts of the reserve that have to be obtainled from the dispatchableunits. 5. All the points in Figures, 5 and 6 with the cover factors lower than 45% correspond to the same operating point. The purpose of including these: points in the diagrams is to
emphasize that the reserve constraints resulting from the cover factors lower than 45% are redundant as this level of cover comes at no additional cost.
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6 . The point we were making in the conclusion of the paper was that a probabilistic analysis can and should be used to derive the unit cover factors. The analysis itself appears to us as a very interesting research subject, and we hope that this paper will provide a motivation for it. Regarding the sensitivity of the results presented, drawing on some well known results of stability of solutions of the convex programming we can state the standard “smoothness” claim that the changes in results will be small if so are the changes in the unit data are small as well. Manuscript received November 18, 1996.
