Chronological simulation for transmission reliability margin evaluation with time varying loads

Electrical Power and Energy Systems 33 (2011) 1054–1061

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Chronological simulation for transmission reliability margin evaluation with time varying loads A.B. Rodrigues 1, M.G. Da Silva ⇑,1 Federal University of Maranhão, Department of Electrical Engineering, 65080-040 São Luís Maranhão, Brazil

a r t i c l e

i n f o

Article history: Received 23 March 2007 Received in revised form 14 December 2010 Accepted 8 January 2011 Available online 12 February 2011 Keywords: Available transfer capability Monte Carlo method Linear programming Composite system Power system reliability

a b s t r a c t This paper has as objective to assess the chronological variations in the Available Transfer Capability (ATC) caused by uncertainties associated with hourly load fluctuations and equipment availabilities. The system states resulting from these uncertainties are generated using the Monte Carlo Method with Sequential Simulation (MCMSS). The ATC for each generated state is evaluated through a linear optimal power flow based on the Interior-Point Method. These ATC values have been used to generate the probability distribution of the hourly ATC. This probability distribution enabled to estimate the Transmission Reliability Margin (TRM) for a specified risk level. The results, with a modified version of the IEEE Reliability test System, demonstrate that the time dependent uncertainties have a significant impact on the TRM. Ó 2011 Elsevier Ltd. All rights reserved.

1. Introduction The restructuring of the electric sector has been stimulated by the economic benefits to society resulting from the deregulation of other industries such as telecomunications and airlines. Currently, electrical utilities around the world are undergoing a radical transformation from an essentially regulated and monopolistic industry to a new model characterized by competition in generation with guaranteed access to open transmission [1]. The supply competition and the open transmission services have caused an increase in the number of transactions among market agents such as generation/distribution companies, pool companies and brokers [2,3]. The transactions carried out among these agents are defined by market forces without considering engineering problems of controlling, operating and planning of an electrical power system. Consequently, there may exist transactions that violate system operation constraints, that is, unfeasible transactions. Nevertheless, the transmission expansion has not been stimulated. This situation has been caused by right-of-way constraints and investment limits for the power sector budget due to economic difficulties. These constraints have compelled transmission providers to operate the interconnections near to their limits. Due to this

⇑ Corresponding author. E-mail addresses: [email protected] (A.B. Rodrigues), [email protected] (M.G. Da Silva). 1 Supported by FAPEMA (Fundação de Amparo à Pesquisa e ao Desenvolvimento Científico e Tecnológico do Estado do Maranhão in Brazil) and ELETRONORTE (Centrais Elétricas do Norte do Brasil SA). 0142-0615/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijepes.2011.01.024

operating condition new indices have been developed with the objective of providing a quantitative assessment of the power transfer reliability, among them the ATC. The ATC is a measure of the transfer capability remaining in the physical transmission network for future commercial activity over and above already committed uses [4]. Consequently, the ATC is subject to uncertainties in system parameters such as: generation dispatch patterns, load fluctuations and equipment (lines and generators) availabilities. These uncertainties may cause significant variations in the ATC. Due to this, the North American Electric Reliability Council (NERC) has recognized the importance of including the system uncertainties in the ATC evaluation [4]. Usually, the impact of system uncertainties on the ATC has been assessed using probabilistic methods [5–13]. These methods have been preferred due to their capacity to model, not only the severity of a state or event and its impact on system behavior and operation, but also the likehood or probability of its occurrence. The combination of severity and probability provides indices that really express the system risk [14–17]. The probabilistic assessment of power transfers consists basically of two main steps: the selection of a system state and the power transfer evaluation for the selected state. The power transfer evaluation for a system state was carried out using both nonlinear [5–10] and linear [11–13] models of the electrical network. On the other hand, the most used technique for the state selection is the Monte Carlo Method (MCM) with non-sequential simulation [7–13]. The MCM with non-sequential simulation can accurately model uncertainties associated with equipment availabilities and load forecast errors. These uncertainties can be modeled without

