Optimizing the Spinning Reserve Requirements Using a ... - IEEE Xplore

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IEEE TRANSACTIONS ON POWER SYSTEMS, VOL. 22, NO. 1, FEBRUARY 2007

Optimizing the Spinning Reserve Requirements Using a Cost/Benefit Analysis Miguel A. Ortega-Vazquez, Member, IEEE, and Daniel S. Kirschen, Senior Member, IEEE

Abstract—Spinning reserve (SR) is one of the most important resources used by system operators to respond to unforeseen events such as generation outages and sudden load changes. While keeping large amounts of generation in reserve protects the power system against the generation deficits that might arise from different contingencies, and thus reduces the probability of having to resort to load shedding, this reserve provision is costly. Traditional unit commitment (UC) formulations use deterministic criteria, such as the capacity of the largest online generator to set the SR requirements. Other UC formulations adjust this requirement based on probabilistic criteria but require iterative processes or approximate calculations of the level of risk associated with the provision of reserve. This paper describes an offline method for setting the SR requirements based on the cost of its provision and the benefit derived from its availability. Index Terms—Expected energy not served (EENS), loss-of-load probability (LOLP), mixed integer programming (MIP), power generation dispatch, power generation scheduling, spinning reserve (SR), value of lost load (VOLL).

I. INTRODUCTION OWER system operators across the world try to keep a certain amount of generation capacity as spinning reserve (SR) to ensure that the power system is able to withstand the sudden outage of some generating units or an unforeseen increase in load without having to resort to load shedding. The traditional criterion for setting the minimum amount of SR (the spinning reserve requirement) is that it should be greater than or equal to the capacity of the largest online generator [1]. This criterion ensures that no load has to be curtailed if any single generating unit is suddenly disconnected. On the other hand, this criterion does not guarantee such a positive outcome if two generating units trip almost simultaneously. Increasing the SR requirement using a similarly simple criterion would reduce the probability of an unprotected contingency and, at the same time, lower the overall risk of not meeting the demand. However, providing SR has a cost because it requires committing additional generating units and operating other units at less than optimal output. Power system operators should therefore not only schedule generating units to minimize the cost of providing the required SR but also determine what this requirement should be to achieve the optimal level of risk. This paper proposes a technique that balances the cost of providing SR against the benefits of this reserve in pool-based elec-

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tricity markets with unit commitment (UC). The benefits of the reserve are measured in terms of reduction in the expected energy not supplied (EENS) because of a lack of generation capacity. As the SR increases, the cost of operating the system increases while the expected socioeconomic cost of interruptions decreases, because it is less likely that load will have to be shed in response to outages of generating units. In this paper, the scheduling of reserves is performed in two steps. First, the amount of SR that minimizes the sum of these two costs is calculated for every period of the scheduling horizon taken separately. This calculation takes into account the load as well as the cost and reliability characteristics of the available units. Second, these hourly SR requirements are used as input to a traditional reserve-constrained UC program that produces a generation schedule that takes into account all the standard constraints and the inter-temporal constraints and couplings. Throughout this paper, SR is used to refer to the capability of the power system to respond voluntarily to contingencies within the tertiary regulation interval with the already synchronized generation [2]. II. METHODS USED TO SCHEDULE SPINNING RESERVE This section reviews the different formulations that have been developed to produce optimal and secure generation schedules. The aim of all these methods is to determine the commitment and dispatch of a set of generating units to supply the load forecast over a short-term horizon (usually one day to one week). These methods differ in their objective function and the formulation of the constraint responsible for guaranteeing the provision of SR. A. Direct Enforcement of the SR Constraint The conventional UC formulation falls in this category. The objective function considers only the sum of the running and startup costs of all units over all periods of the scheduling horizon (1) where

Manuscript received February 9, 2006; revised September 8, 2006. This work was supported in part by CONACyT, Mexico. Paper no. TPWRS-00076-2006. The authors are with the School of Electrical and Electronic Engineering, University of Manchester, Manchester M60 1QD, U.K. (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TPWRS.2006.888951 0885-8950/$25.00 © 2007 IEEE

running cost of unit during period ; cost of a possible startup of unit during period ; power produced by unit during period ; status of unit during period ; number of periods in the optimization horizon; number of available generating units.

