Power handling capabilities of transmission systems using a temperature-dependent power flow

Electric Power Systems Research 169 (2019) 241–249

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Power handling capabilities of transmission systems using a temperaturedependent power flow

T

Mahbubur Rahmana, , Valentina Cecchia, Karen Miub ⁎

a b

Electrical and Computer Engineering, University of North Carolina at Charlotte, NC, USA Electrical and Computer Engineering, Drexel University, PA, USA

ARTICLE INFO

ABSTRACT

Keywords: Conductor temperature Transmission line Temperature dependent power flow Transmission line capacity Thermal limit Dynamic line rating

The paper presents an approach to determine the maximum power transfer capabilities of electric transmission systems by coupling dynamic line ratings with temperature-dependent line modeling. This approach avoids using a set of predefined, conservative weather conditions often assumed uniform along lines. Specifically, a temperature-dependent power flow algorithm was been developed to account for weather conditions, their effects on conductor temperature, line parameters and line models. Moreover, coupling temperature-dependent line model structures within the temperature-dependent power flow, captures longitudinal variations in weather conditions (and the subsequent non-uniform distribution of electrical parameters) into the steady-state analysis of the power system. This enables determination of the transmission system loadability in terms of both thermal and voltage stability limits. The proposed methodology is applied to the IEEE 4-bus and 39-bus systems; computational impacts and quantifiable differences in maximum power transfer capabilities determined using the proposed approach and a conventional approach are discussed.

1. Introduction Transmission lines are one of the most critical components of a power system. Often, select lines are operating close to or beyond their nominal ratings [1–4]. Thus, transmission lines are a limiting factor in the reliable and affordable delivery of electric power [5–9]. For example, the line power handling capability is often a limiting constraint in the integration of renewable energy sources such as wind [10,11]; and, in some cases, the generation from wind power plants has to be curtailed because of congestion in existing transmission networks [12]. While construction of new transmission links is an option to increase capacity, building new lines is cost-prohibitive, time consuming, and corridor limited [10,13]. As system reinforcements are considered, it is also sensible to reevaluate historical approaches used to set line capacities. Traditionally, predefined and conservative weather conditions are used to calculate thermal limits known as static line ratings [14]. With the proliferation of weather measurements and the increase in power component sensors and measurements, temperature dependent power system modeling, analysis and metrics are possible: e.g. dynamic thermal line ratings [15–18], multi-segmented line models [19–21] and temperature dependent power flow approaches [22–24]. These separate works have demonstrated that using actual weather measurements may ⁎

significantly increase the power transfer capacity of the overhead lines, e.g. when wind speed is high [15,16], that non-lumped equivalent circuit line models can capture longitudinal variations in weather [19–21] and that a linear approximation [24] of the heat balance equation [25] can be used to incorporate weather conditions into power flow. In this paper, an integrated approach to temperature-dependent power flow has been developed to take into account available weather conditions as well as possible longitudinal changes in conductor temperature. It is noted that previous static and dynamic line rating approaches do not take into account the longitudinal variations in weather parameters because they utilize lumped equivalent circuit models (e.g. π-line models.) As transmission lines span through different geographic locations, they experience a wide range of weather conditions. This impacts conductor temperatures and result in a nonuniform distribution of line parameters. In order to address non-uniformity along lines, differential approaches for electromagnetic transient [26,27] and for steady state studies [28] and multi-segment approaches [19–21,29–31] to line modeling have been presented. Studies show non-uniformity affects the line power handling capabilities, considering conductor thermal limit and voltage stability limit [21,22,29]. This work adopts a multi-segment line modeling approach. The proposed approach determines both thermal and electrical

Corresponding author. E-mail addresses: [email protected] (M. Rahman), [email protected] (V. Cecchi), [email protected] (K. Miu).

https://doi.org/10.1016/j.epsr.2018.12.021 Received 5 November 2018; Accepted 24 December 2018 0378-7796/ © 2019 Elsevier B.V. All rights reserved.