A.B. Rodrigues, M.G. Da Silva / Electrical Power and Energy Systems 33 (2011) 1054–1061

considering the system chronological state transition process. However, the ATC is a function of chronology dependent phenomena such as: hourly load variations, equipment maintenance and limited-energy hydro resources. Furthermore, NERC has recognized the importance of assessing the ATC in a time dependent structure [4]. The impact of time dependent parameters in the ATC may be modeled with the application of the MCMSS [18–23]. The main aim of this paper is to carry out a probabilistic analysis of chronological variations in the ATC using the MCMSS. The ATC associated with each sampled system state will be evaluated by a linear Optimal Power Flow (OPF) algorithm. This algorithm is based on the Interior-Point Method for linear programming. These ATC values have been used to obtain the probability distribution of the hourly ATC. This probability distribution enables to estimate the TRM for a specified risk level. The TRM is defined as the amount of the transmission transfer capability necessary to assure that the interconnected transmission network is secure under a reasonable range of uncertainties in the system conditions [4]. The proposed methodology, to analyse chronological variations in the TRM, has been tested in a modified version of the IEEE Reliability Test System [24], henceforth called MRTS. This assessment was carried out for a weekly study period. The results with the MRTS demonstrate that the uncertainties associated with hourly load transitions and equipment availability cause significant variations in the TRM. The rest of the paper is organized as follows: Sections 2 and 3 describes the methodology used to analyse the chronological variations in ATC and TRM, respectively. The results are given in Section 5. General conclusions are given in Section 6.

2. Outline of methodology In a competitive environment, the generation companies, distribution utilities and brokers market energy through bilateral or multilateral transactions. However, not all energy is marketed through bilateral and multiateral transactions [3,25]. For example, generators and loads may carry out price and quantity bids in a Pool [25]. The bilateral/multilateral transactions and the loads/generators belonging to the Pool use the transmission system in a shared way. Therefore, there must be a coordination strategy between Pool dispatch and bilateral/multilateral transactions. In this paper, the ATC is evaluated considering the simultaneous existence of both Pool and bilateral transacations in the same market structure. The strategy used to coordinate the Pool dispatch and bilateral transactions is the Pool Protection Mode (PPM) [25]. In the PPM, the generators and loads belonging to the Pool have priority over bilateral transactions. That is, bilateral transactions are introduced in the electric network after the load/generation dispatch in the Pool. This priorization procedure guarantees that the power transfers between selling and buying nodes of bilateral transactions do not cause a deterioration of the reliability indices associated with the Pool loads. Nevertheless, other strategies for coordination between Pool and bilateral transactions are possible. For example, bilateral transactions have priority over the load of the Pool when these have declared in advance [25]. By analysing the PPM, it can be noted that an amount of the transmission transfer capability is reserved for the Pool entities. This amount is known as Capacity Benefit Margin [4]. In this way, the PPM models the Capacity Benefit Margin as firm transfers [8]. In other words, curtailments in the Pool loads are not used to improve the ATC. In the PPM, the system operator must evaluate the ATC, associated with the bilateral transactions, after the Pool dispatch has been carried out. The ATC associated with a bilateral transaction

