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Mar 28, 2011 - Abstract: Spinning reserve (SR) is an important resource, which can protect power ... The SR requirement needs to be assigned appropriately.
www.ietdl.org Published in IET Generation, Transmission & Distribution Received on 10th December 2010 Revised on 28th March 2011 doi: 10.1049/iet-gtd.2010.0805

ISSN 1751-8687

Optimising probabilistic spinning reserve using an analytical expected-energy-not-supplied formulation M.Q. Wang1 H.B. Gooi2 S.X. Chen1 1

Research Student, School of EEE, Nanyang Technological University (NTU), Singapore 639798, Singapore Associate Professor, School of EEE, Nanyang Technological University (NTU), Singapore 639798, Singapore E-mail: [email protected]

2

Abstract: Spinning reserve (SR) is an important resource, which can protect power systems without involuntary load shedding. The SR requirement needs to be assigned appropriately. In this study, a probabilistic method is used to optimise the SR requirement. The SR amount is determined by simultaneously optimising the operating cost, reserve cost and the expected interruption cost. Tremendous computational requirements introduced by the expected-energy-not-supplied (EENS) calculation make the major impediment to the optimisation solutions. In this study, a multi-step method is proposed to overcome the computation intractability of the EENS problem. The EENS of each optimisation period is analytically formulated as a piecewise linear function of system spinning reserve of that period. The parameters of the piecewise linear function are derived from the capacity outage probability table, which is established based on the exogenous unit schedule. The efficiency and validity of the proposed method is verified using the IEEE-RTS system.

1

Introduction

Spinning reserve (SR) is the unused capacity of the power system to respond voluntarily to contingencies within a given period of time using the already synchronised devices [1, 2]. It is an important resource that can protect a power system against loss of load after a sudden outage of some generating units or transmission/distribution network facilities and/or a sudden increase of load. Scheduling sufficient SR can reduce the probability and severity of loss of load. However, providing SR has a cost because additional units may be committed and other units may operate less than their optimal output. The SR requirement needs to be assigned appropriately. Traditionally, the SR requirement is calculated by a deterministic approach. The amount of SR is set as a fraction of the total load or the capacity of the largest online unit or their combination [3]. Although this approach is easy to implement, it considers neither the stochastic nature of system behaviour and component failures nor the economics. Various probabilistic approaches [4 – 12] have been developed, which take the stochastic characteristic into consideration. Gooi et al. [4] optimises the probabilistic SR in a unit commitment (UC) problem. It post-processes the UC schedule to compute the system risk. If the risk is greater than the specified target, the SR requirement is adjusted and the UC runs again. In practice, an acceptable risk level may be difficult to define. Chattopadhyay et al. [5 – 8] optimise the SR requirement by solving a reliabilityconstrained UC problem. The reliability metrics, such as loss of load probability (LOLP) and/or expected energy not supplied (EENS), which must fall below a predefined 772 & The Institution of Engineering and Technology 2011

threshold, are incorporated into the UC model. However, it is difficult to design a reliability metric ceiling for different power systems. In [9 – 13], the SR requirement is determined by minimising the operating cost and the expected interruption cost (EIC). The EIC is considered explicitly as an additional term in the objective function of the UC problem. This method is selfcontained. It does not require a system risk or LOLP or EENS target because the SR provision is based on an internal cost/benefit analysis. It can automatically determine the SR requirement by a trade-off between reliability and economics. However, the introduction of LOLP and EENS indices into UC is generally cursed by computational intractability because of their non-linear and combinatorial nature. A very large number of possible combinations of units that need to be calculated prevent their widespread use. In an attempt to overcome this deficiency, Wang et al. [9] calculates EENS based on exogenous energy scheduling, which means that the reserve schedule is independent of the energy schedule. Ignoring the coupling that exists between the energy and the reserve scheduling can lead to suboptimal or infeasible solutions [13, 14]. Wu et al. [10, 11] mitigate the computational burden by considering only a few significant events. Reducing the number of contingency events considered cannot guarantee that the whole spectrum of unreliability is captured. The optimisation may lose considerable computation accuracy and the results are suboptimal. In [12, 13], EENS of every optimisation period is estimated as a three-segment piecewise linear function of the system-committed capacity using a pre-processing auxiliary procedure. However, the proposed EENS approximation is only an upper bound of the real EENS. Using only three IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772 –780 doi: 10.1049/iet-gtd.2010.0805

www.ietdl.org segments to represent the EENS curve is not accurate enough. During the pre-processing optimisation procedure, the intertemporal coupling is not considered [13], which could lead to sub-optimal solutions. In this paper, the SR requirement is computed by simultaneously minimising the operating cost, reserve cost and the EIC. In order to overcome the tremendous computational requirements introduced by EENS, EENS of each optimisation period is modelled by a piecewise linear function of the system spinning reserve (SSR) of that period. The reasons as to why EENS is formulated as a function of the SSR and why a piecewise linear model is used are explicitly investigated. The determination of the parameters of the piecewise linear function is also described. The whole optimisation process is realised by a multi-step method. In the first step, a base UC module is implemented. For each optimisation period, a capacity outage probability table (COPT) is established based on the unit schedule obtained. EENS is then formulated as a piecewise linear function whose parameters are derived from COPT. In the second step, the EENS model obtained in step one is incorporated and the optimisation process runs again. After optimisation, if the convergence criterion is satisfied, the process will stop and the SR requirement can be obtained. Otherwise EENS will be updated based on the newly found unit schedule and a new step is needed. The proposed multi-step method replaces the combinatorial and non-linear function of EENS by a piecewise linear function, and the computation efficiency is greatly improved.

