Edited by: Eleni N. Chatzi, ETH Zürich, Switzerland
Reviewed by: Eliz-Mari Lourens, Delft University of Technology, Netherlands; Costas Papadimitriou, University of Thessaly, Greece
Specialty section: This article was submitted to Structural Sensing, a section of the journal Frontiers in Built Environment
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Transmission line towers are usually analyzed using linear elastic idealized truss models. Due to the assumptions used in the analysis, there are discrepancies between the actual results obtained from full-scale prototype testing and the analytical results. Therefore, design engineers are interested in assessing the actual stress levels in transmission line towers. Since it is costly to place sensors on every member of a tower structure, the best locations for sensors need to be carefully selected. This study evaluates a methodology for sensor placement in transmission line towers. The objective is to find optimal locations for sensors such that the real behavior of the structure can be explained from measurements. The methodology is based on the concepts of entropy and model falsification. Sensor locations are selected based on maximum entropy such that there is maximum separation between model instances that represent different possible combinations of parameter values which have uncertainties. The performance of the proposed algorithm is compared to that of an intuitive method in which sensor locations are selected where the forces are maximum. A typical 220-kV transmission tower is taken as case study in this paper. It is shown that the intuitive method results in much higher number of non-separable models compared to the optimal sensor placement algorithm. Thus, the intuitive method results in poor identification of the system.
Transmission line towers form a significant part of the total cost of power transmission infrastructure. Generally, most of the transmission line towers employ steel lattice structures. Lattice towers are presently designed using traditional stress calculations obtained from linear elastic idealized truss analysis in which nodes are assumed to be concentrically loaded and members are pin connected. However, there are many discrepancies between the actual measurements obtained from full-scale prototype testing and the analytical results. Factors, such as joint eccentricity, connection rigidity, geometric and material non-linearities, uneven foundation, etc., are some of the reasons for variations in these results. Most structures are analyzed, typically using finite element software, by creating simplified models. However, real transmission line tower structures differ from idealized conditions in several aspects. Most of the lattice towers employ angle sections as members. These members that are connected with bolted connections introduce eccentricities between the line of action of the load/force and the longitudinal principal axis of the member. Zhang et al. (
Engineers are interested in experimental results in order to determine the deviation of measured values from the predictions of analysis models. However, it may not be practical to measure the strains or deflections on each and every member. Hence, it is important to select the best locations for placing the sensors such that the real behavior of the structure is adequately explained. This issue had been investigated as early as 1978 (Shah and Udwadia,
Generic sensor placement methods that can be applied to static measurements are also found in the literature. Selvaraj et al. (
Papadimitriou (
A uniform probability distribution gives the maximum value for the entropy. If all the values of a variable lie in the same interval, the entropy of that variable has the minimum value of 0. Hence, the entropy reflects the inhomogeneity, uncertainty, or disorder in a system. In the work of Papadimitriou (
Other authors have also used entropy as a criterion for selecting optimal sensor locations. Lam et al. (
In the model falsification approach, the goal is to identify the candidate models that reasonably explain observations. A model is selected to be a candidate if its predictions match the measurements at each and every sensor location within the threshold of modeling and measurement errors. Measurement errors are estimated using the sensor precision data provided by manufacturers. Estimating modeling errors is more complex and involves specific knowledge about the domain (Vernay et al.,
In order to support model falsification, the best measurement system should produce maximum separation between candidate models. The degree of separation between model predictions is evaluated using the entropy function. Maximum disorder exists when parameter values show wide dispersion. An ideal measurement system results in maximum variation in predicted responses of candidate models at the measured locations. Therefore, the location and type of measurement devices are determined such that the entropy of the set of model predictions is the maximum.
Model falsification approach has been applied to many full-scale structures details of which can be found in Goulet and Smith (
Wang et al. (
Another class of methods for sensor placement uses the concept of probability updating (Beck,
One of the difficulties with Bayesian updating is in estimating
The overall goal of this work is to develop a generic methodology to determine optimal sensor locations for the system identification of transmission line towers. Design engineers are interested in assessing the actual stress levels in transmission line towers. However, it will be costly to place sensors on each and every member of a tower structure since transmission towers typically consist of a few hundred members and joints. Therefore, a scientific methodology for determining sensor locations is required so that the real behavior of the structure can be explained with minimum number of sensors. Since the probabilities are difficult to establish, Bayesian updating procedures are not considered for system identification. Instead, the model falsification approach is adopted and an entropy-based sensor placement strategy is used. In order to bring out the advantages of this strategy, the performance of the optimal sensor configuration is evaluated by comparing it with a traditional intuitive method. The traditional rule of thumb is to place sensors where the stresses are the maximum. In many experimental methods, this rule of thumb is followed and no systematic sensor placement method is followed. Therefore, in this work, we aim to show that the maximum stress criterion is not appropriate for identifying the best sensor locations.