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is the maximum amount of power that a transmission system can transport in addition to the already committed transmission services, when power is injected at one location (selling node) and the same amount of power is extracted at the same time at another location (buying node) without violation of transmission constraints [26,27]. The ATC values evaluated by the system operator are posted in an OASIS (Open Access Same-Time Information System) aiming to help power marketers, sellers and buyers to schedule their energy interchanges. In this paper, the bilateral contract ATC is evaluated considering uncertainties associated with chronological load variations and equipment availabilities. These uncertainties have been modeled using the MCMSS. The MCMSS is the natural tool for estimating probabilistic indices in systems with time correlated or time dependent operation [18–23]. The MCMSS consists basically of generating one sample of system scenarios to estimate probabilistic indices from this sample. A system scenario is composed of a collection of system states in chronological order. That is, a system state can be considered as ‘‘a conceivable static photo of the system’’, while a system scenario can be considered as ‘‘a video containing a conceivable story of the system, covering the study period (e.g. a week or a year)’’ [20]. Usually, the system operating scenarios are generated using a State Duration Sampling Technique [15]. In this approach, the state transition sequences of the components are firstly generated by sampling the duration of the up and down states. These durations are sampled using the probability distributions that model the operation and repair times. Secondly, the state transition sequence of the components are combined to generate a system scenario. In this paper, the duration of up and down states associated with generators and circuits have been sampled using the Inverse Transform Method [15]. In this method, the state durations are sampled as follows: (1) Generate a uniform distribution random number U between [0,1]; (2) Calculate the duration of the current state by X = F1(U), where F1(U) is the inverse of the cumulative probability distribution F(X) that models the duration X of the current state. If the durations of the up and down states are exponentially distributed, then X ¼ T lnðUÞ. If the present state is the up-state, T is equal to the Mean Time to Failure (MTTF). On the other hand, if the current state is the down state, T is equal to the Mean Time to Repair (MTTR). After the system scenario has been generated, the ATC assessment of each system state of the scenario is carried out in chronological order. This assessment consists of two main steps. First, the Pool dispatch is carried out with the objective of minimizing the generation costs and maximizing the customers worth subject to the following constraints: power balance equation, power injection limits and circuits loading [1]. Secondly, the ATC is evaluated by maximizing the power injections in the selling and buying nodes of the bilateral transactions, in accordance with the Willingness to Pay to Avoid Curtailments (WPAC) of each transaction [2]. The ATC evaluation must satisfy the same constraints of the Pool dispatch. Furthermore, the objective function of the ATC evaluation minimizes the costs of the available generation resources that may be used to improve the ATC. The Pool dispatch and the ATC evaluation are solved using the version of the Predictor-Corrector Primal-Dual Interior-Point Method proposed in [28]. The method proposed in Ref. [28] has been chosen because: (1) variables with upper and lower bounds are easily modeled; (2) unified modeling of inequality and equality constraints is considered;

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(3) unfeasibility cases are easily identified. This characteristic enables that problems diagnosed as unfeasible can be solved using the following strategies: (a) reduction in the step length; (b) constraint perturbation; (c) objective function normalization; (d) adaptive tolerances. (4) feasible starting-points are not required. In OPF problems, such as the Pool Dispatch and the ATC evaluation, the number of active constraints is relatively small. Due to this, in this paper the Pool dispatch and the ATC evaluation have been solved combining the Interior-Point Method with Active-Set Techniques [15,29]. The algorithm of the Active-Set Technique used in this paper is presented in Fig. 1. The ATC values obtained for each system state can be used to estimate the ATC associated with a given time interval as follows:

ATC jit ¼

Nj 1 X F i ðsjk Þdðsjk Þ tf  t i k¼1

ð1Þ

where

In this paper, the width of the time interval t (tf  ti) is equal to 1 h. This value has been chosen aiming to evaluate the ATC for each hour of the chronological load curve (hourly ATC). The use of this time interval enables to correlate the hourly ATC with the chronological load curve. The sample of system scenarios can be used to estimate the expected value of the ATC for a given time interval as follows: NS 1 X e ATC jit E½ATC it  ¼ NS j¼1

Where e E½ATC it  is the expected value of the ATC for the transaction i, within an interval t and NS is the pre-specified number of simulations, that is, the sample size of system scenarios. The uncertainty of an estimate obtained with the MCMSS, can be assessed using the coefficient of variation [15]. In the ATC assessment, this coefficient is given by:

b½ATC it  ¼

r~ ½ATC it  e E½ATC it 

where

ATC jit

is the ATC associated with the bilateral transaction i, in the time interval t belonging to the scenario j; ti and tf are the initial and final times, respectively, associated with the time interval t; Nj is the number of system states of the scenario j; sjk is the state k of the scenario j; dðsjk Þ is the duration of the state sjk ; F i ðsjk Þ is equal to ATC for the transaction i, in the state sjk if this state belongs to the interval t. Otherwise, F i ðsjk Þ is equal to zero.

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi e V ar½ATC it  ; NSP NS j 2 1 e e V ar½ATC it  ¼ NS1 j¼1 ðATC it  E½ATC it Þ ; e ar½ATC it  is the estimated variance of ATCit; V r~ ½ATC it  is the estimated standard deviation of ATCit.

r~ ½ATC it  ¼

A flowchart of the proposed algorithm is shown in Fig. 2.

START

OBTAIN A INITIAL

FLOWS SET

NO

OPTIMAL SOLUTION OBTAINED

YES OPTIMIZATION SUBPROBLEM

Fig. 1. Active-set technique algorithm.

ð2Þ

Fig. 2. Algorithm for the ATC evaluation using the MCMSS.