ORTEGA-VAZQUEZ AND KIRSCHEN: OPTIMIZING THE SR REQUIREMENTS USING A COST/BENEFIT ANALYSIS

This objective function must be minimized subject to a number of constraints, the most important of which is clearly at that the total generation must match the system demand each period (2) To ensure that the schedule provides at least the specified amount of SR during period , the UC program enforces the following constraint: (3) where is the contribution that unit makes to the SR during period . This contribution is given by (4) is the ramp-up rate of unit , and is the amount in which of time available for the generators to ramp-up their output and deliver reserve capacity. Besides these two “system” constraints, each generating unit is subject to its own operating constraints, which include minimum and maximum production levels ( and ), minimum up- and down-time constraints, and maximum ramp-up and ramp-down constraints. In terms of reserve, the crucial element in this formulation is the SR requirement . A commonly used deterministic criterion sets the desired amount of SR at the capacity of the largest committed generating unit at period [1] (5) This criterion ensures that no load will need to be disconnected in the event that any single unit suddenly trips. However, this amount of SR could be excessive if the largest online generator is highly reliable and/or if the value that the customers attach to a continuous supply of energy is low. Even though a single contingency might result in load shedding, the risk associated with such disconnection may not be sufficiently large to justify the continuous expense of providing this amount of reserve. In other words, the provision of this amount of SR may not be economically justified. On the other hand, this criterion does not guarantee that the entire load would be served if two or more generating units were to trip nearly simultaneously. In essence, this criterion deems such simultaneous outages so much less likely than the outage of a single unit that it ignores the associated probability of occurrence. However, in power systems with a large number of generating units, this probability is not negligible. While setting the SR requirement at the capacity of the largest committed unit is the basic requirement, a number of variations around this criterion have been developed offline to achieve a level of risk that is deemed acceptable in a particular system [3]–[7]. While simple and practical, setting the SR requirements on the basis of these simple criteria is suboptimal because they do not balance the cost of providing

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reserve at all times against the occasional socioeconomic losses that consumers might incur if not enough reserve is provided. Anstine et al. [8] were the first to consider the probabilistic nature of the outages in the calculation of the SR provision. They proposed a technique that takes into account the forced outage probabilities of the generating units. Then, they establish the level of risk probability that should be maintained throughout the scheduling horizon. The SR requirement is adjusted at each scheduling period to maintain a uniform level of reliability. The advantage of this approach is that the provision of reserve can be reduced or increased compared to the amount specified by (5) according to the reliability requirements of the system. The drawback of this approach is that the risk is a quantity that lacks an intuitive interpretation and thus does not per se tell how much SR should be scheduled. The risk in one system could represent a completely different level of reliability in other systems because it depends on the number, capacity, loading, and reliability of the units as well as the value that the consumers attach to an uninterrupted supply. This technique does not optimize the SR provision itself but simply increases the committed capacity until the target risk is attained. From an economic perspective, this is suboptimal because the cost associated with the provision of reserve is not taken into account. Furthermore, insisting on a uniform risk level at all periods of the optimization horizon may require the commitment of expensive generating units when such expensive reserve might not be justified by the benefit that it provides. Gooi et al. [9] are the first to have considered how the SR could be optimized within the UC problem. Their approach consists in postprocessing the UC schedule to compute the level of risk of consumer disconnection at each hour. If this risk is not within a certain range of a prespecified target for some periods, the SR requirement is adjusted for these periods and the UC is run again. The first advantage of this approach is that it optimizes the SR provision, keeping intact the reserve-constrained UC formulation of (1)–(4). A further contribution is that since the risk level is an abstract concept that lacks an intuitively quantifiable interpretation, they introduce another reliability metric that considers not only the probability of having to shed load in response to a unit outage but also the amount that may have to be shed. At each period, an external cost/benefit analysis is then used to compute the level at which the marginal cost of SR matches the benefit it provides, i.e., the reduction in the expected socioeconomic cost of energy not supplied. The drawback of this approach is that it is computationally intensive because several UC computations may have to be performed before the target risk is achieved. It also should be noted that this approach considers the cost characteristics of generating units but ignores their individual reliability. This model was then extended to include the ramp-rate limits of the generating units in [10]. B. Enforcement of the SR Constraint Using a Risk Proxy In order to keep the risk of disconnection below a predefined threshold Chattopadhyay and Baldick [11] have developed a system-dependent exponential function that approximates the loss-of-load probability (LOLP) for any generation schedule. This approximation is incorporated in a linear