2

Problem formulation

2.1

EENS by a value of lost load (VOLL). VOLL is defined as the average value that customers attach to the loss of one kW for one hour [15]. The SC and EIC are conflicting objectives. When the SR increases, the SC increases whereas the EIC decreases. The SR requirement can be automatically determined by a trade-off between reliability and economics. The formulation of EENS is shown in Section 2.2. The objective function must be minimised subject to a number of constraints: Power balance constraint NG 

where PD t is the load demand during period t. SR constraints 

+

NG NT  



: maximum power output of unit i; Pmax i Rup i : ramp up rate of unit i; t: amount of time available for the generators to ramp up their output for delivery of reserve capacity [16]. In this paper, t is assumed to be 0.5 h.

qi,t Ri,t + EENS × VOLL

(1)

where NT: number of periods in the optimisation horizon; NG: number of available generating units; Ui,t , Pi,t: status (0/1) and power output of unit i during period t; qi,t , Ri,t: reserve bid price and reserve bid amount of unit i during period t; Ci,t(Pi,t , Ui,t): running cost of unit i during period t represented by a three-segment piecewise linear function; SUCi: start-up cost of unit i; K i,t: a binary variable that satisfies Ki,t ≥ 0 Ki,t ≥ Ui,t − Ui,t−1

∀i, ∀t

(5)

The block of constraints (5) generally includes upper- and lower-generation MW limits, minimum-up and down-time constraints, initial conditions, ramp-up and ramp-down rate constraints [17]. They are all considered in this paper. 2.2

t=1 i=1



(4)

where

(Ui,t , Pi,t ) [ C,

[Ci,t (Pi,t , Ui,t ) + SUCi Ki,t ]

t=1 i=1

Ri,t ≤ Pimax Ui,t − Pi,t Ri,t ≤ Ui,t (tRup i )

Unit operating constraints

In the proposed method, the SR requirement is determined by simultaneously optimising the operating cost, reserve cost and the EIC. The objective function is

min

(3)

i=1

Mathematical model

N N G T  

Pi,t = PtD

Formulation of EENS

If the unit schedule, Ui,t , Pi,t and Ri,t , are known, EENS can be easily calculated based on COPT [3]. For each optimisation period, a COPT can be generated. COPT gives the probability that the total outage capacity will be greater than or equal to a certain value. EENS is equal to the summation of the products of the probabilities and the associated energy curtailed when the outage capacity exceeds the SR requirement. EENS will be zero when the loss of capacity is less than the SR requirement. If the unit schedule Ui,t , Pi,t and Ri,t cannot be determined a priori, COPT cannot be formed. EENS can be explicitly formulated by unit commitment variables based on every possible outage event [6] EENSt =



ps,t bs,t (DPs,t + DRs,t − SSRt )DT

(6)

s[St

(2) where

The first term in the objective function is the operating cost, which includes the running cost and start-up cost. The second term is the reserve cost. The sum of the operating cost and reserve cost is called the schedule cost (SC). The final term is the EIC, which is obtained by multiplying IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772– 780 doi: 10.1049/iet-gtd.2010.0805

St: set of all possible events during period t; ps,t: probability of event s during period t; DPs,t , DRs,t: power curtailment and reserve curtailment of event s during period t, respectively; 773

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www.ietdl.org reliability, the outage probabilities p1i,t, p2i,j,t and p3i,j,k,t can be formulated as [6]

SSRt: SSR during period t, that is, SSRt =

NG 

Ri,t

(7)

NG 

p1i,t = ui Ui,t

i=1

(1 − uj Uj,t )

(13)

j=1,j=i

DT: time duration of each period where DT ¼ T/NT; T is the entire period of observation and NT is the number of optimisation periods.

bs,t =

1, 0,

if DPs,t + DRs,t − SSRt . 0 otherwise

NG NT  

(8)

p2i, j,t b2i, j,t (Pi,t + Ri,t + Pj,t + Rj,t − SSRt )

t=1 i=1 j.i

+

NG  NG  NG NT  

p3i, j,k,t b3i, j,k,t (Pi,t + Ri,t + Pj,t

t=1 i=1 j.i k.j

+ Rj,t + Pk,t + Rk,t − SSRt )

(9)

where p1i,t: probability when only outage of unit i occurs during period t; p2i,j,t: probability when simultaneous outage of units i and j occurs during period t; 3 pi,j,k,t : probability when simultaneous outage of units i, j and k occurs during period t. 3 satisfy Binary variables b2i,j,t, b2i,j,t and bi,j,k,t

 b1i,t  b2i,j,t =  b3i, j,k,t =

1, 0,

=

1, 0,

if Pi,t + Ri,t − SSRt . 0 otherwise

ui = gi · DT

(15)

(10)

1, if Pi,t + Ri,t + Pj,t + Rj,t + Pk,t + Rk,t − SSRt . 0 0, otherwise (12)

When a two-state model is used to represent the unit

(16)

where gi is the failure rate of unit i.