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states of the power system. A non-linearized version of the heat balance equation is utilized, reducing potential inaccuracy from a linear approximation. Using the line currents and a set of weather parameters, the conductor temperatures and resultant line resistances are updated iteratively. Subsequently, this approach is applied to identify statically secure, power handling capabilities of the transmission system with respect to improved determination of both thermal and voltage stability limits of the transmission network. Specifically, the contributions of this work can be summarized as follows:

accommodate for the longitudinal variations in weather conditions and resultant conductor temperature. 2.2. Temperature-dependent model of transmission lines 2.2.1. Temperature-dependent electrical parameters The conductor temperature for known values of line current and weather conditions can be calculated by the steady-state heat balance Eq. (1). Once the conductor temperature (Tc) is obtained, it can be used to calculate the line series resistance. The linear relationship (2) between temperature and resistance is used for this purpose:

• A temperature-dependent power flow method has been developed to • •

incorporate weather parameters into power flow analysis and to determine information about thermal and electrical conditions of a system. A temperature-dependent line modeling approach has been integrated into the proposed temperature-dependent power flow to study the impact of the longitudinal variations in weather conditions. System-level impacts of the proposed power flow algorithm on the determination of the power handling capabilities of transmission systems have been studied.

R = Rref [1 +

Tref )]

(2)

Where α Temperature coefficient of resistance of the conductor Tref Reference temperature RRef Resistance per unit length at the reference temperature 2.2.2. Longitudinal non-uniformity in line electrical parameters Transmission line models that are currently used in power systems cannot take the longitudinal variation in conductor temperature into account. A transmission line can experience substantial variation in weather conditions and resulting conductor temperatures along its length. Therefore, the line impedance is a function of position along the line, i.e. non-uniformly distributed, as follows (3):

This paper is organized as follows: Section 2 gives an overview of the impact of weather conditions on conductor temperature, temperature-dependent transmission line modeling, and power transfer limiting factors. The temperature-dependent power flow method is discussed in Section 3 including: an overview of the proposed power flow method and the integration of temperature-dependent line modeling into the temperature-dependent power flow method. Section 4 discusses the power handling capabilities of transmission systems using the proposed temperature-dependent power flow. The comparison of the results between conventional power flow and proposed temperature-dependent power flow and the power handling capability of a transmission system using different line modeling approaches is investigated using a small (4-bus) and a large (39-bus) transmission network in Section 5 Finally, a conclusion and summary of future work are presented in Section 6.

Z (Tc (x )) = R (Tc (x )) + jXL

(3)

Where Tc(x)Tc as a function of position along the line, x Z (Tc(x)) impedance as function of Tc(x) R (Tc(x)) resistance dependent on Tc(x) XL inductive reactance, independent of Tc(x) In order to consider this non-uniformity in line conductor temperatures, and therefore in line impedance, different approaches have been proposed in previous studies [21,27–31]. A temperature-dependent line model segmentation approach was proposed in Refs. [19–21]. Due to the computational complexities of the differential modeling approaches, the multi-segment line model is used in this work to consider the spatial variation of weather conditions. This approach divides the line into multiple lumped parameter segments, each of which has parameters calculated at the weighted average conductor temperature for that specific section of the line. Line model segmentation is performed in such a way that no segment within a line experiences a temperature differential greater than a pre-set threshold value. Each of the segments has its own weighted average conductor temperature, from which line impedance for each segment can be calculated as is depicted in Fig. 1.

2. Background 2.1. Ambient conditions and conductor temperature In contrast to traditionally used static line ratings, where pre-set values of weather conditions are used, dynamic line ratings consider real time weather conditions to determine the current carrying capacity of a transmission line. It was shown that dynamic line ratings can accommodate substantially more energy yield from distributed generation compared to static line ratings [32–34]. The steady-state heat balance equation from IEEE Std. 738 is used to calculate the line ampacity or conductor temperature for known weather condition [25,35]. A conservative and fixed set of weather parameters, i.e. low wind speed and high ambient temperature, are used in this equation to calculate the conductor temperature during the static line rating approach, while dynamic line rating approaches use measured data from weather stations or installed sensors as inputs for the heat balance Eq. (1) [36].