The research methodology consists of the following steps:
Selection of a case study Generation and analysis of model instances Identification of optimal sensor locations using entropy-based sensor placement strategy Computation of the performance metric for the optimal sensor configuration Computation of the performance metric for the intuitive sensor configuration Comparison of results from steps 4 and 5. These steps are described in more detail below.
A typical 220-kV transmission tower with square base configuration is modeled (Figure
SL no. | Description | Design parameter |
---|---|---|
1 | ||
Small angle tower – DB | 0–15° deviation | |
2 | 320 m | |
3 | ||
(a) Number and code name of conductor per phase | Single conductor |
|
(b) Diameter | 28.14 mm | |
(c) Unit weight | 1.624 kg/mm | |
(d) Cross-sectional area | 468.5 mm2 | |
(e) Ultimate tensile strength | 14175 kg | |
(f) Modulus of elasticity | 7730 kg/mm2 | |
(g) Coefficient of linear expansion (°C) | 19.0 E−06 | |
4 | ||
GSW | ||
(a) Number and code name of earth wire | GSW 7/3.15 | |
(b) Diameter | 9.45 mm | |
(c) Unit weight | 0.428 kg/m | |
(d) Cross-sectional area | 54.55 mm2 | |
(e) Ultimate tensile strength | 5710 kg | |
(f) Modulus of elasticity | 19330 kg/mm2 | |
(g) Coefficient of linear expansion (°C) | 11.5 E−06 | |
5 | ||
(a) Wind zone | Wind zone (2): 39 m/s | |
(b) Terrain category | 2 | |
(c) Reliability level | 1 | |
(d) Minimum temperature for conductor/ground wire | 10°C | |
(e) Everyday temperature for conductor/ground wire | 32°C | |
(f) Maximum temperature for conductor | 85°C | |
(g) Maximum temperature for earth wire GSW | 53°C | |
6 | ||
(a) Normal ground clearance | 7 m | |
(b) Allowance for sag error | 4% of maximum sag | |
(c) Shielding angle | 30° | |
(d) Midspan clearance | 8.50 m | |
7 | ||
Tension string | ||
(i) Swing of jumper | ||
for 0° swing | 2130 mm | |
for 10° swing | 2130 mm | |
for 20° swing | 1675 mm | |
for 30° swing | 915 mm | |
8 | 3345 mm | |
Insulator diameter | 255 mm | |
Number of insulator string | 2 | |
Weight of insulator string max | 300 kg | |
Weight of insulator string min | 150 kg | |
9 | ||
(i) Wind span | ||
(a) Normal condition | 320 m | |
(b) Broken condition | 192 m | |
(ii) Weight span (max/min) |
||
(a) Normal condition | 640/−640 m | |
(b) Broken condition | 384/−384 m |
Variable | Normal/broken condition | Loads for all load cases (kg) |
|||
---|---|---|---|---|---|
Transverse | Vertical |
Longitudinal | |||
V-max | V-min | ||||
Earth wire | NC | 940 | 277 | −276 | 0 |
Conductor | NC | 2652 | 1640 | −739 | 0 |
Earth wire | NC | 940 | 277 | −276 | 0 |
BWC | 510 | 166 | −166 | 2084 | |
Conductor | NC | 2652 | 1640 | −739 | 0 |
BWC | 1536 | 1224 | −324 | 5627 | |
Earth wire | NC | 285 | 706 | −552 | 0 |
Conductor | NC | 926 | 3942 | −1478 | 0 |
Earth wire | Intact | 285 | 706 | −552 | 819 |
Broken | 143 | 485 | −332 | 1091 | |
Conductor | Intact | 926 | 3942 | −1478 | 2658 |
Broken | 463 | 3111 | −648 | 3544 | |
Earth wire | NC | 285 | 277 | −277 | 1091 |
Conductor | NC | 926 | 1640 | −740 | 3544 |
From Table
Application of the model falsification approach involves generating multiple model instances by selecting different combination of values of model parameters that are uncertain. One model instance represents one set of combination of values of parameters and the corresponding results of the analysis. Linear static analysis of the transmission tower is performed. Uncertainties, such as support settlements, variations in material properties of steel, variations in joint connection rigidity, and support fixity conditions are modeled by selecting a range of values for the model parameters representing these effects. A series of model instances are generated by taking different combinations of values of these parameters as given in Table
Joint and support fixity | Support settlements (mm) | Young’s modulus (GPa) |
---|---|---|
0.0 (Pinned) | 1 | 180 |
0.5 | 1.5 | 220 |
0.75 | 2 | |
1.0 (Rigid) | 2.5 |
Four conditions of connections are used, namely, pinned, rigid, and partial moment resisting connections of 50 and 75% rigidity. In a model, all the joint connections have the same fixity condition except at the supports. Assuming the same type of connections for all the joints is justified because all the connections involve similar design details and are fabricated using standard mechanized procedure. Different combinations of fixity conditions are used for the four supports. Four values of support settlements are used, namely 1, 1.5, 2, and 2.5 mm. Each value of support settlement is used for each support that is assumed to have settled. Different combinations of the four supports are assumed to have settled in different model instances. Two extreme variations of Young’s modulus for steel are used. They are 180 and 220 GPa.