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constraint that replaces the traditional SR constraint in the UC formulation (6) The advantage of this approach is that, since the probability of load disconnection is represented explicitly, postprocessing of the results and iterations are avoided. On the other hand, the drawback of this approach is that it is not self-contained because it requires the selection of an appropriate risk criterion. LOLP is also not particularly suited to the computation of the socioeconomic cost of outages because it measures only the probability that the load exceeds the generating capacity but does not quantify the extent of the disconnections that might result from by cost/benefit analsuch deficits. Thus, setting the ysis would require the computation of an extra reliability metric that accounts also for the extent of the contingencies. Setting the arbitrarily could lead to generation schedules that are not economically optimal. If this target is set too high, there will be on occasion insufficient reserve to cover unforeseen generation deficits. On the other hand, if this ceiling is set too low, the increase in the cost of the generation schedule will exceed the potential economic benefits of avoiding load disconnections. Furthermore, the arbitrary selection of the ceiling can compromise the feasibility of the problem. Bouffard and Galiana [12] have proposed a pool market clearing process that includes a probabilistic reserve determination. This UC formulation considers two reliability metrics, the expected load not served (ELNS) and the LOLP. These authors emphasize the advantages of the ELNS over the LOLP in this context. They argue that the provision of reserve should be such that the scheduling provides lower ELNS and/or LOLP than a fixed target

problem on a uniform clearing price follows:

is formulated as

(9) where is the amount of SR provided by unit (contained in the set of selected units ). VOLL is the value of lost load [14]. is justified Purchasing an increment of spinning reserve if and only if: , where is the cost increment. This process is repeated for each of the periods of the optimization horizon. However, in this process, the individual reliability of the generating units is ignored. Furthermore, it assumes that the reserve market is independent of the energy market; thus, the bidders are limited to be already synchronized generating units. Furthermore, ignoring the coupling that exists between the energy and the reserve scheduling can lead to suboptimal or infeasible results [15]. In [16], it is suggested that the socioeconomic cost of outages could be considered explicitly under the form of an additional term in the objective function of the UC problem (10)

C. SR Optimization

The optimization then automatically determines the amount of reserve that minimizes the sum of the operating cost and the expected cost of outages caused by failures of generating units at each period of the optimization horizon. The expected cost of outages is equal to the value that the average consumer places on unserved energy (VOLL), multiplied by an approximation of the EENS as a function of the committed capacity. This EENS approximation must be calibrated for each load level. This approach has the advantage of being self-contained in the sense that it does not require a LOLP or EENS target because the SR provision is based on an internal cost/benefit analysis. References [17] and [18] discuss the difficulties associated with the selection of appropriate ELNS ceilings and propose to penalize the ELNS inside the objective function to avoid choice of arbitrary ELNS and/or LOLP ceilings. It is also mentioned that the problem might become infeasible if there are insufficient reserve resources to attain such ceilings or if these resources are very unreliable. The stochastic programming used in these papers creates some dimensionality problems that force the authors to assume that only one contingency can occur during the time horizon. Note that the approaches proposed in [16]–[18] optimize the risk within the multiperiod optimization process. By doing so, risk can be “traded” among periods. For instance, if at a given period the system risk can be reduced by committing additional expensive generation, but with large minimum up times, the optimization process might consider procuring less reserve and accepting a slightly higher EENS cost at such a period.