Qc + Qr = Qs + I 2R (Tcmax )

(Tc

2.3. Power transfer limiting factors Each type of overhead line conductors has its own maximum allowable conductor temperature to ensure safe electrical operation and integrity of the thermal properties of the conductors. When the temperature of a line conductor reaches its maximum allowable limit, the line is considered to have reached its thermal limit. Using the steadystate heat balance Eq. (1), the maximum allowable line current for a given set of weather conditions can be determined for different types of line conductors [25,35]. The voltage stability limit is another major restricting factor for transmission systems. A transmission system reaches its voltage collapse point when it is no longer capable of delivering power to the connected loads [37,38]. Traditionally, continuation power flow (CPF) methods are used to determine the voltage collapse point of a transmission system. For example, CPF is able to determine the maximum loading point of a system by systematically increasing the system load under constant power factor [39].

(1)

Where Qc Convective cooling (W/m) Qr Heat radiation (W/m) Qs Solar heat gain (W/m) Tcmax Temperature of the conductor (°C) I Line current (A) R(Tc)Resistance at Tc (Ω/m) The steady-state heat balance equation does not account for the spatial variation in the weather parameters along the line. Therefore, temperature-dependent modeling of transmission lines is necessary to 242

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Fig. 1. Multi-segment lumped parameter line model.

3. Proposed temperature-dependent power flow 3.1. Overview of TD-PF Conventionally Newton-Raphson power flow methods use a predefined set of line parameter values to perform the conventional power flow analysis. Therefore, the traditional load flow studies can generate a substantial margin of error when compared to real-time measurements [24,40]. There have been a few approaches developed to consider the weather conditions into power system state estimation, and power flow studies. Researchers in [24], incorporated the heat balance equation into the conventional power flow algorithm. However, linear approximation of the heat balance equation was used in that study which can introduce deviation from the actual thermal conditions of the line conductors. Instead, in this work, the solution of the conductor temperature from the heat balance equation (HBE), and the solution of the power flow equations are determined sequentially within each iteration of the proposed method. The proposed power flow method provides the following outputs: i) Steady-state conductor temperature of each line in the system, ii) Voltage magnitude and phase at each bus, as in traditional power flow. Fig. 2. Proposed temperature-dependent power flow.

A flowchart that summarizes the proposed power flow approach is shown in Fig. 2. An initial guess of the conductor temperature (Tc) is made. For that Tc, line resistances are determined and an initial power flow is performed. The line current(s) are then calculated and fed into the heat balance Eq. (1), together with the weather parameters (which can be measured, given, or predicted), to obtain the updated Tc. It is noted that the heat balance equation is highly nonlinear [25,35]. A Newton codebased solution method to solve non-linear equations [41] was used in this work to determine line conductor temperatures from the heat balance equation. In the power flow Eq. (4), it can be seen that all the elements of the Y matrix are functions of conductor temperatures (Tc) of the respective iteration steps, and the Tc are functions of the locations along the line, x. Therefore, the values of Tc is used to update the line resistance at each iteration. The process continues until the difference in Tc for two subsequent iterations falls within a predefined tolerance, and the value of steady-state conductor temperature, Tcss, is determined.

Pi = |Vi |2 Gii (Tc ) + Qi =

|Vi |2 Gii (Tc )

N |V V Y Eq (Tc (x ))| cos( inEq (Tc ) n = 1; n i i n in N |V V Y Eq (Tc (x ))| sin( inEq (Tc ) n = 1; n i i n in

i

+ i

temperature and line parameters). The next subsection details how the TD-PF algorithm can incorporate these temperature-dependent line models. 3.2. TD-PF including longitudinal variations in conductor temperature In order to incorporate the variation of weather conditions along the line into the proposed temperature-dependent power flow, the previously described line model segmentation approach (2.2.2) is used.

n)

+

n)

(4) The above described approach of temperature-dependent power flow considers ambient conditions and obtains conductor temperature, assumed constant along each line. However, temperature-dependent line modeling (2.2) is needed to account for effects of longitudinal variations in ambient conditions (and consequently in conductor

Fig. 3. Tc profile and line model segmentation. 243

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Fig. 4. Outcomes of temperature-dependent line modeling.