By taking different combinations of the above variables, 555 model instances are generated. The model responses considered are strains in the members of the transmission line tower in the axial direction. Then for each model instance, different strain values are tabulated for different members in the tower. All the members are considered as probable locations where sensors could be placed. In total, 320 probable sensor locations are considered.
The entropy function is used to evaluate the degree of separation between the model predictions. Putting sensors at locations where model predictions have the maximum variation helps to eliminate maximum number of candidate models after a measurement is taken. Hence, sensor configuration is selected such that the entropy of the set of model predictions is the maximum. A greedy algorithm is adopted in which sensors are added one at a time based on maximum entropy. This is a variation of the joint entropy algorithm used by Papadopoulou et al. ( Step 1: Initialize a list to store the sensor locations that have been selected. To start with, this list is empty. Initialize a list to store sets of model instances that cannot be separated further with the current sensor configuration. To start with, the list contains a single element, which is the initial set of model instances. Step 2: Loop over each model set Step 2.1: Loop over each sensor location Step 2.1.1: Compute the histogram of predictions of all the models in the current set at the current sensor location. Step 2.1.2: Compute the entropy of Step 2.1.3: Move to the next sensor location and continue the loop 2.1.1. Step 2.2: Select the next model set and continue the loop 2.1. Step 3: Select the sensor location Step 4: Loop over all the sets in the list of non-separable model sets. Divide each set into subsets that cannot be separated further using the sensors that are currently selected. Remove the parent set from the list and add the subsets into the list if the number of model instances in the subset is more than one. Step 5: If there is any more sensor location that has not been selected, repeat from Step 2.
Additional details about these steps are provided in the following sections.
In order to compute the entropy of predictions at a given location, a histogram needs to be created. The interval width of the histogram is chosen based on this principle: when the measured value is at the midpoint of an interval, all the model predictions that are within the threshold of errors should lie within that interval. Thus, the half width of the interval is equal to the error threshold. There are errors in modeling as well as measurements. Finite element analysis does not give accurate results because of effects that are not modeled and the assumptions involved in formulating the mathematical model. Similarly, there are errors in measurements because of the precision and resolution of sensors. Hence, the error threshold is computed as the sum of the measurement and modeling errors. This is taken as the half width of the histogram.
In order to create the histogram, the range of predictions (
The probability
In order to estimate modeling errors, effects due to eccentricity and P-Delta effect are considered. Separate analysis was conducted with and without these effects and a rough estimate of modeling errors was made. Mean variation due to eccentric connections was found to be 6.73% and that due to P-Delta effect was found to be 7.05%. Measurement errors depend on the accuracy of sensors used. Here, HBM strain gage sensors are used with an accuracy of 0.1%. The total error is estimated as the sum of the absolute values of modeling and measurement errors (Vernay et al.,
After a sensor is chosen, the measurement from that sensor can be used to eliminate model instances whose predictions lie within other intervals. The model instances whose predictions lie within the same interval as the measurement cannot be separated with this sensor. Therefore, each set of model instances is further subdivided after a sensor is selected. Since the measured value could be within any interval, one new subset is created for each interval which contains more than one model instance. Since the initial set of model instances is hierarchically divided after each new sensor is added, this procedure automatically takes care of redundant sensor information (mutual information between sensors). That is, only those model instances that cannot be separated by the previous sensor configuration are subdivided by the new sensor. Since the entropy is calculated separately for each subset, it is conceptually the same as the joint entropy calculation used in Papadopoulou et al. (
There are many possible ways of evaluating the performance of a sensor configuration. Here, the objective is to identify the state of the system. Hence, the metric chosen is the number of models that cannot be separated with the sensor configuration. An ideal configuration should be able to separate all the models and the number of non-separable models should be 0. However, this may not be possible in practice because of the low accuracy of sensors and uncertainties in modeling. In general, the number of non-separable models decreases with the addition of a new sensor. Hence, the performance of the sensor configuration can be compared only for a fixed number of sensors. The lower the number of non-separable models with a specified number of sensors, the better is the sensor configuration.
In practice, there are many other criteria that are important for selecting sensor types and their locations. Cost and feasibility of installation are important considerations. These criteria are not included in the present work.