3

EENS approximation Relationship between EENS and SSR

From (13) – (15), one sees that the outage probabilities are functions of statuses of all units. They are highly non-linear. So the formulation of EENS is highly non-linear. From (9) one sees that because of various possible permutations of units, combinatorial nature exists. Combined with the UC problem, the curse of dimensionality occurs and the solution space is tremendously large. The calculation of EENS is an impediment to the optimisation process. New methods need to be developed to compute EENS efficiently. From (6) one sees that EENSt is a function of ps,t , bs,t , DPs,t , DRs,t and SSRt . The dependent variables ps,t , bs,t , DPs,t and DRs,t are functions of independent variables Ui,t , Pi,t and Ri,t . These functions are directly related to every outage event. If any of the four variables, ps,t , bs,t , DPs,t and DRs,t are kept in the formulation of EENSt , EENSt cannot be greatly simplified since it is still directly related to every possible outage event and the combinatorial nature still exists. However, SSRt has no relationship with outage events. It can be concluded that if EENSt can be analytically expressed as a function of only SSRt , the formulation of EENSt can be greatly simplified. 3.2

if Pi,t + Ri,t + Pj,t + Rj,t − SSRt . 0 (11) otherwise

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(1 − ul Ul,t )

where ui is the outage replacement rate (ORR). For short-term operation problems, ORR is used to compute EENS. It is assumed that unit failures are exponentially distributed and the time to repair is so long that repairs can be ignored. It is also assumed that if a unit is failed during an optimisation period, it will not be available for the subsequent periods. The ORR of unit i during DT is given by [3, 6]

3.1 p1i,t b1i,t (Pi,t + Ri,t − SSRt )

NG  NG NT  

(14)

l=1,l=i, j,k

t=1 i=1

+

NG 

p3i, j,k,t = ui uj uk Ui,t Uj,t Uk,t

In this paper, DT is assumed to be 1 h. Only random outages of units are considered. Load forecast errors and the effect of transmission and distribution networks are not taken into account. The outage events can be classified by outage orders. First outage order means outage of one unit; second order means simultaneous outage of two units and third order means simultaneous outage of three units. EENS can be explicitly formulated as below where superscripts ‘1’, ‘2’ and ‘3’ represent first, second and third order of outage events, respectively. For formulation simplicity, EENS caused by fourth- and higher-order outage events are not shown here. EENS ≃

(1 − uk Uk,t )

k=1,k=i,j

The binary variable bs,t satisfies 

NG 

p2i,j,t = ui uj Ui,t Uj,t

Piecewise property of the EENS curve

When EENSt is formulated as a function of SSRt , extreme turning points occur on the EENSt(SSRt) curve. An extreme turning point occurs at the place where the rate of change in EENSt has a drastic decrease when SSRt increases. This phenomenon is illustrated by an example. Consider the IEEERTS system [18, 19] where the hydro units are excluded. This system consists of 26 units. For simplicity only the first hour is considered in this subsection. VOLL is 1000 $/MWh. The EENSt(SSRt) curve of Fig. 1 is obtained by a sensitive analysis. When SSRt increases from zero to the maximum available value, the corresponding EENSt can be obtained for each fixed SSRt by optimising (1) with EENS of (9). In Fig. 1, IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772 –780 doi: 10.1049/iet-gtd.2010.0805

www.ietdl.org extreme turning points suggests that the relationship between EENSt and SSRt should be modelled by a piecewise function. To facilitate the integration of EENS in UC problem, the EENS curve will be linearised starting from SSR at 0 MW to the left-most extreme turning point and between any two successive extreme turning points. Then EENS of an optimisation period can be modelled by a piecewise linear function of SSR of that period. 3.3 Formulating EENS as piecewise linear function of SSR