Using the weather data and calculating line currents from an initial power flow, a conductor temperature (Tc) profile is created for each line using the steady-state heat balance equation [19–21]. If available, measurement from conductor temperature sensors can be used directly. Once a Tc profile is defined for each line, the temperature-dependent line model segmentation is performed. Each line model is divided into multiple lumped parameter segments depending on the threshold value of the difference of conductor temperature (Tcthres) along the line [21]. Fig. 3 shows an example Tc profile along a transmission line and its segmentation using the temperature-dependent line modeling approach with a threshold value of 5 °C. The temperature-dependent line modeling approach and its outputs are presented briefly in Fig. 4. Segment lengths, lseg, and weighted average weather parameters (Vw seg, Ta seg) for each segment are calculated from line model segmentation. The weighted average weather parameters are required to update the conductor temperatures in the iterative process of the temperaturedependent power flow method. Once the line models are divided into multiple segments, the transmission system model is updated accordingly. The start/end of a segment within a line model is considered a load-less PQ bus. The introduction of these load-less PQ buses effectively increases the size of the Y-bus matrix, which can impact computational time. However, this increased number of buses would also increase the sparsity of the matrix which can be exploited using, for example, the techniques described in [42], or Kron’s reduction [43] can be employed. The proposed power flow approach is performed on the updated network model. Fig. 5 graphically summarizes the procedure of taking the longitudinal variations in weather conditions into account via temperature-dependent line models, and incorporating it with the temperature-dependent power flow shown in the previous subsection.

4. Power handling capability studies using temperaturedependent power flow

Fig. 5. Integration of temperature-dependent line modeling with temperaturedependent power flow.

Fig. 6. Power handling capability studies using temperature-dependent power flow.

With the integration of renewable sources of energy, especially the interconnection of wind energy with the existing grid, it is becoming increasingly difficult for the existing transmission lines to accommodate the energy generated from wind sources [3,10,34]. However, the high wind speed, which would result in high wind power output, would also cool the overhead lines, resulting in potential line uprating. Instead of using predefined conservative values of weather conditions, measured values of weather parameters provide a more realistic estimate of the power handling capability of the system. The proposed power flow approach incorporated with the longitudinal variation of weather conditions delivers the steady-state thermal conditions of transmission lines. Subsequently, its application can inform users if lines are close to their thermal limit. It also provides a more accurate representation of the electrical line parameters, which impacts the determination of the voltage stability conditions of the system. In order to determine the effective power transfer limit of a system, both of the limiting factors need to be considered. Continuation Power Flow (CPF) provides the maximum power transfer capability of a system based on voltage stability limit. Each line current corresponding to the maximum loading point (λmax) in CPF is determined and compared to the line current needed to reach the line thermal limit (Tcmax) under the given weather conditions to determine the critical power transfer limiting factor. Let, Ii-j(λmax) = line current to reach λmax, and Ii-j(Tcmax) = line current to reach Tcmax, If (Ii-j(λmax) > Ii-j(Tcmax)), Then Critical limiting factor = Thermal limit Else Critical limiting factor = Voltage stability limit The loadability limiting factor for each line, and therefore for the whole transmission system, can be defined, as shown in Fig. 6. A temperature-dependent continuation power flow method (TD-CPF) was

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used to obtain the thermal and voltage stability limits [44]. Conventionally, maximum loading parameter (λmax) is used to represent the voltage stability limit of the system using CPF. In this work, a new parameter named ‘effective maximum loading parameter’ (λmaxeff) is introduced in order to incorporate the thermal limit with the voltage stability limit of a transmission system. Though CPF does not provide conductor temperature (Tc), the line currents for each step increase in loading parameter, λ, are obtained and then used in the HBE (1) to get the respective Tc. As the system load is gradually increased during the CPF, the line currents increase at each step, affecting Tc. The value of the loading parameter, λ, for which the Tc of a line reaches its maximum allowable conductor temperature (Tcmax), is the effective maximum loading parameter for the system. The procedure can be summarized as below: If (Tc(λmax) < Tcmax) Then λmaxeff = λmax Else λmaxeff = λ(Tcmax) When a transmission line is divided into multiple segments, different segments will have different conductor temperatures. In that case, the segment reaching the maximum allowable conductor temperature first is considered for determining the line thermal limit.