Proposed optimal sensor placement methodology is compared with the intuitive method of sensor placement, that is, by locating members having maximum stress. This comparison is performed as follows:
Sensors are added one by one according to the optimal sensor placement algorithm. For each configuration, the number of non-separable models is computed. The locations having maximum stress are selected one by one. By placing a sensor on each of these members, the number of non-separable models using the intuitive method is computed. The number of non-separable models using the two methods is compared for the same number of sensors selected.
The histograms for the two locations 38 and 84 are shown in Figures
In the first iteration of optimal sensor placement, the entropy is computed for all the locations. The first 10 locations in decreasing order of entropy are shown in Table
Locations | Entropy |
---|---|
38 | 2.774 |
32 | 2.773 |
31 | 2.754 |
284 | 2.739 |
In Table
Repeating the steps, the best locations selected subsequently are the members 253, 257, 201, 34, and 1. The maximum number of non-separable models decreases to two, after selecting six sensors. It does not decrease further after adding more sensors. This is summarized in Table
Location | Max non-separable model instances |
---|---|
38 | 121 |
253 | 68 |
257 | 21 |
201 | 10 |
34 | 3 |
1 | 2 |
2 | 2 |
The first three sensor locations are on the second segment of the transmission tower from the bottom. The first sensor is on a transverse bracing and the other two on a K-bracing. The fourth sensor location is at the top of the tower, where the wires are supported. The fifth sensor location is again on the second segment, where the first three sensors are located. The sixth sensor is on a leg that is connected to the foundation. This configuration of six sensors is able to separate out the model instances in the initial set such that there are at most two model instances in a subset. It should be noted that all the locations in Table
Generally, engineers would choose sensor locations by considering either maximum forces or stress values or both. Table
Beam | Axial force (kN) |
---|---|
2 | 334.49 |
215 | 333.89 |
22 | 302.19 |
248 | 301.73 |
214 | 287.31 |
4 | 286.54 |
254 | 259.18 |
24 | 258.54 |
410 | 254.62 |
44 | 220.71 |
The heavily loaded members in a transmission tower are generally leg members. Hence, when we place sensors based on maximum forces, possible locations will be at leg members as shown in Figure
The performance of the intuitive method is evaluated using the metric of the maximum number of model instances that cannot be separated. It is compared with that of the optimal sensor placement methodology. For each selected number of sensors, the metrics for the two algorithms are plotted in Figure
The conclusion related to the superiority of the optimal sensor placement algorithm is based on a single case study. By repeating the study using towers with different geometries and loading conditions, the generality of the conclusion could be verified. However, it is expected that the present sensor placement strategy will have superior performance since it has firm foundations on information theory, whereas the intuitive method lacks scientific basis. Future work involves comparing the performance of other sensor placement strategies and conducting full-scale experiments to validate the results.
This paper presented a methodology for the placement of sensors on transmission line towers for explaining its actual structural behavior. The methodology is based on the concepts of entropy and model falsification. Sensor locations are selected based on maximum entropy such that there is maximum separation of model instances that represent different possible combinations of parameter values that are uncertain. Thus, the optimal combination of sensor locations helps to narrow down to a few possible explanations of structural behavior.
The performance of the proposed algorithm is compared to that of an intuitive method in which sensor locations are selected where the forces are maximum. It is shown that the intuitive method results in much higher number of non-separable models compared to the optimal sensor placement algorithm, especially when fewer sensors are used. The following are the specific conclusions made from the present study:
Shannon’s entropy function is a useful tool which can identify the variability between the candidate models at possible sensor locations. The methodology using the entropy function provides support for sensor placement in the condition assessment of transmission towers. The part below the waist of the transmission tower is prone to significant variations under the considered modeling assumptions, which is evident from the fact that the top sensor locations are almost always below the waist of the tower. Proposed method of placing sensors helps to identify behavior models that can explain the real behavior of transmission towers, which cannot be expected from the conventional method. This methodology can minimize unnecessary data collection and interpretation by avoiding redundant sensors that provide no additional information.
The limitations of the present study are summarized as follows:
Factors such as cost and ease of installation have not been included. The interval width of the histogram depends on estimates of modeling and measurement errors; hence, the optimality of the proposed sensor network is sensitive to the accuracy of these estimates. The conclusions are drawn using a single case study. Other sensor placement algorithms have not been compared. Actual experiments have not been carried out and the results are based purely on theoretical analysis.
Despite these limitations, the proposed methodology is expected to be a valuable tool to engineers in their decision-making process.
The first author proposed the methodology used in the research and supervised the graduate student. The second author collected the data and performed the analysis.
The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.
The authors wish to thank the help provided by Mr. K. S. Ranjith and other colleagues at the institute.
The postgraduate study of the second author is supported by funding from Larson and Toubro (L&T).