Fig. 1 Variation of EENS against SSR in the first hour

two insets show the magnifications of EENSt when SSRt is larger than 400 and 800 MW. The variations of EENSt when SSRt is around 400 and 800 MW in the first hour are shown in Fig. 2. From Figs. 1 and 2, it can be concluded that EENSt decreases drastically when SSRt initially increases. The rate of decrease in EENSt also decreases as the SSRt increases. From Fig. 2 one sees that the rate of decrease in EENSt decreases sharply when the SSRt is around 400 and 800 MW. For example, when the SSRt increases from less than 400 MW to larger than 400 MW, the rate of decrease in EENSt changes from 1.821 to 0.0155. When the SSRt increases from less than 800 MW to larger than 800 MW, the rate changes from 8.332 × 1024 to 1.143 × 1025. Here 400 MW is equal to the largest committed capacity of a single unit. The committed capacity of a unit is equal to the output power plus the committed SR. Likewise, 800 MW is equal to the sum of the committed capacities of two largest units. When the SSRt is smaller than 400 MW, EENSt is relatively large because EENSt is caused by some first-order outage events, and all second- and higher-order outage events. When the SSRt becomes larger than 400 MW, the system is able to withstand any first-order outage events. The outage probabilities of the second- and higher-order outage events decrease several orders of magnitude compared with those of the first-order outage events. However, the outage capacities caused by the second- and higher-order outage events increase linearly compared with those of the first-order outage events. Obviously, the effect of decrease in outage probabilities on EENSt outperforms the effect of the increase of outage capacities. That is why the rate of decrease in EENSt drops sharply when the SSRt is around 400 MW. A similar situation occurs when the SSRt is around 800 MW. The existence of

When the unit schedule is determined, the calculation of EENSt through (6) and COPT is essentially the same. An event in (6) corresponds to an outage capacity line in COPT. If some outage events have the same outage capacity, their total effect corresponds to one outage capacity line. The sum of power curtailment and energy curtailment, namely DPs,t + DRs,t in (6), corresponds to the outage capacity in COPT. The variable ps,t in (6) corresponds to the outage probability in COPT. From the analysis of comparison, it can be found that the outage capacity, outage probability and number of outage capacity levels in COPT are the parameters of the EENS piecewise linear function, which is shown in (17). In this paper, based on the given unit schedule, COPT can be established and EENSt can be formulated from COPT as EENSt =

N (t) 

{pi,t bi,t (DCi,t − SSRt )}

(17)

i=1

where N(t): number of outage capacity levels during period t; DCi,t: amount of the ith outage capacity during period t; pi,t: probability of the ith outage capacity level during period t; bi,t: a binary variable which is introduced to model the presence or absence of loss of load owing to the ith outage capacity during period t. It satisfies  bi,t =

1, 0,

if DCi,t − SSRt . 0 otherwise

(18)

The parameters N(t), DCi,t and pi,t are all derived from COPT. The number of segments of the piecewise linear function N(t) is equal to the number of outage capacity levels in COPT. From (17) one sees that EENSt has been formulated as a piecewise linear function of only SSRt . Compared with (6) and (9), the formulation of EENS is greatly simplified and the computation efficiency is greatly improved. Equation (18) can be linearised [6, 20] to DCi,t − SSRt NG max Si=1 Pi

≤ bi,t ≤ 1 +

DCi,t − SSRt N

G Si=1 Pimax

(19)

Finally, EENS can be formulated as the summation of the products of some binary variables and a bounded continuous variable, which can be linearised [6, 20]. After EENS is linearised, a mixed integer linear programming (MILP) can be used to solve the proposed model. 3.4 Fig. 2 Variations of EENS when SSR is around 400 and 800 MW in the first hour IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772– 780 doi: 10.1049/iet-gtd.2010.0805

Cluster property of EENS

From the preceding subsection, one sees that EENSt can be formulated as a piecewise linear function of SSRt based on a 775

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www.ietdl.org given unit schedule. However, the unit schedule Ui,t , Pi,t and Ri,t cannot be obtained before optimisation. If Ui,t , Pi,t and Ri,t take arbitrary values, EENSt may have a large variation. As far as our problem is concerned, the unit schedule Ui,t , Pi,t and Ri,t are constrained to a small feasible region because they must satisfy the power balance constraint, SR constraint and unit operating constraints. Besides, the total cost, a function of Ui,t , Pi,t and Ri,t , needs to be minimised. When Ui,t , Pi,t and Ri,t vary in this small feasible region, it can be found that the EENSt(SSRt) curves have a cluster property. Take the IEEE-RTS system as an example. For simplicity only the first hour is considered. Fig. 3 shows EENSt(SSRt) curves under different feasible unit schedules. These unit schedules are obtained based on a family of reserve constrained unit commitment (RCUC). The RCUC in which SSR must satisfy the SR requirement is only used in this subsection. The purpose of introducing a family of RCUC is to produce a series of representative unit schedules. Then EENSt(SSRt) curves can be plotted and analysed based on these unit schedules. The objective function of RCUC is min

N N G T  

[Ci,t (Pi,t , Ui,t ) + Ki SCi,t ] +

t=1 i=1

NG NT  

 qi,t Ri,t

t=1 i=1

(20) Equation (20) is the same as the first and second terms of (1). The objective function must be minimised subject to constraints (2) – (5). Besides, the following constraint should also be satisfied in the RCUC problem SSRt ≥ RRCUC t