Wang et al. [13] propose an independent paid-as-bid operating reserve market in which a function that represents the socioeconomic benefits or losses is maximized or minimized. This function combines two conflicting objectives: the cost of reserve and the expected cost of interruptions. The minimization

III. PROPOSED APPROACH The method proposed in this paper determines the SR requirement for each period of the optimization horizon in an auxiliary computation prior to the UC solution. This auxiliary computation balances the benefit derived from SR against an estimate of

(7) (8) This approach is the first that includes the calculation of the reliability metrics within the optimization process. It has the advantage of enabling the inclusion of the ELNS target. However, the solutions obtained are not economically optimal because the selection of an arbitrary ELNS ceiling entails the same problems as the arbitrary selection of LOLP ceilings. While in this formulation ELNS and LOLP are estimated within the UC calculation, the computation of these estimates is truncated to the consideration of the simultaneous outages of only two units to avoid a combinatorial explosion. This approach requires a significant number of extra constraints and integer variables that can have a significant impact on the computational speed.


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subject to the load/generation balance and reserve constraints and

Fig. 1. Algorithm flowchart of the proposed approach.

the cost of its provision. To reduce the computational burden, this auxiliary computation neglects the inter-temporal coupling that affects the UC solution. Ideally, the SR optimization should be carried out in an inter-temporal optimization; however, this complete optimization is not possible unless the contingency enumeration is truncated and/or a risk proxy is used. In this approach, once the optimal SR requirement has been determined for each hour, these values are used in the standard SR constraints of a conventional UC (see Fig. 1). Fig. 1 shows that the optimization of the SR requirement is decoupled from the optimization of the generation schedule. It must be emphasized that, while the inter-temporal couplings are neglected in the auxiliary computation, they are taken into account fully in the UC solution. The generation schedules that are produced are therefore completely feasible. This two-step approach is an example of a bilevel optimization problem [19] and is described in more detail in the remainder of this section. A. SR Requirements Optimization The benefit of SR is measured by the reduction in the expected socioeconomic cost of supply interruptions achieved through the provision of SR. This quantity will be called the EENS cost. The optimal amount of SR is the amount that minimizes the sum of the EENS cost and the operating costs. This minimum, however, is a complex function not only of the amount of SR but also of the load, of the particular combination of units used to meet that load and of the reliability characteristics of these units. This combinatorial dependence gives the function a number of local minima that must be distinguished from the global optimum. Formally, the auxiliary time-decoupled optimization problem for the SR requirements estimation is expressed as follows: (11) where is the amount of SR required at period is the running cost of serving the demand and procuring the amount of SR , and is the EENS cost for the given scheduling period. For a given SR requirement, the running cost for each load level must be minimized with respect to the committed units and their dispatch using a single period UC (12)

respectively. The dispatch of the generating units is also constrained by their minimum and maximum generating limits. Even though this auxiliary UC solution is time-decoupled, the ramp-up limits of the generating units’ must be taken into account in order to guarantee that the SR can actually be delivered as denoted by (4). The socioeconomic cost of a particular supply interruption is defined as the product of the energy not supplied (ENS) and a VOLL determined using a survey [14]. Since it is impossible to predict the generation outages (or if any) during the implementation of a particular generation schedule, only an expected cost of interruptions can be computed. The EENS cost is thus given by (13) where EENS is the expected energy not served that would result from the generation schedule. A technique for computing EENS is described in [20]. This technique generates a capacity outage probability table (COPT) that takes into account the committed generating units, the probability of forced outage of each generating unit, the amount of SR that each unit can provide, and the load. This COPT gives the probability that the total capacity on outage during this period will be greater than or equal to a certain value. Using these cumulative probabilities, it is then possible to calculate the probability that a certain amount of load cannot be served because the capacity on outage exceeds the SR. Summing over the possible outages the product of these probabilities with the associated energy curtailed gives the EENS for this combination of generating units and this load level. Since it is impossible to predict which load will have to be curtailed, the use of an average value such as VOLL is appropriate. For more details on the components of these costs and how they are derived, the reader is advised to go to [21]. Fig. 2 illustrates this auxiliary optimization for the IEEE-RTS system [22]. Omitting the hydro generating units, this system consists of 26 units with a total capacity of 3105 MW. The quadratic approximation of the cost functions and ramp-up limits were taken from [23]. A load level of 1690 MW and a $/MWh were assumed. As the SR requirement increases, the dispatch cost increases while the EENS cost decreases. The sum of these two costs exhibits a global minimum at the optimal amount of SR. A number of search-based techniques have been developed to solve efficiently such univariate optimization problems. In this case, the total cost curve is unimodal but non-convex because the dispatch cost changes suddenly every time an additional unit is committed to meet the SR requirement. The search technique should therefore be able to avoid getting trapped in a local minimum. Furthermore, the number of cost calculations should be kept small because each evaluation requires one single-period unit commitment and the calculation of the corresponding COPT. Among the various search techniques that have been developed to solve this type of optimization problem, the iterated grid search with three-grid points [24] provides good results in

