Congestion management enhancing transient stability of power systems

Applied Energy 87 (2010) 971–981

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Congestion management enhancing transient stability of power systems Masoud Esmaili a,*, Heidar Ali Shayanfar a, Nima Amjady b a b

Centre of Excellence for Power System Automation and Operation, Iran University of Science and Technology, Tehran, Iran Department of Electrical Engineering, Semnan University, Semnan, Iran

a r t i c l e

i n f o

Article history: Received 1 June 2009 Received in revised form 26 September 2009 Accepted 28 September 2009 Available online 25 October 2009 Keywords: Congestion management Power market Corrected transient stability function Transient stability margin Security cost

a b s t r a c t In a competitive electricity market, where market parties try to maximize their profits, it is necessary to keep an acceptable level of power system security to retain the continuity of electricity services to customers at a reasonable cost. Congestion in a power system is turned up due to network limits. After relieving congestion, the network may be operated with a reduced transient stability margin because of increasing the contribution of risky participants. In this paper, a novel congestion management method based on a new transient stability criterion is introduced. Using the sensitivity of corrected transient stability margin with respect to generations and demands, the proposed method so alleviates the congestion that the network can more retain its transient security compared with earlier methods. The proposed transient stability index is constructed considering the likelihood of credible faults. Indeed, market parties participate by their security-effective bids rather than raw bids. Results of testing the proposed method along with the earlier ones on the New-England test system elaborate the efficiency of the proposed method from the viewpoint of providing a better transient stability margin with a lower security cost. Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved.

1. Introduction A competitive electricity market is operated with the bids submitted by market participants to the Independent System Operator (ISO). In such a competitive environment, all participants including generators and demands try to maximize their profit using available bidding strategies [1]. This trend leads to a more stressed power system. That is, the security margin of power systems may be reduced in such a circumstance. Recent major blackouts in North America and Europe [2] have remembered us the importance of security requirements of power systems, a matter that has been regretfully neglected in favor of more financial concerns in recent years. It is the responsibility of ISO to keep an acceptable level of security in the power system fairly for the continuity of electricity services to customers [3]. The transmission network, as a medium between generations and consumptions, has not only a limited capacity but also some security concerns. Congestion in an electricity market occurs when the transmission network is unable to accommodate all of the desired transactions due to a violation in power system operating limits. Especially, in systems having weak connections among different areas, the congestion problem frequently occurs due to overloading or security requirements. Usually, there is a spot market for short term electricity transactions [4]. In the spot market, market participants submit their * Corresponding author. Tel.: +98 21 88891890; fax: +98 21 88731293. E-mail address: [email protected] (M. Esmaili).

next-day hourly generation or demand bids to ISO. The ISO anticipates the demand level in electricity markets using load forecasting methods [5]. Generation companies also forecast their future generation using available methods for a given number of hours [6]. With submitted bids from participating generators and demands, the ISO clears the energy market using available methods [7] to schedule the powers and determine the Market Clearing Price (MCP) [8]. In case of congestion, ISO employs a congestion management method to make possible transactions as much as possible. Rescheduling generations and demands is an effective tool to relieve congestions assuming that system configuration has already been set. Generators and demands participate in the congestion management by bidding for up and down their production or demands, respectively. While choosing generators or demands for re-dispatching, the least cost option is picked up to minimize total rescheduling cost or to maximize social benefit. After rescheduling, the network is operated with no violation or congestion. It is a common practice to use the DC model for congestion management [9]. In the DC congestion management method, sometimes called the classical congestion management model, some assumptions are made to get a simple model. All voltage magnitudes are considered one per unit. Also, all network impedances are assumed to be pure reactance without any resistive part. Besides, reactive powers are ignored and only active powers are considered. The AC model, based on the AC power flow equations and more accurate than the DC congestion model, is also used to

0306-2619/$ - see front matter Crown Copyright Ó 2009 Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.apenergy.2009.09.031

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relieve the congestion [9]. However, neither the DC nor AC congestion management method considers the stability margin of power systems. In view of the fact that congestion management is a mathematical nonlinear optimization problem satisfying a set of constraints, some variables will hit their limits. For instance, a few branches can be fully loaded; or, some voltages can be set at their lower limit; or, the generation of critical generators can be increased. Although there is no violation, the system may be vulnerable against disturbances. In other words, the stability margin of the network may be reduced after relieving congestion. In such a network, if any potential instability due to a prospective contingency is expected especially in vulnerable areas of the network, some preventive actions, e.g. load shedding or startup of new units, have to be done to retain the system security [10]. Therefore, it is essential to mitigate congestion by means of a method so that the stability margin of the network is retained. Rotor angle stability refers to the ability of synchronous machines of an interconnected power system to remain in synchronism after being subjected to a disturbance [11]. Large disturbance rotor angle stability, as commonly referred to transient stability, is concerned with the ability of the power system to maintain synchronism when subjected to a severe disturbance, such as a short circuit on a transmission line. Instability is usually in the form of non-periodic angular separation due to insufficient synchronizing torque, manifesting as first swing instability. Recently, some techniques are presented for congestion management considering transient stability. Efficiency of these techniques depends on whether they can include the detailed generator models with exciter and governor. Furthermore, the application of some techniques is limited to only the power system DC model, ignoring power system details compared with the AC model. Generation rescheduling has served as an efficient tool to enhance transient stability. In [12], a method using trajectory sensitivity is proposed to reschedule power generation to ensure system transient stability for a set of credible contingencies while satisfying the economical goal. However, its instability criteria are only based on generators’ angle differences. In [13], the re-dispatch of generations is undertaken if Transient Energy Margin (TEM) is inadequate after congestion management. The approach, handling the power system DC model, is based on the classical simple model of generators. In [14], a structure preserving energy margin sensitivity-based analysis is presented to determine the amount of preventive generation rescheduling to stabilize a transiently unstable power system. An expression using a simplified model is derived to relate the change in the energy margin to change in generation. In [15], congestion management is so done that the desired enhancement in the transient stability margin is achieved through shifting power from critical generators to noncritical ones using a sensitivity analysis. The contribution of this paper is to relieve congestion providing a sufficient level of transient stability margin. The proposed transient stability index is constructed considering credible faults with their probabilities. Also, in addition to generators, demands are so participated into the congestion market that transient stability is retained. As a fact, different loads and generators have their own effects on the transient stability margin. For instance, if vulnerable generators of the network were known, the system operator could more take care of generation increment at these units than others. The method proposed in this paper provides a mechanism by means of which the system operator reduces the contribution of risky market participants from the transient security viewpoint even if their bids are more cost-effective. Indeed, market parties are participated into the congestion management by their security-effective bids rather than their raw bids. Some earlier methods [15] were introduced to reduce the participation