(21)

is the predefined SR requirement during period t where RRCUC t in the RCUC problem. Fig. 3 is obtained using the following procedure: 1. Specify the SR requirement RRCUC to be enforced. At the t beginning, RRCUC is 0. t 2. Perform RCUC and the unit schedule can be obtained. If RCUC is feasible, go to Step 3. Otherwise, RRCUC exceeds t the maximum reserve supported by the system and the iterative process aborts. 3. Calculate N(t), DCi,t and pi,t using COPT based on the unit schedule obtained in Step 2. Plot EENSt by varying SSRt via (17). 4. Increase RRCUC and go to Step 2 above. t

In Fig. 3, the maximum RRCUC is 540 MW. When RRCUC is t t larger than 540 MW, the RCUC becomes infeasible. For clarity the inset shows the magnification of EENSt when the SSRt increases from 400 to 1000 MW. From Fig. 3 one sees that EENSt is sensitive to the SSRt . When the SSRt increases from 0 to 1000 MW, EENSt decreases several orders of magnitude. However, EENSt is not so sensitive to the variation of unit schedules. For a fixed SSRt , EENSt does not change too much. Thus, EENSt(SSRt) curves under different feasible unit schedules have a cluster property, that is, EENSt varies within a narrow band as seen in Fig. 3. It should be noted that when plotting EENSt(SSRt) curves via (17) in Step 3 above, SSRt is seen as the independent variable and all other constraints in the optimisation are relaxed during this step. So the EENSt (SSRt) curves shown in Fig. 3 are relaxed curves. The feasible EENSt (SSRt) of the whole optimisation model belongs to a subset of the plotted curves of Fig. 3. From Fig. 3, one sees that even the relaxed EENSt(SSRt) curves satisfy the cluster property. As a subset of the plotted curves, the feasible EENSt(SSRt) should also satisfy the cluster property. The EENSt (SSRt) curve that corresponds to the optimal unit schedule also belongs to the cluster curves. Hence, it is logical to formulate EENSt as a function of only SSRt based on a feasible unit schedule.

4

Implementation of the proposed method

4.1

Proposed multi-step method

From the above section, it can be seen that EENS for each optimisation period can be formulated as a piecewise linear function shown in (17). However, when employing this piecewise linear function, two problems occur. The first is that a feasible unit schedule must be known before the optimisation can proceed. The second is that even though the cluster property exists, the EENS calculated using (17) may have some approximation errors. Accordingly, a multistep method is proposed. A feasible unit schedule is produced in Step 1 and the approximation error of EENS is gradually eliminated in the following steps. In the first step, a base UC module without reserve is used to produce a feasible unit schedule. Then for each optimisation step, a COPT is established. In the second step, the EENS for each optimisation period is formulated as a piecewise linear function of SSR via (17). Then (1) is optimised and a new unit schedule can be obtained. A convergence criterion is introduced   |EENSdruing − EENSafter | t t DEENS = max ≤1 t EENSafter t

(22)

where : EENS of period t calculated during optimisation EENSduring t via (17); EENSafter : EENS of period t calculated after optimisation via t COPT based on the optimisation results and e : convergence tolerance.

Fig. 3 EENSt (SSRt) curves under different feasible unit schedules, which are produced by a family of RCUC in the first hour 776 & The Institution of Engineering and Technology 2011

If (22) is satisfied, the SR requirement is obtained and the optimisation will stop. Otherwise, the EENS is updated based on the COPT formed using the newly found unit schedule. A new step will be implemented till (22) is satisfied. IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772 –780 doi: 10.1049/iet-gtd.2010.0805

www.ietdl.org The proposed multi-step method replaces the and combinatorial EENS formulation by a linear formulation. The computation efficiency improved. The flowchart of the proposed method is shown in Fig. 4. 4.2

non-linear piecewise is greatly multi-step

Convergence of the multi-step method

In the proposed multi-step method, a UC module without reserve is implemented in Step one. In Step 2, EIC is considered and the system committed capacity increases in the most economical way. The SR requirement is computed and the system operation becomes more reliable. In Step 3, the EENS curve is updated based on the unit schedule of Step 2. More units are scheduled and more capacity is committed for the scheduled units. The EENS curve used in Step 3 moves slightly upwards compared with that used in Step 2. A higher SR requirement is scheduled and the system reliability is further improved. As the iteration continues, the sytem committed capacity gradually increases and the EENS curve moves monotonously and asymptotically close to the final curve owing to the cluster property of the EENS curves. The variations of unit status and output power of two successive steps become smaller and smaller. The difference in EENS values of two successive steps becomes negligible. The proposed multi-step method will finally converge when criterion (22) is satisfied. 4.3

Accuracy of the multi-step method

The accuracy of the proposed multi-step method depends on whether the piecewise linear formulation of EENS can model the real EENS curve exactly. When criterion (22) is satisfied, the EENS caluclated using (17) during optimisation is approximately the same as that calculated using COPT after optimisation. This means that the accuracy of the approximation is guaranteed. Take the IEEE-RTS system as an example and only the first hour is considered. VOLL is 1000 $/MWh. The convergence