ð3Þ

A.B. Rodrigues, M.G. Da Silva / Electrical Power and Energy Systems 33 (2011) 1054–1061

3. TRM evaluation In the Ref. [30], the TRM is evaluated as the difference between ATC values associated with the base case condition (without uncertainties) and with a percentile of the cumulative distribution (CPD) of the ATC (probabilistic assessment with uncertainty). This percentile is evaluated using Monte Carlo Method with non-sequential simulation. In this paper, the proposed methodology in the Ref. [30], has been expanded to assess the temporal variations in the TRM. These chronological variations are obtained by generating the CPD of the hourly ATC for a given bilateral transaction. This CPD is used to evaluate the hourly ATC for a specified risk level (percentile). In this way, the TRM for the transaction i in the time interval t (TRMit) is given by:

TRMit ¼

ATC 0it



F 1 it ðzÞ

ð4Þ

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Where ATC 0it is the ATCit value for the base condition and F 1 it ðzÞ is the inverse of the CPD associated with the random variable ATCit for a risk level z. From now the function F 1 it ðzÞ will be called Risk Based ATC. The Eq. (4) establishes that there is for each hour in the study period, an ATC value in which the curtailment probability is lower or equal to the specified risk level. Due to this, buying and selling entities that are willing to avoid curtailments in their transactions, must specify low risk levels. Consequently, a more constrained and reliable power transfer is scheduled for these transations. 4. Characteristics of the test system The models and proposed techniques described in the previous sections have been tested in the MRTS [24]. The MRTS is composed of two control areas: one area of 230 kV and other of 138 kV. The

Fig. 3. Single-line diagram of the MRTS, where the gray dashed arrows indicate the transactions 1 (T #1), 2 (T #2) and 3 (T #3).

A.B. Rodrigues, M.G. Da Silva / Electrical Power and Energy Systems 33 (2011) 1054–1061 Table 1 Main characteristics of the MRTS.

Table 3 Bilateral transactions used in ATC evaluation.

Installed capacity Load peak Number of bus Number of circuits Number of generation units Number of generation plants

4304 MW 3562.5 MW 24 31 40 10

Transaction

Selling node

Buying node

WPAC ($/MWh)

1 2 3

123 121 123

115 116 109

1500.0 1500.0 1500.0

3600

Table 2 Price functions for the MRTS generator’s.

12 20 50 76 100 155 197 350 400

Type

Thermal Thermal Hydro Thermal Thermal Thermal Thermal Thermal Nuclear

3400 CðPg i Þ ¼ Cg 0i þ Cg 1i Pg i

350

3200

Cg 0i

Cg 1i

25.1817 120.0000 0.0000 83.2091 219.0476 143.6261 255.4434 179.0267 202.9916

25.2452 37.5000 0.0000 13.1714 20.6562 10.6609 20.8166 10.8274 5.4830

Hourly Load Curve (MW)

Capacity (MW)

400 Load Curve Base Case ATC

300 3000 250

2800 2600

200

2400

150

Hourly ATC (MW)

1058

2200 100 2000

single-line diagram and the main characteristics of the MRTS are presented in Fig. 3 and Table 1, respectively. This system has been used aiming to assess the ATC in a stressed transmission network. The ATC evaluation is carried out after the pool dispatch in the proposed method. This dispatch has as objective the maximization of the social welfare, that is, the minimization of the generation costs and the maximization of the customers worth [1]. Due to this, the price functions associated with generators and loads are required to carry out the pool dispatch. In this paper, it is assumed that the price functions to sell power associated with generators have the following form:

CðPg i Þ ¼ Cg 0i þ Cg 1i Pg i where Pgi is the power output for the generator i; Cg 0i ðCg 1i Þ is the zero (first) degree coefficient in the cost function associated with the generator i; C(Pgi) is the price to produce Pgi MW in the generator i. The coefficients Cg 0i and Cg 1i were estimated applying linear regression techniques in the generation unit operation cost data of the MRTS. The coefficients Cg 0i and Cg 1i for all generators of the MRTS are presented in Table 2. On the other hand, the power purchase price for all loads in the MRTS is equal to 1500.0 $/MWh. This data has been obtained from the Ref. [11]. However, it is possible to use different power purchase prices for each system bus. Finally, it must be mentioned that detailed information about the MRTS data (such as buses load values, generation site and size and the maximum capacity of the circuits) can be found in Refs. [24,31]. 5. Results This section presents the results obtained with the application of the proposed model in the MRTS. This system has been used to evaluate the hourly ATC for the base condition and to estimate the TRM for a specified risk level. The results presented in the following subsections are organized as follows:  In Subsection 5.1, the correlation between the hourly load curve and hourly ATC is analysed for a weekly period;  In Subsection 5.2, the TRM for a specified risk level is evaluated using the proposed methodology.