Table 1 Line model segmentation for 4 bus network. Branch #

From Bus

To Bus

Length (km)

# of Segs

Seg Length, lseg (km)

Seg. vw (m/s)

Seg. Ta (°C)

1

1

2

130

3

2 3

1 1

3 4

80 140

1 2

4

2

4

110

3

5

3

4

70

1

27 34 69 80 75 65 50 15 45 46 24

1.24 2.40 5.3 0.86 1.38 3.37 6 3.35 1.7 1.5 2.45

33.2 32.66 32.75 33.63 32.25 32.75 33.1 32.7 32.35 33 33.25

which account for the lowest line ampacity (or highest Tc) are used. Case C: The temperature-dependent power flow approach is now coupled with temperature-dependent line modeling (TD-LM). Using a threshold value of 5 °C for the conductor temperature difference between two points, line model segmentation is applied. The lengths of each segment, the corresponding weighted average wind speed and the ambient temperature for each segment (resulting from the line model segmentation as discussed in Section 3.2) are presented in Table 1. Using the TD-LM, the 4-bus network is transformed into a 10-bus network, where the 6 new buses are zero-PQ buses. For example, the details of segmentation of a select line (branch 4) was presented in Fig. 3 of Section 3.2. Table 2 denotes resulting conductor temperatures, Tc. Values for case B are lower than those of Case A, which is expected as even the lowest measured wind speed (vw) for each line is still higher than the value of vw used in static line rating, and the highest measured ambient temperature (Ta) for each line is lower than 40 °C. Case C takes the variation of the measured weather parameters into account. Therefore, the respective line Tc for case C are lower than the Tc from Case A and Case B. Table 3 represents the comparison of branch losses for the 4-bus network. As the branch losses are directly related to Tc and resultant line resistance, Case A has the highest branch loss for individual lines compared to the other cases. For Case C, branch losses for each segment of a line are added together to obtain the total branch loss of respective branches. The abovementioned three different approaches result in different line electrical parameters for the 4-bus system. In order to study the impact of the weather conditions on the power handling capability of the system, continuation power flow (CPF) is used on the cases above. Fig. 8 shows that for Case C, the value of the maximum loading

5. Case studies & observations 5.1. 4-Bus system First, a 4-bus transmission network is used for the case study. The network is composed of five transmission lines of different lengths and all the lines are made of ACSR Rook conductor. The electrical parameters for the Rook conductor can be found in Ref. [45]. Each of the branches is considered to have multiple weather stations along its length as presented in Fig. 7. For this work, realistic weather data was mimicked using regional weather station data from the CharlotteDouglas International Airport in North Carolina [46]. Three different cases will be studied for the 4-bus system – conventional power flow with π-line models, the proposed TD-PF approach with π-line models, and the TD-PF with the temperature-dependent line modeling. Case A: Conventional power flow method is used in this case, which does not take the measured weather conditions into account. Commonly used predefined weather conditions are considered in this case to calculate the conductor temperatures [47]. Specifically, an ambient temperature of 40 °C and a wind speed of 0.6 m/s are used [45]. Case B: The proposed temperature-dependent power flow for steadystate analysis is used but the longitudinal variation of weather conditions is not considered. The measured weather conditions along a line

Table 2 Conductor temperatures (Tc) for 4 bus network from TD-PF.

Fig. 7. IEEE 4-bus system.

245

Branch #

Conventional PF Case A (°C)

Proposed TD-PF Case B (°C)

Proposed TD-PF with TDLM Case C (°C)

1

67

59.9

2 3

62 56

54.8 48.2

4

66

55.5

5

59

48.8

53.0 48.0 43.2 50.7 42.8 39.2 41.9 44.8 49.1 44.6 41.9

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Table 3 Branch losses for 4 bus network. Branch #

Conventional PF Case A (MW)

Proposed TD-PF Case B (MW)

Difference between Case A and Case B

Proposed TD-PF with TD-LM Case C (MW)