of critical generators in the congestion management. To make a comparison, results of the proposed method are compared in detail with those of the earlier methods in the section of Numerical results. The proposed congestion management is discriminated from Security Constrained Unit Commitment (SCUC) and Security Constrained Optimal Power Flow (SCOPF) in diverse aspects although all of them are optimization problems. Some electricity markets rely on SCUC for security constrained market-clearing, while some others clear the market and then alleviate the possible congestions based on a congestion management tool. It depends on the market design and some examples can be found in [9],[16]. Another important difference between SCUC and congestion management is their time frameworks. While SCUC usually considers day-ahead horizon, the time framework for congestion management is usually one hour, as congestion-relieving actions are considered hour by hour [9]. As the market is getting closer from day-ahead cleared results to real time operation, some changes such as generator tripping or transmission branch outages may be observed. These changes after market-clearing can be considered into the congestion management procedure. Another important discrimination between the proposed congestion management procedure and both SCUC and SCOPF frameworks is that the most of previous SCUC and SCOPF methods only consider static security constraints (such as branch flow limits), while the proposed congestion management procedure includes dynamic security constraints in addition to static security constraints. For this purpose, the new concept of TSM is introduced in this research work. Moreover, many of previous SCUC and SCOPF methods are based on the DC network model, while AC network model is included in the proposed congestion management procedure. The remaining parts of the paper are organized as follows. In Section 2, the transient stability margin is illustrated as a quantitative measure indicating how much the system is transient stable. In Section 3, the proposed method is elaborated. In Section 4, numerical results of testing both the earlier and the proposed method on a test system are presented and compared thoroughly. Section 5 concludes the paper.

2. Transient stability margin Although the analytical sensitivity approach has been developed for transient stability margin analysis, it can only be used for systems represented by classical models [17]. To handle a detailed generator representation with exciters and nonlinear load models, hybrid methods were developed for on-line transient stability assessment. In these methods, a time-domain simulation is first carried out beyond fault clearing and then a TEM, called hybrid TEM, is computed using the concept of potential energy boundary surface (PEBS) crossing [17]. However, it is observed that the variation of hybrid TEM versus some system key parameters often exhibits erratic non-linearity around its critical value. This phenomenon might make the results either highly optimistic or conservative. To correct the hybrid TEM, the definitions of the Corrected Transient Kinetic Energy (CTKE) and Transient Potential Energy (TPE) are used. The CTKE is the part of Transient Kinetic Energy (TKE) that contributes to a system separation leading to instability. However, the sum of CTKE and TPE along post-fault trajectories is not of conservation. To solve this, the corrected TPE (CTPE) function and corrected potential energy boundary surface (CPEBS) are proposed and used for the definition of Corrected hybrid TEM (CTEM). The sum of CTKE and CTPE function is defined as the Corrected Transient Energy Function (CTEF). Theoretically, it is proved that CTEF is of conservation during post-fault transient period and there is

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no erratic non-linearity exhibited on the variation of CTEM both for plant and inter-area mode disturbances [17]. The motion between the set of advanced generators A and the set of non-advanced generators B with respect to Center Of Inertia (COI) can be written as [18]: 

M eq xAB ¼

nA nB M eq X Meq X ðPmi  Pei Þ  ðPmi  P ei Þ  PAB M A i¼1 M B i¼1

ð1Þ

where nA X

M i xi =M A ;

xB ¼

nB X

i¼1

MA ¼

nA X

Mi xi =M B ;

xAB ¼ xA  xB

ð2Þ

i¼1

Mi ;

MB ¼

i¼1

nB X

Mi ;

M eq ¼ M A M B =ðMA þ MB Þ

ð3Þ

i¼1

where Mi is the inertia constant of machine i, xi the speed of machine i with respect to COI, Pmi the mechanical power input of machine i, and Pei is the electrical power output of machine i. Then, CTKE is given by:

CTKE ¼

1 M eq x2AB 2

ð4Þ

As shown above, xAB is the speed difference between two sets of advanced and non-advanced generators. Also, CTKE is the kinetic energy resulted from the speed difference. PAB as given by (1) is the accelerating power between two sets of advanced and non-advanced generators. A positive PAB implies a positive derivative of xAB or an increasing angular separation and vice versa. To introduce the proposed transient stability margin, consider Fig. 1, wherein fault is applied at S0. If the fault is cleared at time tA, the system goes on the stable trajectory of TRA ending to a new stable equilibrium point of SA. If the fault is cleared at time tB, the system going on the instability boundary trajectory of TRB is still stable (marginally stable). The CTKE of the trajectory B, called as the critical trajectory, is the maximum value that the power system can absorb without losing transient stability. In the CTEM method, the fault is reinserted at the point of PA, where xAB = 0. After reinserting the fault, system goes on the trajectory TRF, making the system unstable at the point of PB crossing CPEBS. CPEBS is boundary surface where the accelerating power changes from negative to positive values. CTEM is defined as the CTPE change along trajectory TRF from PA to PB [17]. On the other hand, for transient stable cases, transient stability margin (TSM), which is proposed in this paper, is defined as the

1. Monitor xAB after clearing fault. The post-fault system is stable if the sign of xAB changes from positive to negative at the end of the first swing. Otherwise, the system is unstable and xAB passes a positive minimum at the end of the first swing. 2. For a stable case, TSM is defined as the distance of the first swing CTKE from the critical first swing CTKE. In this situation, TSM is positive and represents the transient stability margin. 3. For an unstable case, TSM is defined as the minus of CTKE at the moment of passing CPEBS (exit point). For this situation, TSM is negative and represents the transient instability depth. It should be noted that both CTEM and proposed TSM are quantitative indices of system transient security. In treating unstable 100 Maximum CTKE that can be absorbed