50

1800 1600 0

20

40

60

80

100

120

140

160

0 180

Time (Hours) Fig. 4. Load curve and base case ATC for the transaction 1.

5.1. Base case ATC This section presents an ATC assessment for the base case condition in MRTS. That is, all components are in up-state and there are not uncertainties associated with the hourly load forecast. This analysis has been carried out under the following conditions: (1) The study period for the base case analysis is one week (168 h); (2) The chronological curve used to model the load fluctuations in the study period is the hourly load curve of 51 week (peak winter week) in the MRTS load data [15]; (3) The ATC evaluation has been carried out considering that the bilateral transactions described in Table 3 and showed in Fig. 3 are simultaneously added in the transmission network. The hourly ATC associated with the transactions 1, 2 and 3 are showed in Figs. 4–6. In these figures the hourly load curve for the study period is also shown. From Figs. 5 and 6, it can be noted that the ATC associated with transactions 2 and 3 are inversely correlated with the load curve. That is, high load levels are associated with low values of ATC. On the other hand, Fig. 4 shows that the ATC associated with transaction 1 is directly correlated with the system load. These effects are caused by the fact that the ATC is highly dependent on the current load/generation patterns. The diversification of these patterns is increased by the Pool dispatch. The objective of the Pool dispatch is to minimize the generation costs to supply the Pool load. In this way, when the system load is reduced, the output power of the most expensive generator is decreased. Consequently, the generation pattern of a system state with low load level does not correspond to the scaled-down generation pattern of a system state with high load level. This procedure stems from flows patterns in the circuits that may be opposite or have the same direction of

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3600

1200

1300 Load Curve Base Case ATC

3400

1000

1100

3200

1000 900

2800

800 2600 700 2400

600

2200

800

ATC (MW)

3000

Hourly ATC (MW)

Hourly Load Curve (MW)

1200

400

500

2000

400

1800

300

1600

200 180

0

20

40

60

80

100

120

140

160

200

0 0

Base case ATC Risk Based ATC

20

40

60

80

100

120

140

160

180

Time (hours)

Time (Hours)

Fig. 8. Risk based ATC (lower curve with dashed line), base case ATC (upper curve with full line) and TRM (light gray area associated with the difference between the upper and lower curves) for transaction 2.

Fig. 5. Load curve and base case ATC for the transaction 2.

3600

600

550 600

3400

Base case ATC Risk Based ATC

500 500 450 400

2800 2600

350

2400

300

2200

400

ATC (MW)

3000

Hourly ATC (MW)

Hourly Load Curve (MW)

3200

300

200

250 2000

100 200

1800

Load Curve Base Case ATC

1600

0

20

40

60

80

100

120

140

160

150 180

0

0

20

40

60

80

100

120

140

160

180

Time (hours)

Time (Hours)

Fig. 9. Risk based ATC (lower curve with dashed line), base case ATC (upper curve with full line) and TRM (light gray area associated with the difference between the upper and lower curves) for transaction 3.

Fig. 6. Load curve and base case ATC for the transaction 3.

400 Base case ATC Risk Based ATC

350

the flow components associated with bilateral transactions. Therefore, it is not possible to guarantee that a load reduction will result in favorable conditions for a given power transfer.

ATC (MW)

300 250

5.2. TRM analysis

200

This subsection has as objective to assess the impact of system uncertainties on the TRM. This assessment has been carried out using qualitative and quantitative indices such as chronological curves and sample statistics. Initially, the chronological curves of the TRM associated with the transactions 1, 2 and 3, are presented in Figs. 7–9, respectively. The chronological curves showed in Figs. 7–9 have been obtained under the following conditions:

150 100 50 0 0

20

40

60

80

100

120

140

160

180

Time (hours) Fig. 7. Risk based ATC (lower curve with dashed line), base case ATC (upper curve with full line) and TRM (light gray area associated with the difference between the upper and lower curves) for transaction 1.