Fig. 4 Flowchart of the proposed multi-step method

Table 1

tolerance 1 is 0.5% and the MILP duality gap is 0.01%. The results obtained by the proposed multi-step method and results obtained by optimising (1) with EENS of (9) are shown in Table 1. When the multi-step method is implemented, three steps are needed. The value of DEENS in Steps 2 and 3 are 4.07 and 0.02%, respectively. As the iteration continues, more SR is scheduled. Finally, solution converges at Step 3 and the SR reaches the optimal SR requirement. The SR requirement and various costs obtained by the multi-step method are approximately the same as those obtained by minimising (1) using EENS (9). When VOLL is extremely large, say 10 000 $/MWh, the EENS obtained using the multi-step method is somewhat overestimated compared with that obtained by optimising (1) using EENS (9). However, the SC obtained by the multi-step method is smaller than that obtained by optimising (1) using EENS (9). The increase of EIC is mainly compensated by the decrease in SC and finally the total cost computed by the multi-step method is only slightly larger. The small error (about 2.1%) occurs because EENSt is only formulated as a function of SSRt . The direct interaction between EENSt and Ui,t , Pi,t in the EENS formulation is neglected. This error is the price one pays for using an approximated EENS formulation, which greatly improves the computation efficiency. In normal system operation, an extremely large VOLL is seldom used.

5

Case studies and discussion of results

The proposed multi-step method is tested using the IEEE-RTS system without the hydro-generation. This system consists of 26 units. The UC data and ramp rate limits were obtained from [18], and the start-up costs and reliability data were obtained from [19]. The power generated by the units committed at t ¼ 0 is given by the economic dispatch of the committed units for the first hour at a load level of 1700 MW. The system lead time is 1 h. For simplicity, it is assumed that all generating units offer SR at prices equal to 10% of their highest incremental cost of energy production. COPT is truncated by omitting the MW outage levels in which the cumulative probabilities are less than 10210. The outage capacity levels are rounded to a fixed rounding increment [3] of 1 MW. VOLL is 1000 $/MWh. The convergence tolerance 1 is 0.5% and the MILP duality gap is 0.01%. After the third step, the convergence criterion is satisfied and the optimisation procedure stops. The value of DEENS in Steps 2 and 3 are 8.22 and 0.02%, respectively. The unit output power and reserve are shown in the Appendix. Units 4 – 9 and 23 are not scheduled during the entire optimisation period. Their output power and reserve are always zero and they are not listed in the tables. The SR requirement of Steps 2 and 3 are shown in Fig. 5. From Fig. 5 it can be found that in Step 2, the approximated SR amount is determined. In Step 3, the SR amount is adjusted and finally the optimal SR value is obtained.

SR requirement, SC, EIC and TC obtained by multi-step method and optimising (1) with EENS of (9)

SR requirement, MW SC, $ EIC, $ TC, $

Results of minimising (1) with EENS (9)

Results of step 1 in the multi-step method

Results of step 2 in the multi-step method

Results of step 3 in the multi-step method

155.00 18 677.05 618.12 19 295.17

0.00 18 242.93 0.00 18 242.93

142.95 18 601.05 671.86 19 272.91

155.00 18 678.48 618.11 19 296.58

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www.ietdl.org 5.1

Effect of system size and VOLL

In order to illustrate the effect of system size, systems with similar characteristics but different sizes were created by duplicating the IEEE-RTS and by proportionally scaling the load profile. Three larger systems with 3, 5 and 10 times the number of units in the base system were created. The rounding increment is 10 MW and VOLL is 1000 $/MWh. The convergence tolerance is 0.5% and duality gap is 0.1%. Fig. 6 shows how the SR requirement varies under different system sizes. Fig. 7 shows how the SR requirement varies under different VOLL. As expected, for large power systems, more SR is needed. As VOLL increases, the tradeoff balance between SC and EIC tend to make the system more reliable and more SR is scheduled. 5.2

Fig. 7 Variations of SR requirement under different VOLL

Computation time

Owing to the cluster property of EENS, an approximated EENS can be achieved in the second step. The unit status and output power have been approximately determined. After Step 2, the status and output power need little adjustment to satisfy the accuracy requirement. Usually, three or four steps are all it takes for the problem to converge. Compared with the base UC problem, (17) was added in the objective function in the second and subsequent steps. Only N(t) binary variables are introduced in (17) for each optimisation period. These binary variables are only related to SSR and it has no relationship with unit commitment variables. The computation time does not increase too much. The total time required is related to the maximum acceptable MILP duality gap, convergence tolerance,

Fig. 5 SR requirement of every hour

Fig. 8 Run times vary as a function of system size

rounding increment of COPT, VOLL, system size, load condition and type of generators. Relaxing the MILP duality gap, convergence tolerance or COPT resolution will reduce the computation time. In order to illustrate the effect of system size on computation time, systems with similar characteristics but 3, 5 and 10 times the number of units in the base system were used. The rounding increment is 10 MW and VOLL is 1000 $/MWh. The convergence tolerance is 1% and the MILP duality gap is 0.1%. Fig. 8 shows how the run times vary as a function of system size. Fig. 8 shows that with the increase of system size, the computation time tends to increase. High-order outage events are all considered in the proposed multi-step method and the proposed method is not time consuming. For the 26unit IEEE-RTS system, if the proposed multi-step method is not used and the SR requirement is determined by minimising (1) using EENS (9), too many binary variables are introduced and the system will run out of the memory soon. The model is coded in GAMS [21] and solved using a largescale MILP solver CPLEX 11.2 combined with Visual C. The CPU solution time was recorded on a Windows-based server with 2.6-GHz processors and 3.3 G bytes of RAM.