90

PAB=0

PB E

80

TSM

TRF

TRC

CPEBS

TSM

70 First swing CTKE peak

60

PA tC

TRB

CTKE

xA ¼

CTKE distance between the two trajectories TRA and TRB as shown in Fig. 1. For clearing times beyond tB, the system will be transient unstable as shown by trajectory TRC. For a first swing unstable post-fault, the post-fault trajectory crosses CPEBS as shown in the figure. The crossing point called exit point indicated by E in Fig. 1. For unstable post-fault trajectories, the proposed TSM is defined as the minus CTKE at the exit point. It is noted that for transient unstable cases, the proposed TSM indicates instability depth. After crossing CPEBS, the separation between advanced and nonadvanced generators is more continued due to a positive accelerating power. For a better illustration of the proposed TSM, its representation in the time-domain simulation is shown in Fig. 2. An individual power system is first swing stable as long as it is able to absorb the CTKE and convert it to the potential energy [15]. The first swing peak of the CTKE increases by increasing fault duration. The system can retain its stability provided that the fault clearing time is less than the critical clearing time (tB in Fig. 1), leading to the critical first swing peak of the CTKE. Thus, the system can absorb kinetic energy as much as the critical CTKE without losing transient stability. This concept is shown in Fig. 2. In this figure, the fault is applied at t = 1 s and cleared at t = 1.2 s. As depicted in Fig. 2, TSM is defined as the distance between the first swing CTKE peak and the critical first swing CTKE peak. It is noted that the term of first swing refers to the first swing of difference between the COI angles of the advanced and non-advanced generator groups [17]. A larger TSM makes the power system be more robust from transient stability viewpoint. Calculation of the proposed TSM for transient stable and unstable cases can be summarized as the following algorithm:

50 40

tB

30

TRA

20

tA

10

SB Fault applied S0

SA

Fig. 1. Different trajectories for transient stability.

0

0

0.5

1

1.5

2

2.5

3

3.5

Time (Sec.) Fig. 2. A typical first swing CTKE and TSM.

4

4.5

5

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cases, both are defined as the minus of CTKE at the moment of CPEBS crossing. However, they treat differently for stable cases. In the CTEM, after reinserting the fault at the point of PA, CTEM is calculated as the integral of PAB from fault insertion to CPEBS crossing point. However, the TSM is defined as the CTKE margin between the current and critical trajectories. To compare CTEM and TSM, CTEM relies on a fictitious trajectory of reinserted fault to calculate the energy margin, while TSM is based on the real margin of the current trajectory from CPEBS boundary; this is what happens in a real power system. Besides, calculation of CTEM for stable cases using the integral of PAB from fault insertion to exit point requires high computational burden when detailed generator models with exciters are included. This makes it less efficient in on-line applications. Furthermore, this integral is path dependent for a power network model with transmission losses [17]. On the other hand, the proposed TSM is independent from CTPE and does not involve with the computation of the path dependent integral. The TSM is useful as long as it bears a linear relationship with respect to operating parameters such as pool generation shift in congestion management. This linearity is examined in the section of Numerical results.

tively. SG indicates the set of accepted generators in the electricity down are used market. It should be noted that the same DP up Gj and DP Gj for all generation units. Dividing the Eqs. (5) and (6) by that of the base case TSM yields: up

@TSM TSMGj =TSMBC  1  DPup @Pup Gj Gj =TSMBC @TSM @Pdown Gj

qup ij ¼

3.1. The sensitivity of transient stability margin

@TSM @Pdown Gj



DTSM DPdown Gj

¼

j 2 SG

TSMdown  TSMBC Gj

DPdown Gj

j 2 SG

ð5Þ

ð6Þ

down where TSMBC represents the base case TSM. Also, DPup Gj and DP Gj are the applied perturbations to generator j active power generation down are the in up and down directions, respectively. TSMup Gj and TSMGj power system TSM after the up and down perturbations, respec-

j 2 SG

DPdown =TSMBC Gj

TSMup Gj TSMBC

;

qdown ¼ ij

ð8Þ

TSMdown Gj TSMBC

i 2 SF; j 2 SG

ð9Þ

In (9), qij indicates sensitivity factor for generator j considering fault i. SF indicates the set of credible faults. A value of sensitivity greater than one implies that the transient stability margin of the network is improved with the change. In contrast, a value less than one means a decrease in system security with the change. It is noted that the effect of all credible faults should be included into the sensitivity factor. Considering n credible faults and m generators participating in congestion management, the matrix of sensitivity factors for both directions of generation change can be written as:

0

1

0

.. C C . A;

B SMdown ¼ B @

qup . . . qup 11 1m

B . SM up ¼ B @ ..

..

.

qup    qup nm n1

To derive the sensitivity of the proposed TSM with respect to each generation or load, the first order approximation of the Taylor series around the operating point of the power system is used. In other words, each sensitivity factor is approximated by its first order term. The index that is used in the proposed method must have a linear behavior in order to yield enough accuracy with its sensitivities. As it is shown in the section of Numerical results, the TSM bears linearity with respect to power shifts in a wide range. Since there is no analytical expression for TSM while detailed models of generators and loads are considered, to numerically compute the sensitivity of TSM with respect to the generation j for a given fault i, the change in TSM is calculated after applying a small perturbation in the generation. The sensitivity factors for generation change in both directions are obtained as: up @TSM DTSM TSMGj  TSMBC  ¼ DPup DPup @Pup Gj Gj Gj

TSMdown =TSMBC  1 Gj

ð7Þ

The purpose of introducing TSM sensitivities in this paper is to rank generators from the viewpoint of their effect on transient stability. Eqs. (7) and (8) consist of three terms as TSMup Gj =TSMBC ; 1, for (7) and TSMdown =TSMBC ; 1, and and DPup Gj Gj =TSMBC DPdown =TSMBC for (8). Considering the same perturbation for all Gj generators, the second and third terms in both (7) and (8) are constant for all generators and do not contribute to discriminating the generators from transient stability viewpoint. Therefore, to derive a more tangible index, we can retain only the variable terms and omit the constant terms. The resulted sensitivities for a given fault become as:

3. The proposed congestion management method The proposed congestion management is implemented here with the sensitivities of TSM as a simple and concrete index. However, it can also be implemented with other dynamic stability indices like CTEM. The transient stability index of the proposed method is constructed considering all credible faults with their likelihood. The proposed method can handle various types of faults that occur and are cleared after a duration period. For instance, they include faults on transmission lines, transformers, and generators. The fault is cleared as the result of acting protective relays by opening the line, transformer, or generator step-up transformer. The powers are so rescheduled that the transient stability of the system is more retained than the previously presented congestion management methods.



j 2 SG

qdown . . . qdown 11 1m .. .