Difference between Case A and Case C

1 2 3 4 5

11.36 3.98 1.34 8.25 1.84

10.945 3.85 1.29 7.83 1.80

3.65% 3.27% 3.73% 5.09% 2.17%

10.74 3.8 1.3 7.76 1.79

5.46% 4.52% 2.98% 5.94% 2.72%

system reaches λmax. Using the approach discussed in Section 4, Tc at each step of CPF is calculated. In Fig. 9, the asterisk marks (*) indicate the loading point for which at least one branch of the system reaches Tcmax. However, it must be noted that line currents required to reach thermal limits are not the same for different cases. For conventional power flow (Case A), the system reaches the maximum power handling point whenever the line current for branch 1 or 4 reaches 850 amps, even though according to CPF, branch 1 and 4 can carry line currents of 1163 and 1213 amps respectively. Therefore, for Case A the transmission system cannot reach the point of λmax. The value of λ for which the Tc for branch 1 or 4 reaches Tcmax would effectively be the λmax. The effective λmax, (λmaxeff) for Case A is found as 1.595. Similar outcomes are obtained for Cases B and C. The line currents required to reach the Tcmax are higher in Case B than in Case A. In Case B, the system reaches its maximum loading point when branch 1 current reaches 980 amps or branch 4 current reaches 1080 amps. The λmaxeff for this case is obtained as 1.715, which is 7% higher than the case where measured weather conditions are not taken into account. Furthermore, for Case C, where the longitudinal variation of weather conditions is taken into account the value of λmaxeff is the highest, and it is more than 12% higher than that of case A. Table 6 shows the comparison among the λmaxeff values for three different cases.

Fig. 8. P–V plots for bus 2 for each case.

Table 4 Comparison of λmax (4-bus system). Case #

λmax

% diff in λmax from case A

Case A Case B Case C

1.7340 1.7855 1.8100

– 2.97 4.38

5.2. 39-Bus system The IEEE 39-bus network is considered in this case study. All lines are Rook type conductor. Out of the 46 branches in the system, only the ones under the yellow shaded region in Fig. 10 have weather measurements stations on or near it. Three approaches described as Case A, B and C in the 4-bus case study, are also used for the 39-bus system. Table 7 contains the complete weather station locations and measurements. Similar to the 4-bus system Tc for the selected lines in the 39-bus system were higher for Case A, compared to Cases B and C. Table 8

parameter, λmax, is the highest since segment conductor temperatures are lower than the corresponding Tc’s for Case A and Case B. For Case A the value of λmax is substantially lower than Cases B (˜3%) and C (˜4.4%). Table 4 shows the tabulated form of the results from the CPF. Table 5 shows the line currents when the system reaches the voltage stability limits, as well as the line currents for individual lines to reach the thermal limits. For all the cases, the branch 1 and branch 4 line currents required to reach the thermal limit are lower than the currents required to reach the voltage stability limit, λmax. Therefore, branch 1 and branch 4 will reach their individual thermal limits before the Table 5 Line currents to reach thermal and voltage stability limits. Branch #

1 2 3 4 5

Case A

Case B

Case C

I for λmax (A)

I for Tcmax (A)

I for λmax (A)

I for Tcmax (A)

I for λmax (A)

I for Tcmax (A)

1163 727 378 1213 470

850 850 850 850 850

1165 738 376 1213 470

980 960 980 1080 1065

1182 740 377 1230 480

1100 1000 1110 1170 1135

Fig. 9. P–V plots with thermal limits.

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Table 6 Comparison of λmaxeff for 4-bus system.

Table 8 Comparison of conductor temperatures for 39 bus network.

Case #

λmaxeff

% diff in λmaxeff from case A

Branch #

Case A Case B Case C

1.595 1.715 1.7925

– 6.99 12.38

To

From

Conventional PF Case A (°C)

Proposed TD-PF Case B (°C)

Proposed TD-PF with TD-LM Case C (°C)

1

2

75.03

68.8

2

30

81.2

74

10 6

32 31

79.7 74.5

68.5 64.8

16 22

24 35

55.3 77.6

55 69

23

24

70.8

63

25

37

71.8

66.8

57.8 63.8 66.9 66.5 70.9 73.4 65.8 58.7 61.2 53 49 61.2 65.7 53.6 57.6 61.6 57.3

Table 11. For both case studies, the observations can be summarized as follows:

• The results from the temperature-dependent power flow may vary from the conventional power flow method by a substantial margin. • Consideration of the longitudinal variation of weather conditions

Fig. 10. IEEE 39 Bus System.