6

Fig. 6 Variations of SR requirement under different system sizes 778 & The Institution of Engineering and Technology 2011

Conclusions

In this paper, a new method is proposed to estimate the SR requirement. The optimal SR amount is determined by minimising the operating cost, reserve cost and EIC. In order to overcome the computation intractability introduced by EENS, a multi-step method is proposed and EENS is formulated as a piecewise linear function of the SSR. The parameters of the piecewise linear function are derived from COPT which is established based on the results of last step. IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772 –780 doi: 10.1049/iet-gtd.2010.0805

www.ietdl.org The proposed approach has been successfully tested using the IEEE-RTS system. Acceptable SR requirements can be obtained within reasonable run time.

7

9 Wang, J.X., Wang, X.F., Wu, Y.: ‘Operating reserve model in the power market’, IEEE Trans. Power Syst., 2005, 20, (1), pp. 223 –229 10 Wu, L., Shahidehpour, M., Li, T.: ‘Cost of reliability analysis based on stochastic unit commitment’, IEEE Trans. Power Syst., 2008, 23, (3), pp. 1364– 1374 11 Bouffard, F., Galiana, F.D., Conejo, A.J.: ‘Market-clearing with stochastic security—Part I: formulation’, IEEE Trans. Power Syst., 2005, 20, (4), pp. 1818– 1826 12 Ortega-Vazquez, M.A., Kirschen, D.S., Pudjianto, D.: ‘Optimising the scheduling of spinning reserve considering the cost of interruptions’, IEE Proc., Gener. Transm. Distrib., 2006, 153, (5), pp. 570– 575 13 Ortega-Vazquez, M.A., Kirschen, D.S.: ‘Optimizing the spinning reserve requirements using a cost/benefit analysis’, IEEE Trans. Power Syst., 2007, 22, (1), pp. 24– 33 14 Galiana, F.D., Bouffard, F., Arroyo, J.M., Restrepo, J.F.: ‘Scheduling and pricing of coupled energy and primary, secondary, and tertiary reserves’, IEEE Proc., 2005, 93, (11), pp. 1970–1983 15 Kariuki, K.K., Allan, R.N.: ‘Evaluation of reliability worth and value of lost load’, IEE Proc., Gener. Transm. Distrib., 1996, 143, (2), pp. 171–180 16 Fotuhi-Firuzabad, M., Billinton, R., Aboreshaid, S.: ‘Spinning reserve allocation using response healthy analysis’, IEE Proc., Gener. Transm. Distrib., 1996, 143, (4), pp. 337–343 17 Venkatesh, B., Yu, P., Gooi, H.B., Choling, D.: ‘Fuzzy MILP unit commitment incorporating wind generators’, IEEE Trans. Power Syst., 2008, 23, (4), pp. 1738– 1746 18 Wang, C., Shahidehpour, S.M.: ‘Effects of ramp-rate limits on unit commitment and economic dispatch’, IEEE Trans. Power Syst., 1993, 8, (3), pp. 1341–1350 19 Grigg, C., Wong, P., Albrecht, P., et al.: ‘The IEEE reliability test system – 1996’, IEEE Trans. Power Syst., 1999, 14, (3), pp. 1010– 1018 20 Floudas, C.A.: ‘Nonlinear and mixed-Integer optimization: fundamentals and applications’ (Oxford University Press, New York, USA, 1995, 1st edn.) 21 Brooke, A., Kendrick, D., Meeraus, A., Raman, R.: ‘GAMS: a user’s guide’ (GAMS Development Corp., Washington D.C., 1998)

Acknowledgments

Financial support from the Intelligent Energy Distribution Systems Project (No. 072133 0038), Agency for Science, Technology, and Research (A∗ STAR), Singapore is gratefully acknowledged. The authors would also like to express their appreciation to the Nanyang Technological University, Singapore, for the support of research student scholarship.