..

.

.. .

1 C C A

qdown    qdown n1 nm ð10Þ

To define a unique sensitivity factor for generator j as an overall sensitivity including n credible faults, column j of the matrices are combined using the weighting factor of faults as:

SF up Gj ¼

n X

W i  qup ij ;

SF down ¼ Gj

i¼1

n X

W i  qdown ; ij

i¼1

n X

Wi ¼ 1

i¼1

ð11Þ The weighting factors of Wi are determined by contingency ranking based on the severity and likelihood of the credible faults [19]. In down are the overall sensitivity of the TSM consider(11), SF up Gj and SF Gj ing all credible faults with respect to change in generation j in up and down directions, respectively. In a power market with demand side bidding, demands like generators can participate into the congestion management market. Similar to generators, the TSM overall sensitivity with respect to demand k can be driven as:

SF up Dk ¼

n X

W i  qup ik ;

i¼1

SF down ¼ Dk

n X i¼1

W i  qdown ; ik

n X

Wi ¼ 1

i¼1

ð12Þ SF up Dk

SF down Dk

where and show the overall sensitivity of TSM considering all credible faults with respect to change in demand k active power for up and down directions, respectively.

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M. Esmaili et al. / Applied Energy 87 (2010) 971–981 up down PDk ¼ PMC Dk þ DP Dk  DP Dk ;

3.2. The proposed method The proposed method employs the AC model of power system, which is more accurate than the DC model. Furthermore, there is no limitation for the degree of detail in modeling power system components such as generators in time-domain simulation. The proposed method mitigates congestion minimizing the following objective function:

Bup Gj SF up Gj

X

Minimize :

j2SG

DPup Gj þ

Bdown Gj SF down Gj

!

DPdown Gj

! Bup Bdown up down Dk Dk D P þ D P Dk Dk SF up SF down Dk k2SD Dk  X VOLLDk DPLS þ j 2 SG; k 2 SD; Dk ;

þ

X

ð13Þ

k2SD

where the cost consists of three parts. The two first parts are the payment that the system operator pays to generators and demands to alter their powers as per their bid. The third part is related to the payment of involuntary load shedding that the ISO may apply to loads to manage the congestion in some difficult cases. SG and SD are the set of in service generators and demands, respectively. Bup Gj and Bdown are the bid prices of generator j to increase and decrease Gj up down its power to relieve congestion, respectively. Also, DPGj and DP Gj are up and down generation shifts of unit j that will be determined down by the congestion management procedure. Similarly, Bup Dk , BDk , down DPup , and D P are analogous parameters of demand side bidding. Dk Dk Also, DP LS Dk and VOLLDk are the amount of involuntary load shedding and the Value Of Lost Load (VOLL), respectively. The VOLL, paid to demands for the load shedding, depends on the power market policy and is usually much higher than the bids offered by demands to participate in the congestion management market. The optimization is solved subject to following constraints: max Pmin Gj 6 P Gj 6 P Gj ;

j 2 SG

max Q min Gj 6 Q Gj 6 Q Gj ; max Pmin Dk 6 P Dk 6 P Dk ;

Q Dk ¼ PDk tanð/Dk Þ; PGn  PDn ¼ jV n j

ð14Þ

j 2 SG

ð15Þ

k 2 SD

ð16Þ

k 2 SD

X

ð17Þ

jY n;h jjV h j cosðdn  dh  hn;h Þ;

n 2 SN

ð18Þ

n 2 SN

ð19Þ

h2SN

Q Gn  Q Dn ¼ jV n j

X

jY n;h jjV h j sinðdn  dh  hn;h Þ;

h2SN

PGn ¼

X

PGj ;

n 2 SN

ð20Þ

j2SGn

Q Gn ¼

X

Q Gj ;

n 2 SN

ð21Þ

P Dk ;

n 2 SN

ð22Þ

j2SGn

PDn ¼

X k2SDn

Q Dn ¼

X

Q Dk ;

n 2 SN

ð23Þ

k2SDn

6 V n 6 V max ; V min n n jSm ðV; dÞj 6 Smax m ;

n 2 SN

ð24Þ

m 2 SB

up down PGj ¼ PMC ; Gj þ DP Gj  DP Gj

ð25Þ j 2 SG

ð26Þ

LS PLS Dk ¼ P Dk  DP Dk ; max 0 6 P LS Dk 6 P Dk ;

k 2 SD

k 2 SD

ð28Þ

k 2 SD

down down DPup ; DPup P 0; Gj ; DP Gj Dk ; DP Dk

ð27Þ

ð29Þ j 2 SG; k 2 SD

ð30Þ

In the above equations, SN and SB are the set of nodes and branches of the power system, respectively. Also, SD is the set of demands. Also, SGn and SDn indicate the set of on-line generators and demands connected to bus n, respectively. Eqs. (14) and (15) set active and reactive power limits for generators, respectively. Of course, reactive limits should have been determined using generator capability curve. However, it is common to use constant values for generator reactive limits in congestion management applications [20]. This is due to the fact that the power rescheduling of generators in congestion management is usually performed in a narrow band around the market-clearing results, and then, we can approximately assume constant reactive limits. Eq. (16) sets active power limits for demands. Eq. (17) expresses reactive power constraints for demands; it relates active and reactive powers of demands considering a constant power factor [20]. Eqs. (18)–(23) represent the nodal power balance for active and reactive powers. Voltage magnitude limits for all buses are set by Eq. (24). The thermal rating of branches is limited by (25) in apparent power (MVA). MC In (26) and (27), PMC Gj and P Dk represent the active power generation and consumption determined by the market-clearing procedure, respectively. Also, PGj and PDk are final rescheduled active powers of generators and loads due to congestion management, respectively. In (28), P LS Dk gives demand k final rescheduled power after relieving congestion considering the effect of both demand side bidding and involuntary load shedding (DP LS Dk ). Eq. (29) sets the limits of P LS Dk . Finally, (30) confines all up and down power changes due to congestion management to positive values. It is worthwhile to note that the objective function as formulated by (13) uses the effective bids of generators and demands up up (such as Bup Gj =SF Gj ) rather than raw bids (such as BGj ). For instance, suppose that all generators bid the same price to increase their generation. However, the generators have different SF up Gj sensitivities and so differently affect the TSM. Since the system operator uses effective bids, generators with a smaller sensitivity value will result in a greater effective bid. As a result, the participation of generations/demands, which reduce the network TSM, decreases. Likewise, by applying this mechanism, the participation of loads and generations improving the stability margin will increase. This finally leads to more improvement of the network transient stability margin than the previous methods. Specially, this mechanism is more useful when the power system is operated near its instability boundary, where critical generators and demands with low sensitivity factors will have large effective bids and this reduces their participation into worsening transient stability. The underlying idea of the proposed congestion management framework is to mitigate congestion with maintaining the dynamic security of the power system as much as possible. For this purpose, at first a new and efficient criterion, i.e. TSM, is proposed to evaluate the transient stability status of the power system. Based on the idea, the participation of the risky participants, which can deteriorate the dynamic security status of the system (or equivalently decrease the TSM), should be limited as much as possible. So, the bids of the risky participants should be penalized considering the amount of their negative effect on the TSM. A simple way is to add penalty factors to the bids of these participants. However, the correct tuning of these penalty terms for each power system is complex and even ambiguous. Moreover, these penalty factors