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Fig. 3. Equidistant grid search.

8) Repeat steps 3–7 until the difference between the upper and lower bound in which the solution is contained is smaller than the defined length of uncertainty. Step 5 is required from the second iteration step on, because all three test points are needed as the middle test point in the next iteration coincides with one of the previous test points due to the equidistant grid. Only two new test points are required at each iteration, except the first. Note that an appropriate selection of the span in which the optimal SR requirement is contained (i.e., search upper and lower bound) increases the speed of convergence of the optimization process and reduces the number of COPT calculations. Fig. 2. Dispatch, EENS, and total costs for the IEEE-RTS for L = 1690 MW and VOLL = 1000 $/MWh.

a reasonable time. This search algorithm operates according to the following steps. 1) Select an acceptable length of uncertainty “err,” that is, a maximum span of SR in which the solution is contained. 2) At the first iteration , select the interval of SR in which the solution is contained (i.e., an upper and lower bounds for the search), . 3) If the difference between the upper and lower bounds is smaller than a defined length of uncertainty , then the minimum cost lies in the span [ ]; else, continue. 4) Select three test points: such that ; then the set of values gives some indication of the shape of . Since the shape of is unknown, it is suggested that the test points are equidistant (see Fig. 3), and then, they can be computed as (14) , check for the values of at that repeat 5) If at in order to avert repeated evaluations of . 6) Let be the set of values of evaluated at each test point; thus, . 7) Apply the following conditions to update the search interval: If has the smallest value among , then take the interval AB for the next iteration, i.e., . If has the smallest value among , then take the interval BC for the next iteration, i.e., . If has the smallest value among , then take the interval CD for the next iteration, i.e., . That is, take the surrounding cell where the smallest value is located and use it in the next iteration. To take care of equality, ; then take ABC.

B. Reserve-Constrained Unit Commitment This part of the process is a standard optimization of the generation schedule as described in Section II-A and in [1] and [25] with the exception that the SR requirements are determined by the auxiliary calculation described above. This UC therefore takes into account all the inter-temporal couplings and respects all the inter-temporal constraints that were neglected during the auxiliary calculation. It should be noted that the SR provision of unit at period might be limited by the ramp rates of the generating unit as shown by (4).

IV. TEST RESULTS The proposed technique has been tested with a UC program implemented using mixed integer programming (MIP) and a branch and bound technique [26]. The search was stopped when an MIP gap no larger than 0.008% was achieved. The basis for these tests was the IEEE-RTS [22], which consists of 26 units. The UC data and the load profile used for testing were obtained from [23]. Note that the hydro generating units were removed from this base system and were not considered in any of the tests. The power generated by the units committed at is given by the economic dispatch of the committed units for the first period for a load level of 1700 MW. All the results presented in the following sections are for an optimization horizon of 24 h considering all the inter-temporal constraints, unless stated otherwise. For the EENS cost calculation, the COPTs were computed considering probabilities down to and with 1 MW resolution. The maximum uncertainty length for the grid search is set to 0.5 MW; and the search span (i.e., the initial range for the search) extends from 100 to 1000 MW. A. Effect on the Generation Schedule Fig. 4 contrasts the generation schedules obtained with the traditional approach and the proposed technique and shows that optimizing the SR requirements can have a significant effect on the UC and the SR provision.


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Fig. 4. (a) Traditional approach. (b) Proposed VOLL = 1000 $/MWh.