(1) The number of simulations (NS) is 125; (2) The uncertainties associated with load forecast have been included in the ATC assessment using the Gauss Model [32]. In this model, it is assumed that the load at each hour follows a normal distribution with known mean and vari-

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Table 4 Total time in which the risk based ATC is lower than 1.0 MW for a weekly period. Transaction

Case I Percentage

1 2 3

39.2857 2.3810 2.3810

Case II Hours 66.0 4.0 4.0

Percentage 0.0 0.0 0.0

Hours 0.0 0.0 0.0

ance. Furthermore, the load in each hour t is uncorrelated with the load at another hour s. In this paper, the mean and standard deviation of the normal distribution used to model the hourly load forecast uncertainty, are equal to the load level at an hour t and 5% of this load level, respectively. (3) Circuit and generator outages have been considered in the ATC assessment; (4) The operation and repair times of the components are exponentially distributed. However, other distributions can be used, e.g., the repair times can be modeled with a log-normal distribution [15]; (5) The risk level used to estimate the TRM is 5.0%. The simulation of the 125 scenarios has required 12,160 Pool dispatches and 22,130 ATC evaluations. That is, the total number of OPF assessment is equal to 34290. The run time for this simulation was 17.3 min. This time has been obtained using a PC with Pentium processor of 1.133 GHz and 128 Mb of RAM. Furthermore, the maximum values of the coefficients of variation for the expected value of the hourly ATC, associated with the transactions 1, 2 and 3, are equal to: 7.1%, 3.1% and 3.6%, respectively. From Figs. 7–9, one can be noted that the system uncertainties have higher impact on the TRM associated with the transaction 1. This fact demonstrates that the system random behavior has different effects on the simultaneous power transfers. Furthermore, it can be noted that the total time in which the power transfer, associated with the transaction 1, underwent severe reductions is greater than those associated with the transactions 2 and 3. This result can be assessed quantitatively by evaluating the total time in which the risk based ATC for transactions 1, 2 and 3 is lower than 1.0 MW. These total times have been evaluated for two case studies:  Case I: recognizing the uncertainties associated with equipment failures (assumption used to obtain the curves showed in Figs. 7–9)  Case II: without considering the components outages in the probabilistic model. The total times in which the risk based ATC is lower than 1.0 MW are presented in Table 4 for the case studies I and II. The results presented in Table 4, indicate that the ATC for transaction 1 is very constrained for a risk level of 5.0% in the case I. Due to this, the buying entity associated with transaction 1 must seek a power supplier in other points of the electrical network to improve its power interchanges. Furthermore, it can be noted that the equipment failures have a large impact on the time in which the risk based ATC is lower than 1.0 MW. For example, there is no risk based ATC values lower than 1.0 MW when only peak load forecasting uncertainties are included in the model (case II). At this point, it must be noted that the planning horizon is weekly (short-term). Nevertheless, the contingencies in generators and circuits cause severe reductions in the risk based ATC. The chronological curves of the TRM provide a qualitative assessment of the impact of the system uncertainties on the TRM. However, it is also important to analyse quantitatively the

Table 5 Statistical indices for the percentage variation between the TRM and base case ATC in the case study I. Indices (%)

Mean Standard deviation Minimum Maximum Lower quartile Median Upper quartile

Transactions 1

2

3

272.6606 28.5767 6.8916 100.0000 45.9638 80.7264 100.0000

29.6406 15.1018 7.8270 100.0000 19.8351 25.6560 36.8837

29.9441 22.0361 3.3234 100.0000 13.9296 19.2220 33.0924

Table 6 Statistical indices for the percentage variation between the TRM and base case ATC in the case study II. Indices (%)

Mean Standard deviation Minimum Maximum Lower quartile Median Upper quartile

Transactions 1

2

3

40.5173 17.7418 10.1197 87.8069 29.4562 35.4186 43.2173

15.4625 7.7002 0.4868 36.5277 13.0580 16.3207 18.7925

13.9953 6.7279 2.6486 30.7539 9.0142 12.2046 19.7153

Table 7 ETR values associated with the transactions 1, 2 and 3 for the case studies I and II. Transaction