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depend on the power system operating conditions. For instance, the values of these penalty factors near the stability boundary should be larger than their values in a secure operating point. The quotient combination proposed here to combine the bid prices and sensitivity factors can satisfactorily adjust bids considering transient stability status. Based on this combination, a perturbation in a critical operating point (near the stability boundary) affects the bids much more than a secure operating point far from the stability border. It is noted that TSMBC appears in the denominator of the sensitivity factors shown in (9). For a better illustration, consider this simple example. Suppose two operating points including one critical point near the stability boundary with TSMBC = 0.5 Per Unit (pu) and the other secure point far from the stability boundary with TSMBC = 5 pu. Also, suppose that a same perturbation is applied to both these operating points, approximately decreasing the TSM by 0.1 pu. For the critical operffi ð0:5  0:1Þ=0:5 ¼ 0:8 pu, while for ating point, we obtain qdown ij ffi ð5  0:1Þ=5 ¼ 0:98 pu. So, the the secure point we have qdown ij proposed sensitivity factor penalizes the bid of the risky participant near the stability boundary more than a secure operating point, since its bid is divided by 0.8 and 0.98, respectively (or equivalently increases by 1/0.8 = 1.25 and 1/0.98 = 1.02, respectively). This is in accordance with the ISO needs, because in the critical operating points, more attention should be paid to the security level of the power system than the secure operating points. In other words, the proposed sensitivity factor adaptively penalizes the bids of risky participants considering the dynamic security status of the power system. Previously introduced methods [15] mainly shift power generation from advanced to non-advanced generators to enhance dynamic security in congestion management. In view of the fact that the advanced generators are varied with the fault, shifting generation power will be ambiguous when multiple faults should be considered. The proposed method is advantageous over the earlier ones from some points. The first point is that it is able to consider all credible faults by combing all of them into one set of sensitivity indices. Then, it takes into account all credible faults, based on their weight factor obtained from the contingency ranking technique [19], rather than a single one. The second point is that the proposed method is able to consider the effect of not only

generators but also demands on the dynamic stability, while the previous methods consider only generators. 4. Numerical results The proposed method along with the previous methods is examined on the New-England power system, a well-known test system with 39 buses, 34 lines, 2 shunt capacitors, 12 transformers, 19 loads, and 10 machines. The single-line diagram of the network is depicted in Fig. 3. Data of the test system can be obtained from [21]. In addition to the data, the rating of branches 1–39, 2–3, and 26–27 is set to 100 MVA in this paper. Also, market data for this test system are listed in Appendix. In simulations performed in the paper, all loads are considered as constant impedance. Also, generator reactive limits are taken into account. The dynamic behavior of the test system is investigated using the time-domain simulation of PSS/E 30 software [22], a powerful package developed by the PTI Inc. USA. To get more accurate results, the integration time step is reduced to 0.1 ms. To determine the transient stability behavior, a credible fault (n = 1), i.e. a short circuit on bus 26, is selected here, which is also considered in [15]. The fault is applied at t = 1 s and cleared at tcl = 1.1 s with the duration of 100 ms by the tripping of line 25– 26. In this situation, generators at buses 31 through 38 are determined as critical ones while generators at buses 30 and 39 are non-critical. To determine the critical CTKE that the power system can absorb without losing stability, the fault clearing time is increased as long as the system keeps synchronism and remains stable. The TSM is defined as the distance of the first swing CTKE peak from the critical absorbable CTKE. Using the base case configuration, there are some overloaded branches. Lines 2–3, 1–39, and 26–27 are overloaded to 386.7%, 152.4%, and 285.3% of their rating, respectively. Thus, the system operator has to employ one of the congestion management methods to mitigate the branch overloads. Though all methods relieve the congestion and make feasible power transactions, the network experiences a different post-rescheduling status from the transient stability viewpoint using each method. Depending on the direction of the sensitivity of congested branch flow to generations, increas-

G

37 26

25

G

28

29

30 38

2 24

27

18

G

17

1 16

3

G

35 15

G

21

39

22

4 14

5 6 7 8 9

31

12

19

23

20

11

36

13 33

10

G

34 G

32

G

G

Fig. 3. The single-line diagram of the New-England test system.