Fig. 6. Optimal SR requirements as a function of the normalized load level for systems with similar characteristics but different numbers of units, VOLL = 6000 $/MWh.

TABLE I ITEMIZED COSTS FOR SYSTEMS OF DIFFERENT SIZES

Fig. 5. SR scheduled using the traditional approach and the proposed optimization technique for the base system with VOLL = 1000 $/MWh.

Fig. 5 shows the SR provision obtained when the SR requirement is specified using the traditional approach and the proposed optimization technique. B. Effect of the System Size and Load Level In order to illustrate how the optimal SR requirement varies with the size of the system, systems with similar characteristics but different sizes were created by duplicating the IEEE-RTS and proportionally scaling the load profile. Four larger systems with, respectively, 3, 5, 8, and 10 times the number of units in the base system, were created. Unless otherwise specified, VOLL was set at 6000 $/MWh. Fig. 6 shows the optimal SR requirements for each load level of the scheduling horizon and each of the systems considered. The results are presented as a function of the normalized load, which is obtained by dividing the actual load by the total generation capacity. For the systems with a small number of units, this figure shows that the SR requirement is mostly independent of the load and close to the value given by the traditional approach. However, as the number of generating units increases, the optimal SR requirement generally increases and becomes dependent on the load. This is because the probability of simultaneous outages of generating units increases with the number of units. Fig. 6 also shows that for the large systems at light loads, provision of more SR is not expensive because the marginally inexpensive units are lightly loaded. As the load increases, more expensive units are committed, and the EENS cost does not justify as much SR as when the system was lightly loaded. However, for very large loads, since more outages of generating units would result in load curtailment, the EENS cost increases, and larger amounts of SR are justified, even if this reserve has to be provided by expensive units.

Table I itemizes the costs for the proposed and traditional approaches. While the total cost is always lower with the proposed approach, the savings generally increase with the size of the system. The percent amounts shown under the actual difference correspond to the relative difference over the traditional approach. The largest system attains savings of $5777 (0.08%) per day. Considering that in this method the traditional UC formulation, as given by (1)–(4), remains intact, even the lowest savings justify its implementation. Fig. 7 shows, for each of the test systems, the LOLP achieved using the traditional and the proposed approaches for setting the SR requirements. This figure shows that at every period of the scheduling horizon, the proposed technique achieves a lower LOLP. The LOLP profile is also flatter than what is obtained with the traditional approach. This shows that setting the SR requirement on the basis of a LOLP-based criterion is substantially suboptimal. During the early period of the scheduling horizon, the LOLP is very low because the efficient base units remain committed for a light load, thereby providing an excess of SR. Even for the systems with 26 and 78 units where the difference in SR with the two approaches is very small, the optimization approach achieves a significantly smaller LOLP during some periods.

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Fig. 9. Total cost of supplying a load of 1690 and 2430 MW using the IEEE-RTS as a function of the SR requirement for several values of VOLL.

Fig. 7. LOLP achieved at each scheduling period with the traditional and the proposed approaches to setting the SR requirements for systems with similar characteristics but different numbers of units.

Fig. 10. Total cost of supplying a load of 22 000 MW using the IEEE-RTS escalated ten times as a function of the SR requirement for several values of VOLL.

The reliability of the generating units plays an important role in the SR requirements. Fig. 8 shows the SR requirements for the system with 130 units when the ORR is scaled by factors of 0.5, 1.0, and 1.5 times. As expected, the SR requirements increase with the ORR of the generating units. This is because the probability of unit’s outages increases, and thus, the EENS cost is higher. Conversely, when the units in the system are more reliable, the SR requirements are lower.

generating unit. However, as VOLL decreases, the balance between the dispatch cost and the EENS cost changes and the optimal SR requirement decreases. However, for larger values of load (e.g., 2430 MW), this effect is more pronounced because more expensive generating units must be committed to meet the load and provide SR. In systems where the accepted VOLL is relatively low (for example, in developing countries, where a reliable supply of electricity has not yet become essential to economic activity and quality of life), the traditional approach would schedule an economically excessive amount of SR. Fig. 10 shows a similar family of curves for a single period and a load level of 22 000 MW in the 260-unit system. The optimal SR requirements increase with the system size because the probability of outages (including multiple outages) increases. However, protecting the system against simultaneous outages of two of the largest units is not justifiable until VOLL reaches a large value.