8

References

1 Rebours, Y., Kirschen, D.S.: ‘A survey of definitions and specifications of reserve services’, available at http://www.eee.manchester.ac.uk/research/ groups/eeps/publications/reportstheses/aoe/rebours%20et%20al_tech%20 rep_2005B.pdf, accessed 2005 2 Rebours, Y., Kirschen, D.S.: ‘What is spinning reserve’, available at http://www.eee.manchester.ac.uk/research/groups/eeps/publications/reports theses/aoe/rebours%20et%20al_tech%20rep_2005A.pdf, accessed 2005 3 Billinton, R., Allan, R.N.: ‘Reliability evaluation of power systems’ (Plenum Press, New York, London, 1996, 2nd edn.) 4 Gooi, H.B., Mendes, D.P., Bell, K.R.W., Kirschen, D.S.: ‘Optimal scheduling of spinning reserve’, IEEE Trans. Power Syst., 1999, 14, (4), pp. 1485– 1490 5 Chattopadhyay, D., Baldick, R.: ‘Unit commitment with probabilistic reserve’. IEEE Power Engineering Society Winter Meeting, New York, USA, January 2002, pp. 280– 285 6 Bouffard, F., Galiana, F.D.: ‘An electricity market with a probabilistic spinning reserve criterion’, IEEE Trans. Power Syst., 2004, 19, (1), pp. 300–307 7 Simopoulos, D.N., Kavatza, S.D., Vournas, C.D.: ‘Reliability constrained unit commitment using simulated annealing’, IEEE Trans. Power Syst., 2006, 21, (4), pp. 1699–1706 8 Aminifar, F., Fotuhi-Firuzabad, M., Shahidehpour, M.: ‘Unit commitment with probabilistic spinning reserve and interruptible load considerations’, IEEE Trans. Power Syst., 2009, 24, (1), pp. 388– 397

Table 2

Appendix

The unit output power and reserve are shown in Tables 2 and 3.

Unit output power (MW) of every UC hourly period

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

9

i 1

2

3

10

11

12

13

14

15

16

17

18

19

20

21

22

24

25

26

0 0 0 0 0 0 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 0

0 0 0 0 0 0 0 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 0

0 0 0 0 0 0 0 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 2.4 0

15.2 15.2 15.2 15.2 15.2 51.6 57.4 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 55.8

15.2 15.2 15.2 15.2 15.2 35.5 56.8 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 55.7

15.2 15.2 15.2 15.2 15.2 35.5 56.8 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 65.1 55.7

15.2 15.2 15.2 15.2 15.2 35.5 56.8 76 76 76 76 76 76 76 76 76 76 76 76 76 76 76 56.8 0

0 0 0 0 0 0 0 75 75 78.8 99.8 78.8 78.8 78.8 80.2 79.8 78.8 75.5 78.8 78.8 75 80.9 74.5 0

0 0 0 0 0 0 0 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 74.5 0

0 0 0 0 0 0 0 75 75 75 75 75 75 75 75 75 75 75 75 75 75 75 0 0

127.5 136.7 127.5 127.5 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 140.3

127.5 127.5 127.5 127.5 129.2 132 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 127.5

127.5 127.5 127.5 127.5 127.5 127.5 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 127.5

127.5 127.5 127.5 127.5 127.5 127.5 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 155 127.5

0 0 0 0 0 0 0 123.8 122.2 154.3 169.5 154.3 154.3 128.4 154.3 169.5 128.4 111.6 111.6 128.4 154.3 168 0 0

0 0 0 0 0 0 0 0 111.6 135.7 169.5 125.7 125.7 111.6 154.3 169.5 111.6 111.6 78.4 111.6 139.5 0 0 0

329.2 350 319.2 329.2 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350 350

400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400

400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400 400

IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772– 780 doi: 10.1049/iet-gtd.2010.0805

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www.ietdl.org Table 3

Unit reserve (MW) of every UC hourly period

t

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24

i 1

2

3

10

11

12

13

14

15

16

17

18

19

20

21

22

24

25

26

0 0 0 0 0 0 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 0

0 0 0 0 0 0 0 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 9.6 0

0 0 0 0 0 0 0 9.6 5.8 9.6 9.6 9.6 9.6 9.6 8 9.6 9.6 9.6 9.6 9.6 5.8 9.6 9.6 0

19.3 19.3 19.3 19.3 19.3 19.3 18.7 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19.3

19.3 19.3 19.3 19.3 19.3 19.3 19.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19.3

19.3 19.3 19.3 19.3 19.3 19.3 19.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 11 19.3

19.3 19.3 19.3 19.3 19.3 19.3 19.3 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 19.3 0

0 0 0 0 0 0 0 25 25 21.2 0.21 21.2 21.2 21.2 19.9 20.2 21.2 24.5 21.2 21.2 25 19.2 25.5 0

0 0 0 0 0 0 0 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25.5 0

0 0 0 0 0 0 0 25 25 25 25 25 25 25 25 25 25 25 25 25 25 25 0 0

27.5 18.3 27.5 27.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14.75

27.5 27.5 27.5 27.5 25.8 23 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 27.5

27.5 27.5 27.5 27.5 27.5 27.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 27.5

27.5 27.5 27.5 27.5 27.5 27.5 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 27.5

0 0 0 0 0 0 0 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 0 0

0 0 0 0 0 0 0 0 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 27.5 0 0 0

20.8 0 30.8 20.8 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 0 0 0 0 0

780

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IET Gener. Transm. Distrib., 2011, Vol. 5, Iss. 7, pp. 772 –780 doi: 10.1049/iet-gtd.2010.0805