1 2 3

ETR (%) Case I

Case II

21.3031 71.8845 78.1194

61.7969 85.8870 87.9814

effect of these uncertainties on the TRM. This analysis has been carried out using statistical indices for the percentage variation between the TRM and the base case ATC. That is, the statistical indices (such as mean standard deviation, median, etc.) have been evaluated considering three samples of 168 elements. Each element is given by 100  TRMij =ATC oit , where i = 1, . . ., 3 and t = 1, . . ., 168. These statistical indices were also evaluated for the case studies I and II previously described. The statistical indices associated with the percentage variation of the TRM are presented in Tables 5 and 6 for the case studies I and II, respectively. From Table 5, it can be noted that the mean and percentiles associated with the transaction 1 are greater than those associated with the transactions 2 and 3. That is, the system uncertainties has more impact on the transaction 1. For example, the mean value of the perecentage variation for the transaction 1 is 72.8%. On the other hand, the mean values of the percentage variation for the transactions 2 and 3 are 29.6% and 26.9%, respectively. Additionally, the comparison of Tables 5 and 6 shows that the equipment failures cause significant increase in the percentage variation associated with TRM. For example, the components unavailabilities caused a growth of 79.3323% and 61.0669% in the mean and standard deviation of the transaction 1. Once more, it was demonstrated that the uncertainties associated with system contingencies have large impact on the TRM, in spite of the planning horizon be weekly. Finally, the impact of the system uncertainties on the TRM is assessed using the Energy Transfer Ratio (ETR). The ETR index is defined by the following equation:

P168 1 F it ðzÞ ETR ¼ 100%  Pt¼1 168 0 t¼1 ATC it

ð5Þ

A.B. Rodrigues, M.G. Da Silva / Electrical Power and Energy Systems 33 (2011) 1054–1061

The numerator and the denominator of the fraction in the Eq. (5) are associated with the energy transfers for the risk based ATC and for the base case ATC, respectively. In this way, if the uncertainties have small impact on the power transfers, then the ETR is near to 100%. That is, the ETR index measures the efficiency of the energy transfers. The ETR associated with the transactions 1, 2 and 3 are presented in Table 7 for the case studies I and II. From Table 7, it can be noted that the ETR associated with the transaction 1 is considerably lower than those associated with the transactions 2 and 3. This fact, demonstrate once more that the system uncertainties have significant impact on the power transfers associated with the transaction 1. Finally, it can be noted that the ETR index increases when uncertainties associated only with load forecasting errors are included in the model. In other words, components failures cause significant losses in the energy transfer efficiency. For example, the ETR associated with transaction 1 underwent a reduction of 65.5272% when equipment outages were included in the probabilistic model. 6. Conclusions This paper has described a probabilistic methodology for assessing chronological variations in the ATC. This methodology uses the MCMSS and linear OPF based on Interior-Point Method. The results obtained with the proposed approach in the MRTS demonstrate that: (1) The load and generation patterns have a large impact on the base case ATC. These patterns and the direction of the power transfers can cause variations in the ATC that are directly or inversely correlated with the system load; (2) The uncertainties associated with the load forecast error and equipment availabilities have meaningful impact on the TRM. (3) The energy transfers efficiency is severely reduced by components failures, in spite of the time horizon being of short-term. For example, the minimum reduction in the ETR index casused by the components outages was 11.2092%, regarding to the transactions 1, 2 and 3. These results motivate the development of new methodologies for the ATC probabilistic assessment. An AC model of the electrical network is being developed in order to include reactive power/ voltage constraints in the ATC evaluation. In future publications results involving AC model must be presented. References [1] Christie RD, Wollenberg BF, Wangensteen I. Transmission management in the deregulated environment. Proc IEEE 2000;88(February):170–95. [2] Fang RS, David AK. Optimal dispatch under transmission contracts. IEEE Trans Power Syst 1999;14(May):732–7. [3] Fang RS, David AK. Transmission congestion management in a electricity market. IEEE Trans Power Syst 1999;14(August):877–83. [4] North American Reliability Council (NERC). Available transfer capability – definitions and determinations. NERC report; June 1996. [5] Xia F, Meliopoulos APS. A methodology for probabilistic simultaneous transfer capability analysis. IEEE Trans Power Syst 1996;11(August):1269–78.

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