•

presents Tc for select branches for the three different approaches for the 39-bus system. The plots in Fig. 11 were obtained from continuation power flow for each of the cases. Similar to the 4-bus system, here also, the value of λmax is the lowest for the conventional power flow method, as predefined values of weather conditions were taken into account. Table 9 shows that λmax for Case A is 2% less than that of Case B. The difference is close to 6% when compared to Case C. However, when the line currents to reach the thermal limits were considered, it can be seen from Table 10 that the λmaxeff value for Case A is 8% lower than Case B, which is substantially higher than the difference in λmax. The difference in λmaxeff reaches 15% when the longitudinal variation of weather conditions were taken into account. The comparison of select line currents to reach the thermal and voltage stability limits for different line modeling approaches are shown in

have an effect on the outcomes of the temperature-dependent power flow. The proposed power flow algorithm integrated with temperaturedependent line modeling is capable of providing more up-to-date, actual values in terms of system power handling capabilities, considering both thermal and voltage stability limits.

6. Conclusion A temperature-dependent power flow analysis method is presented in this work which is capable of utilizing the time and space-varying nature of weather conditions under steady-state conditions. The proposed power flow approach coupled with a temperature-dependent line model can better determine the thermal and voltage stability conditions of a power system compared to the conventional power flow method. The benefit of the proposed temperature-dependent power flow integrated with the multi-segment line model has been studied on the analysis of maximum power handling capability of a transmission

Table 7 Measured wind speeds and ambient temperatures for 39-bus system. Branch #

Branch (From- To)

Branch Length (km)

Weather measurement location (from ‘From’ bus (km))

Wind speed, vw (m/s)

Ambient Temperature, Ta (°C)

1 5 14 20 33 34 35 37 38 39 41 46

1–2 2–30 6 – 31 10–32 19–33 20–34 21–22 22–35 23–24 23–36 25–37 29–38

80 70 80 60 80 100 80 80 70 80 70 90

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,

2.3, .85 .85, 2.7, 3.6 2.9, 2, 1.2 1.75, 1.2 3.4, 2, 0.9 2.9, 1.7, 0.75 5.6, 4.4, 1.8, 1.1 1.1, 2.1 3, 2.2, 1.1 3, 5 0.8, 2.2 2.2, 3.1, 4.6

32, 32, 33, 31, 33, 31, 32, 31, 32, 32, 34, 34,

80 30, 40, 60 50 60, 30, 80 30, 80 70 50,

70 80 100 60, 80 70 90

247

32 32, 33, 32 34 33, 32, 33 32 33 33 32

34 33.5 31 32.5, 31

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become variable depending on the utilization of the transmission assets. Future work includes the development of a power flow methodology to study the impact of weather conditions during a line outage or a step change in load in a transmission system. Currently, the authors are working on field data from installed online sensors provided by Southwire Company, LLC. The dataset, which contains real-time conductor temperature, line current, and weather parameters at multiple locations along an operating line, is expected to provide further assessment of the proposed line modeling approach. Declaration of interest The authors declare that there is no conflict of interest regarding the publication of this paper. Funding statement This work was supported in part by the National Science Foundation award #1509681 “A Novel Electric Power Line Modeling Approach: Coupling of Dynamic Line Ratings with Temperature-Dependent Line Model Structures.”

Fig. 11. P–V Plot with thermal and voltage stability limits for 39-bus system.

Table 9 Comparison of λmax for 39 bus system. Case #

λmax

% diff in λmax from case A

Case A Case B Case C

1.3393 1.3659 1.4164

– 1.99 5.76

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Table 10 Comparison of λmaxeff for 39 bus system. Case #

λmaxeff

% diff in λmaxeff from case A

Case A Case B Case C

1.159 1.255 1.331

– 8.28 14.84

Table 11 Selected Line Currents to Reach Thermal and Voltage Stability Limits for 39 Bus System. Branch#

1 5 14 20 39 46

Case A

Case B

Branch (From-To)

I for λmax (A)

I for Tcma (A)

2-3 2-30 6-31 10-32 23-36 29-38

804 1020 1665 1032 1014 954

870 870 870 870 870 870

x

Case C

I for λmax (A)

I for Tcma (A)

816 1033 1655 1043 1032 968

1010 1020 1090 1110 1360 1260

x

I for λmax (A)

I for Tcmax (A)

840 1108 1772 1060 1060 1001

1140 1075 1190 1170 1410 1300

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