G

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M. Esmaili et al. / Applied Energy 87 (2010) 971–981

where, km?n is the sensitivity of CTEM to shift power generation from a critical generator m to a non-critical generator n. A positive

value for km?n indicates that the CTEM is improved by the generation shift. The optimization problem is solved ensuring the desired overall enhancement in CTEM composed of individual generation shifts after congestion management. As seen in Table 1, the CTEM is much improved if generation is shifted from Gen. 38 to Gen. 39. It can be seen that transferring generation to Gen. 39 more improves CTEM than transferring to Gen. 30. The proposed method employs the sensitivity factors of the TSM to get effective bids so that the network TSM is enhanced. To obtain the sensitivity factors of the TSM with respect to an individual generation, a little change is applied to each generation in down ¼ 10 MW and the post-change both directions as DPup Gj ¼ DP Gj TSM is obtained. The same perturbation of the CTEM-based congestion management is considered here. Besides, the same credible fault, i.e. the short circuit on bus 26, is considered for all examined congestion management techniques for the sake of a fair comparison. Also, to make the same framework to fairly compare the four methods, demand side bidding is not considered in the case study. By means of the base case and post-change TSM values, the sensitivities can be obtained according to (11). The results are shown in Table 2. In Table 2, columns 2 and 3 represent transient stability margin after individually increasing and decreasing generation of units, respectively. Also, columns 4 and 5 show generators’ TSM sensitivity factors. These results are in accordance with Table 1. For instance, in the both tables, transient security of the system is much improved with shifting generation from Gen. 38 to Gen. 39. It is necessary to evaluate the linearity of the proposed TSM index. The more linearity of an index makes it more reliable in practical applications. It is expected that critical generators reach first to non-linearity limits. Therefore, to check the linearity of the proposed TSM with respect to generation shifts, the most critical generators are considered in Fig. 4. In the figure, change in TSM is plotted versus positive and negative generation shift from critical generators at buses 35 and 38 to the non-critical generator at bus 39. As depicted in Fig. 4, for the critical generator at bus 35, the proposed TSM gives a highly linear relationship for a wide range of 400 MW generation shift spanning from 200 to 200 MW. For the most critical generator at bus 38, the TSM saves its linearity in the range of 220 MW spanning from 200 to 20 MW. On the other hand, CTEM as investigated in [15] only provides a linear behavior in the range of 100 MW with respect to generation shift at bus 38 of the New-England test system. As a result, the proposed TSM not only derives the real stability margin without any fictitious fault, but also it retains linearity in a much wider range, a property as another benefit of the proposed TSM over CTEM. Generations rescheduled by the implemented methods along with pre-rescheduling powers are shown in Fig. 5. After mitigating congestion using the four mentioned congestion management methods, overloads in branches 2–3, 1–39, and 26–27 are alleviated and these three lines are fully loaded to 100% of their rating.

Table 1 Sensitivities of generation shift from critical generators to non-critical generators for CTEM-based method.

Table 2 The sensitivity of the TSM with respect to each generation for the proposed method.

ing the generation can mitigate or adverse the congestion. The congestion is not mitigated always by increasing the output of noncritical generators as the reviewer commented. If the generation of the non-critical generators cannot be increased by the required amount, the ISO may increase the generation of less critical generators to maintain the dynamic security of the power system as much as possible. Also, the demands can participate in the congestion management based on the proposed framework. If in a serious case, the ISO cannot alleviate the congestion by both generation side and demand side bidding, load shedding is adopted as the last solution to mitigate the congestion. However, the cost of involuntary load shedding is included in the main objective function of Eq. (13) to minimize its possible cost. The congestion in the test system is mitigated using the proposed method as well as the DC congestion management [9,20], AC congestion management [9], and CTEM-based congestion management method [15]. All optimizations are carried out using the CONOPT solver of GAMS 22.7 (General Algebraic Modeling System) software package [23]. The first method implemented here for congestion management is based on the DC model minimizing the total payment to the market participants [20]. This model is a rough estimation of the power system in which reactive powers and voltages are not considered. The second implemented method is the AC congestion management [9]. Despite the fact that the DC and AC method relive the congestion, there is no guarantee to securely operate the power system after rescheduling generations. In fact, after congestion management, the output of critical generators may be increased. This could reduce the transient energy margin of the system and make the system vulnerable. The third method, as referred to the CTEM-based method here, relieves the congestion by transferring generation from critical generators to non-critical ones considering generators’ bids [15]. This method enhances CTEM as much as the desired level after congestion management. However, it is expected that the system operator pays an extra cost, called the security cost, compared to the AC and DC methods because of an enhanced level of transient stability margin. The method is based on the generation shift sensitivities of the CTEM transient stability index. In Table 1, sensitivities of CTEM with respect to generation shift from critical generators to non-critical ones obtained for the New-England test system are shown. CTEM for the base case is obtained as 0.1362 pu. To derive the sensitivity of CTEM to generation shifts, a little generation shift of 10 MW from critical generators G31–G38 to noncritical generators G30 and G39 is carried out. The sensitivities are defined as:

km!n ¼

DCTEM DPGm!n

ð31Þ

Gen. Bus

kðG30G39Þ!G30 (pu)

kðG30G39Þ!G39 (pu)

Gen. Bus

(pu) TSMup Gj

TSMdown (pu) Gj

SF up Gj

SF down Gj

30 31 32 33 34 35 36 37 38 39

– 0.2900 0.4700 1.3425 1.4385 1.4450 1.5255 0.1815 7.2735 0.3455

1.1810 2.6560 3.7800 9.6415 10.2710 10.3155 10.8455 1.9490 41.4950 –

30 31 32 33 34 35 36 37 38 39

12.8393 12.5721 12.6432 12.3646 12.2794 12.2851 12.3462 12.4940 7.8906 12.8280

12.8280 13.0852 13.0127 13.2955 13.3638 13.3666 13.3069 13.1463 17.6004 12.8280

1.0009 0.9801 0.9856 0.9639 0.9572 0.9577 0.9624 0.9740 0.6151 1.0000

1.0000 1.0201 1.0144 1.0365 1.0418 1.0420 1.0373 1.0248 1.3720 1.0000

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M. Esmaili et al. / Applied Energy 87 (2010) 971–981

Delta TSM (PU)

10 5 0 -5 -10 -15 -200

-150

-100

-50

0

50

100

150

200

Delta PG35 (MW)

Delta TSM (PU)

60 40 20 0 -20 -200

-150

-100

-50

0

50

Delta PG38 (MW) Fig. 4. Linearity of the proposed transient stability margin.

As demonstrated visually in Fig. 5, the methods perform rescheduling generations differently to alleviate the congestion and make possible all power transactions. For the sake of simplicity, demand side bidding in not considered in the case study. The optimization problem of (13) subject to constraints (14)–(30) is solved for the case study without any need to load shedding. As a result, demands are not altered after congestion management using the four methods in the case study. After congestion management, the transmission losses may increase or decrease. Before applying congestion management for the case study, total generation and network losses were 6198.7 and 48.2 MW, respectively. After mitigating congestion by the DC, AC, CTEM-based, and proposed methods, network losses are 50.4, 63.2, 44.8, and 62.3 MW. Despite the losses may slightly affect congestion management cost, the main concern of congestion management is to alleviate congestion with the least cost and enough level of security. Difference in network losses obtained by the four examined congestion management methods is so little compared with the total generation before congestion management (6198.7 MW).