D. Effect of VOLL

E. Cost Itemization

Fig. 9 illustrates the effect that VOLL has on the optimal SR requirement when the IEEE-RTS (26-unit system) is used to supply two different values of load for a single period. For a load of 1690 MW and large values of VOLL, the minimum of the total cost curve is obtained for an SR requirement that is close to 400 MW, i.e., the capacity of the largest synchronized

Fig. 11 compares the itemized costs with the optimized and traditional SR requirements for the 26-unit system. For ease of comparison, all these costs have been normalized on the basis of the operating cost (i.e., startup cost plus dispatch cost) for the generation schedule obtained using the traditional SR requirement. These costs are plotted as a function of the VOLL, which

Fig. 8. SR requirements for the 130-unit system for different ORRs, VOLL =

6000 $/MWh.

C. Effect of the Outage Replacement Rate (ORR)


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Fig. 11. Itemization of the normalized costs as a function of the normalized VOLL for the 26-unit IEEE-RTS.

Fig. 13. Normalized total cost and maximum hourly LOLP for the base system as a function of the normalized VOLL for the proposed technique and variants of the traditional SR criterion. The inset in the lower figure shows a magnification around a VOLL of 100 p.u.

Fig. 12. Itemization of the normalized costs as a function of the normalized VOLL for the 78-unit system.

Fig. 14. Normalized total cost and maximum hourly LOLP for the 78-unit system as a function of the normalized VOLL for the proposed technique and variants of the traditional SR criterion. The inset in the lower figure shows a magnification around a VOLL of 1000 p.u.

itself has been normalized on the basis of the average cost of energy with this same generation schedule. Over the whole range of VOLL (500 to 50 000 $/MWh), the total cost is lower with optimized SR requirements. For low VOLLs, this is achieved by an increase in the EENS cost, which is more than compensated by a reduction in the dispatch and startup costs. For VOLLs greater than or equal to 11 000 $/MWh, the difference between the two approaches becomes negligible because the optimization technique determines that the optimal SR at all periods should be 400 MW, which is the value given by the traditional approach for this system. In this system, a VOLL greater than or equal to 11 000 $/MWh is thus implied by the traditional criterion. If the actual VOLL in the system is lower, the economic optimum is not achieved. Fig. 12 shows the same cost itemization for a system that has three times the number of units and a load that is three times

larger. This figure shows that in this case, the savings are particularly significant for large VOLLs and that this is obtained through a reduction in the EENS cost. In this larger system, the traditional approach underestimates the SR requirements. This effect is exacerbated if the reliability of the generating units is poor. F. Comparison With Other Fixed SR Requirements It is useful to explore whether the traditional approach can be enhanced to achieve results that are close to those obtained with the optimized SR requirements. Figs. 13 and 14 compare the proposed technique with the traditional approach and two variants for the base system and the three-area IEEE-RTS. The traditional approach keeps the SR requirement constant at 400 MW. In the first variant, the SR requirement at each period is 420 MW, while in the second, it is set at 380 MW.

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TABLE II ITEMIZED COSTS AND MAXIMUM HOURLY LOLP FOR AN SR CRITERION EQUAL TO A FRACTION OF THE HOURLY DEMAND, 260-UNIT SYSTEM, VOLL = 6000 $/MWH

Fig. 16. Itemized computing times as a function of the system size expressed in terms of the number of generating units. These results were obtained for a VOLL of 6000 $/MWh, a lower and upper bound of 100 and 1000 MW, respectively, a maximum uncertainty length of 0.5 MW, and a COPT that considers probabilities down to 1 10 13 with 1 MW resolution.

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Fig. 15. Hourly spinning reserve requirements for the hybrid approach, 260unit system, and VOLL = 6000 $/MWh.