As seen from Fig. 5, the proposed method increases the output of critical generators at bus 33 and 36 more than other methods. However, by looking at Table 2 of the paper, the sensitivity of TSM for increasing the outputs of generators 33 and 36 is 0.9639 and 0.9624. Since both values are near unity, these two generators are less critical and the TSM are not deteriorated a lot with these generation shifts. On the other hand, the generator at bus 38 is the most critical one with the sensitivity of 1.3720 for decreasing generation. This value means that the TSM is much improved by decreasing this generation. Generation shift of this generator by employing the DC, AC, CTEM-based, and proposed method is obtained 0, 0, 75.7, and 86.3 MW, respectively. In other words, the proposed method can lower the generation of the most critical generator more than the other methods. This results in a larger post-rescheduling TSM by the proposed method. It is worthwhile to mention that the power system experiences a new stability margin after rescheduling generations using different techniques. In other words, as shown in Fig. 2, both levels of the capability of absorbing kinetic energy and the first swing CTKE could be altered. In Table 3, related results before and after rescheduling by the implemented methods are shown. As shown in Table 3, by applying the DC and AC congestion management techniques, the capability of absorbing kinetic energy is reduced from 104.2 to 88.8 and 66.0 pu, respectively, after congestion management. The lower capability of absorbing CTKE leads to a less robust system from transient stability view point. However, the capability of absorbing kinetic energy is increased from 104.2 to 170.8 and 200.1 pu if the CTEM-based and the proposed methods are used, respectively. This implies that the system can bear more stress than the base case using transient stability oriented techniques. This result is more confirmed by the Critical Clearing Times (CCT) values. After congestion management, the system is able to tolerate fault duration as much as 278.2 ms after applying the proposed congestion management, while it tolerates only as much as 203.2, 201.0, and 254.0 ms fault duration using the DC, AC, and CTEM-based congestion management method, respectively. The congestion management cost and stability margin for the implemented methods are shown in Table 4. As illustrated in Table 4, using the AC method, the system operator should pay 21,343.89 $/h to alleviate the congestion while the transient stability margin is dramatically reduced from 12.8 pu in the base case to 1.4 pu. The AC method bearing the lowest cost is treated as the ref-

Generations Rescheduled by Different Congestion Management Methods 1200 Pre-Rescheduling DC AC CTEM-based

1000

Generation, MW

Proposed

800

600

400

200

0 30

31

32

33

34

35

36

37

38

39

Generating Bus Number Fig. 5. Powers rescheduled by the implemented methods along with powers from the market-clearing process.

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M. Esmaili et al. / Applied Energy 87 (2010) 971–981 Table 3 Critical CTKE, first swing CTKE and Critical Clearing Time (CCT) before and after congestion management. Method

Capability of absorbing critical CTKE (pu)

First swing CTKE (pu)

CCT (ms)

Base case DC AC CTEM-based Proposed

104.2 88.8 66.0 170.8 200.1

91.4 87.0 64.6 106.0 82.6

212.5 203.2 201.0 254.0 278.2

Table 4 Congestion management cost and transient stability margin obtained by implemented techniques. Method

Congestion management cost ($/h)

Security cost ($/h)

TSM (pu)

AC DC Proposed CTEM-based CTEM-based

21,343.89 21,433.67 21,362.39 22,075.73 Set to 21,362.39

0 89.78 (0.42%) 18.50 (0.09%) 731.84 (3.4%) –

TSM = 1.4 TSM = 1.8 TSM = 117.5 TSM = 64.8 Infeasible solution

erence method. Using the DC method, the system operator pays an extra cost of 89.78 $/h, 0.42% of the AC one, while the TSM is still very low as much as 1.8 pu. As a result, none of the methods can ensure the security of the system; the transient stability margin is dramatically low. In other words, the system is highly vulnerable against contingencies. This happens since neither the AC nor the DC method takes into account critical generators in rescheduling generations. As a result, the DC method offers no advantage over the AC one. On the other hand, using the proposed method, the system operator pays 21,362.39 $/h to mitigate the congestion. The surplus cost of 21,362.39–21,343.89 = 18.50 $/h, 0.09% of the AC one, is called the system security cost. With this insignificant surplus cost, the TSM is considerably improved from 12.8 pu in the base case to 117.5 pu after congestion management. This increase implies that the system becomes much more robust than the base case after congestion management. This is due to the fact that the proposed method detects generations with a dramatic effect on the stability margin using the TSM sensitivities. That is, the technique is able to manage the congestion so that the stability margin of the network is more retained than the other methods. Actually, all generators participate in the market by their effective bids, not their raw bids. This makes each individual bid be allocated its real value from the stability viewpoint. Generally, since the congestion management problem tries to minimize the cost, the lower bids have more opportunity to be contributed than the higher bids. Using the effective bids, the system operator can prevent increasing the output of the most critical generators. For instance, according to Table 2, generator 38 has the most effect on the TSM. If all generators bids equally for up and down their gen-

erations, the effective bid of generator 38 to increase the generation is the highest of all. Thus, it is less contributed in increasing generation. On the other hand, its effective bid to decrease the generation becomes the lowest of all; therefore, it is more contributed to decrease its output. Since the cost of CTEM-based method depends on the given DCTEM, this method is solved in two cases. In the first case, congestion is relieved by paying 22,075.73 $/h to generators. Compared to the AC congestion management, the system operator pays the extra cost of 731.84 $/h, 3.4% of the AC one, to enhance the transient security to 64.8 pu. Consequently, the proposed method can provide more security margin with much less security cost compared with the CTEM-based method. In the second case (shown in the last row of Table 4) where the congestion is tried to be relieved with the same cost as the proposed method, the GAMS optimization software can not find any feasible solution for the CTEM-based method. So it is seen that the proposed method is more advantageous than the CTEM-based technique. From the price of transient security viewpoint, it is marvelous to compare the CTEM-based technique with the proposed method. The CTEM-based method enhances the network stability margin as much as 64.5  12.8 = 51.7 pu costing 731.84 $/h, while the proposed method improves the security level as much as 117.5  12.8 = 104.7 pu costing 18.5 $/h. That is, the price of providing security for the CTEM-based method is 731.84/ 51.7 = 14.2 $/h, whereas for the proposed method this is 18.50/ 104.7 = 0.18 $/h. As a result, the proposed method has a discernible advantage over the CTEM-based method from the viewpoint of the price of security. To more investigate the performance of the methods, the behavior of the test system after congestion management is evaluated under a few severe credible faults. The results are shown in Table 5. In Table 5, the first contingency, as a short-circuit fault on bus 37 cleared by tripping generator 8, makes the generator at bus 37 be isolated from the network. The remaining system is stable and retains its synchronism. As seen from the table, although all methods make the power system stable after congestion management for the fault, the proposed method provides the power system with a much higher TSM of 163.8 pu than the other methods. For the next fault, which is more severe than the first one, the system is transient unstable if either AC or DC technique is used to alleviate congestion. Instability depth with the AC method, shown with a negative TSM, is more severe than the DC. However, the system rescheduled by both the CTEM-based and the proposed method is still stable and can bear the fault without losing stability. Of course, the proposed method gives a post-fault stability margin about three times as much as the CTEM-based method. The third fault makes the system stable only if the proposed method is used for congestion management. The power system experiences a severe instability using DC and AC methods. Also, it is unstable if the CTEM-based method is applied. To compare the computational burden of the proposed and CTEM-based methods, the average time elapsed to compute stabil-

Table 5 Transient stability strength of the test system after performing congestion management by different methods. Contingency

Congestion management by AC method

Congestion management by DC method

Congestion management by CTEM-based method

Congestion management by proposed method

Fault at bus 37, cleared by tripping trans. 37–25 after 0.25 s Fault at bus 3, cleared by tripping line 3–4 after 0.25 s Fault at bus 26, cleared by tripping line 25–26 after 0.278 s

Stable TSM = 20.6

Stable TSM = 40.4

Stable TSM = 125.8

Stable TSM = 163.8

Unstable TSM = 44.4

Unstable TSM = 11.7

Stable TSM = 36.6

Stable TSM = 78.1

Unstable TSM = 85.3

Unstable TSM = 88.5

Unstable TSM = 6.3

Stable TSM = 7.7

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M. Esmaili et al. / Applied Energy 87 (2010) 971–981

Table 6 Computational burden of the proposed and CTEM-based congestion management methods. Cases

10 Stable cases 10 Unstable cases

Average time of TSM-based method (s)

Average time of CTEM-based method (s)

Time-domain simulation

Calculating TSM

Time-domain simulation

Calculating CTEM

0.8202 0.8925

0.0150 0.0150

0.8745 0.8925

0.0960 0.0150

ity margin by both methods for 10 stable and 10 unstable faults is shown in Table 6. The PC used for this purpose was a P4, 4 GHz CPU. As seen from Table 6, the average CPU time elapsed for running one stable case for TSM and CTEM method was 0.8202 + 0.0150 = 0.8352 s and 0.8745 + 0.0960 = 0.9705 s, respectively. The execution time that is required for the time-domain simulation is reported by the PSS/E software package. Time of calculating TSM and CTEM is reported by MATLAB package. For TSM, this time is the elapsed time to differentiate the first swing and the critical CTKE, while for CTEM this is the time to integrate PAB from PA to PB. This shows the proposed method is 14% more computationally efficient than CTEM-based method. This improvement becomes more marvelous in on-line applications when the system operator analyses a large number of contingencies. Since both methods treat the unstable cases in the same manner, the elapsed time (0.8925 + 0.0150 = 0.9075 s) is the same for the both methods. In view of the fact that in a power system practically most faults are stable, the computational efficiency of the proposed method is retained in practice. It is worthwhile to note that in the case study, the time step of 0.1 s is used for higher accuracy, while a much greater time step can be assumed in practical operations to achieve even more computational efficiency. In case of large scale power systems, a possible solution to avoid recalculating the TSM sensitivities when system operating point is changed is to calculate the sensitivities for the most used state of the power system. In view of the fact that the TSM bears linearity in a wide range of generation changes (220 MW in the case study), the operator can compute the TSM in a new state – after changing generator outputs – by applying the first order linear approximation to the TSM of the most used state instead of recalculating TSM. Moreover, coherency and filtering techniques to reduce the power system model for transient stability analysis [24] and parallel computation methods (e.g. for parallel processing of different contingencies) can be used to speed up the calculations for large scale power systems. 5. Conclusion Congestion in a power market happens because of network limits. In a congested power system, transactions are not feasible unless the system operator uses a method to relieve the congestion. In a deregulated power market, the system operator has to pay to market participants for altering their powers to mitigate the congestion and finally to make feasible all power transactions. After applying congestion management, the system transient security level may be reduced because of increasing critical participants into the market. Such a network is highly vulnerable against even any little disturbance. Thus, it is really beneficial for the system operator to use a method to mitigate the congestion so that the system transient stability margin is more retained after congestion management. The proposed method employs the sensitivity of the proposed transient stability margin with respect to generations and demands. The proposed transient stability margin not only derives the real stability margin without any fictitious fault, but also it retains linearity in a much wider range compared with the conventional CTEM. It has been shown that the proposed congestion management method can provide more transient security margin

with lower security cost. The proposed method makes the network more robust against severe contingencies. Besides, its computation burden is much lower than the CTEM-based method. The research work is under way to incorporate voltage stability in the proposed congestion management framework. Appendix A Market data for generators of the New-England test system are shown in Table A1. In Table A1, PGj represents active powers of generators determined by the market-clearing process. PGjmin and PGjmax indicate the allowable active power range given by generadown are generators’ bids to increase or decrease tors. Also, Bup Gj and BGj the generation to relieve the congestion, respectively.

Table A1 GENCO data for the New-England test system. No.

1 2 3 4 5 6 7 8 9 10

P Gj (MW)

P Gj min (MW)

P Gj max (MW)

Bup Gj

Bdown Gj

($/MWh)

($/MWh)

250.0 572.9 650.0 632.0 508.0 650.0 560.0 540.0 830.0 1000.5

100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0 100.0

1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1000.0 1200.0

20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00 20.00

24.00 24.00 24.00 24.00 24.00 24.00 24.00 24.00 24.00 24.00

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