The amount of reserve scheduled in this hybrid formulation is therefore never less than that specified by the fixed criterion. On the other hand, when VOLL is large and/or the risk of load shedding is significant, this hybrid approach provides more reserve. Fig. 15 shows the hybrid SR requirements for the 260-unit system with a deterministic criterion of 3% of the hourly de$/MWh. mand and H. Computation Time

These figures show the normalized total cost and the maximum hourly LOLP achieved over the scheduling horizon as a function of the normalized VOLL. These results show that adjusting up or down a constant SR requirement does not achieve costs that are significantly closer to the optimum. To reduce costs, the value that consumers place on unserved energy must be taken into account. While increasing the SR requirement to 420 MW reduces the maximum LOLP, reducing it to 380 MW does not decrease it significantly. It might be argued that for large power systems, the requirements imposed by (5) are insufficient and that comparing the proposed method against such criterion is not fair. Another approach that is used in practice is to specify the SR requirements as a fraction of the peak or hourly demand, such as in the Western Zone of PJM [7]. If the SR requirements are given as , then the total cost and maximum hourly LOLP for a $/MWh are shown in Table II when such criterion is applied to the 260-unit system for various . Table II shows that even setting larger amounts of SR for large power systems is not a guarantee that the economical expenses of operating the system will be lower. The LOLP can be dramatically reduced but at a cost that is not justified by a reduction in socioeconomic costs. In this particular system, setting the SR requirement at 2.5% of the hourly demand gives the best results, but this approach does not match the economic benefit achieved by optimizing the SR requirement using the proposed approach.

In addition to a standard UC solution, the proposed approach requires the auxiliary optimization of the SR requirements. The time needed for this auxiliary problem is a function of the COPT resolution, the system size and type, VOLL, maximum acceptable MIP gap, initial SR span for the search (initial upper and lower bounds), and maximum error on the optimal SR requirement. Fig. 16 shows how the itemized computing times vary as a function of the system size. This figure also shows that as the system size increases, the auxiliary optimization takes longer, while in the case of the UC calculation, the solution time is a function of the branching adopted by the optimization software (Xpress ). Part of the delay in the auxiliary optimization is due the communication between the core program (implemented in MATLAB) and the optimization software. This was done using ASCII files through the operating system (Microsoft Windows XP). Fig. 16 shows that the proposed technique is not time consuming, even though no particular efforts were made to reduce the computing times by relaxing the COPT resolution, optimizing the selection of the initial lower and upper bounds of the grid search technique nor the uncertainty length. All the simulations were carried out in a PC with an AMD Athlon clocking at 1.53 GHz and with 512 MB of random access memory (RAM).

G. Hybrid Approach If the optimal amount of SR turns out to be lower than the fixed criterion used in a specific system , the system operator might be reluctant to operate the system with less reserve and a higher LOLP than was previous practice. In such cases, a hybrid approach can be used where the SR requirement is given by

V. CONCLUSION

(15)

This paper presents a new technique to determine the SR requirements at each period of the optimization horizon in a UC calculation. This technique determines the amount of SR that minimizes the total cost of operating the system, i.e., the sum of the actual operating cost and of the socioeconomic cost associated with load shedding. The computing requirements associated with the proposed technique are modest compared with those associated with other techniques that have been proposed


in the literature. In addition, this technique does not require the specification of proxy measurements of risks that inevitably lead to suboptimal solutions. While it might be argued that there is some uncertainty regarding the VOLL that should be selected for a given system, this choice can be guided by solid surveys and international comparisons. On the other hand, choosing a suitable threshold value for LOLP is entirely arbitrary. The proposed technique does not require a reformulation of the UC problem and could therefore be combined with existing computer programs. If a system operator is reluctant to require less SR than is specified by some other criterion, a hybrid approach can be used where the SR requirement obtained using the proposed optimization technique is only used when it is higher than the amount specified by another criterion. This would be the case in systems where the value of lost load is high or the probability of generator outages is high.

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Daniel S. Kirschen (M’86-SM’91) received the electrical and mechanical engineer’s degree from the Université Libre de Bruxelles, Bruxelles, Belgium, in 1979 and the M.S. and Ph.D. degrees from the University of Wisconsin-Madison in 1980 and 1985, respectively. He is currently a Professor of electrical energy systems at the University of Manchester, Manchester, U.K.