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Research on earlywarning index of the spatial temperature field in concrete dams
 Guang Yang^{1, 2}Email authorView ORCID ID profile,
 Chongshi Gu^{1, 3},
 Tengfei Bao^{1, 3},
 Zhenming Cui^{3} and
 Kan Kan^{3}
 Received: 12 June 2016
 Accepted: 4 November 2016
 Published: 14 November 2016
Abstract
Warning indicators of the dam body’s temperature are required for the realtime monitoring of the service conditions of concrete dams to ensure safety and normal operations. Warnings theories are traditionally targeted at a single point which have limitations, and the scientific warning theories on global behavior of the temperature field are nonexistent. In this paper, first, in 3D space, the behavior of temperature field has regional dissimilarity. Through the Ward spatial clustering method, the temperature field was divided into regions. Second, the degree of order and degree of disorder of the temperature monitoring points were defined by the probability method. Third, the weight values of monitoring points of each regions were explored via projection pursuit. Forth, a temperature entropy expression that can describe degree of order of the spatial temperature field in concrete dams was established. Fifth, the earlywarning index of temperature entropy was set up according to the calculated sequential value of temperature entropy. Finally, project cases verified the feasibility of the proposed theories. The earlywarning index of temperature entropy is conducive to the improvement of earlywarning ability and safety management levels during the operation of high concrete dams.
Keywords
 Ward spatial clustering
 Temperature entropy
 Projection pursuit
 Earlywarning
 Synergetics
 Spatial temperature field
Background
Temperature variation is one of main safety monitoring items during the operation of concrete dams. A scientific and reasonable earlywarning index (Wu 2003) of the dam body’s temperature has important significance in accurate hazard recognition and dam security protection. Temperature control is a key factor influencing whether the temperature cracks occur or not. In the process of operation, improper temperature control (Zhu 2014) will lead to dam cracking and even endanger the safety of dam bodies. Generally, the concrete dams belong to the mass concrete structure which has bad temperature conductivity, and they are often exposed to the outside, contacting with water, air and so on. All kinds of factors like temperature change outside, cement hydration heat and constraint stress may produce tensile stress. With the limit ability of concrete tensile strength, cracks tend to appear in the mass concrete structure. Combined with longacting service characteristics of the concrete dams, advanced mechanical and mathematical theories are employed to scientifical theory for earlywarning, as well as to conduct timely and effective determination of the safety status of dams. These activities are critical to perform safe operations and determine crucial topics in dam safety monitoring field research.
To date, the warning methods of dam body’s temperature have obtained a series of research achievements. Temperature earlywarning index (Su et al. 2011, 2012; Zhou et al. 2012) is an important index to control cracks of concrete dam. When the current concrete temperature exceeds earlywarning index of temperature, effective temperature control measures must be taken to control the concrete temperature. In 2004, Wang took some temperature measuring points as constraint conditions, using threedimensional finite element analysis method to represent the characters of the dam’s temperature field (Wang 2004). Wu applied sensitivity analysis method to determine temperature field’s main impact factors (Wu and Song 2011). Combined with the situation of the practical concrete dam temperature control, Qu et al. applied expert evaluation and the theory of entropy weight to establish a multiobjective fuzzy mathematical model for the concrete dams (Qu et al. 2012). Although the abovementioned theories and methods complement and improve traditional methods in solving difficult problems in dam safety, but traditional warnings theories are targeted at a single point which have limitations, and the scientific warning theories on the global behavior of the spatial temperature field are nonexistent. Thus, a scientific and accurate earlywarning index based on the spatial temperature field should be studied to improve the warning ability of the concrete dams.
In this paper, accordingly, the spatial temperature field at different elevations and positions have different behaviors, that is, the regional dissimilarity exist. Through Ward spatial clustering, the spatial temperature field was divided into regions. On this basis, an expression of temperature entropy was proposed based on the synergetics and entropy. This expression can comprehensively evaluate the overall variation of temperature field in concrete dams. With this expression, the sequential value and the earlywarning index value of temperature entropy was analyzed and determined via the small probability method. Finally, project cases verified the feasibility of the proposed theories.
Research on the partitioning method of the spatial temperature field
Measurement method of similarity degree among the monitoring values of the temperature measuring points
N is assumed as the number of all temperature monitoring points, T is the monitoring time series, and \(x_{it} (i = 1,2, \ldots ,N;\, t = 1,2, \ldots ,T)\) represents the temperature data set. For set x _{ it }, S _{ t } is the standard deviation at time t, and the following three distances are defined to describe the similarity degree among the monitoring values of the temperature measuring points.
The “comprehensive distance,” which is a weighted array of “absolute distance” and “incremental distance,” comprehensively describes the similarity of change of the measured values at the measuring points.
Measurement method of similarity degree among the different spatial temperature field partitions
The assumption is that n combinations are conducted in the partitioning process based on the threshold value method, and the ratio of the distance between partitions in the lth partitioning to that in the last partitioning is \(S_{l} = \frac{{D_{l} }}{{D_{n  1} }}\). If the difference between S _{ l } and S _{ l+1} is small and that between S _{ l } and S _{ l−1} is large, then the corresponding distance D _{ l } between the partitions can be the threshold value of the partition.
Partitioning flow

Step 1: Expressions (1) and (2) are used to calculate the “absolute distance” and “incremental distance,” respectively.

Step 2: The calculated coefficient values are substituted into expression (5), and the comprehensive distance d _{ mn }(CED) between every two measuring points among the N measuring points is calculated.

Step 3: Initially, all measuring points selfform a partition, the number of partitions is k = N, distance matrix D ^{(1)} between partitions is built, and the ith partition is \(G_{i} = \left\{ {X_{(i)} } \right\}(i = 1,2, \ldots ,N)\).

Step 4: According to the principle of the minimum sum of the squares of deviations, two partitions with the minimum comprehensive distance are combined as a new partition, the comprehensive distance d _{ ij }(CED) between the new partition and other partitions is calculated, a new distance matrix is obtained, and Steps 4 and 5 are repeated until the partitioning ends.

Step 5: The optimal partition combination of the measuring points is obtained based on the threshold value method. Thereafter, the optimal number K of partitions is obtained.
Temperature entropy
Previous studies have reported that the temperature field changes gradually, whereas the concrete structure has an obvious nonlinear selforganization mechanism and a multiscale coupling effect during the operation of concrete dams (Andrie and Chen 1975; Xie 2004; Bai 2008). The macromechanical properties of the concrete structure have a collaborative selforganized phenomenological response of internal multiscale physical quantities to various effects. Currently, the multiscale synergic evolution of a system is difficult to elaborate because of the limited theoretical basis and poor calculation methods. Although the system has different evolution equations at different scales and levels, energy is a general physical quantity that could run across scales and levels. To date, some researchers have studied system evolution from the perspective of energy. Reference analyzed the macro physical significance of entropy based on the available energy of closed thermal systems (Yu 1995). Reference discussed the variation law of deformation energy and the mechanism of sudden energy changes during the gradual evolution of concrete dams (Wu and Guo 2010). Reference studied the analytical methods of multiobjective decisionmaking during concrete dam operation based on the entropy weight (Maken et al. 2014). The abovementioned studies imply that the orderly evolution of systems can be effectively studied from the perspective of energy.
Entropy features of the spatial temperature field
Methods of characterizing contributions of single observation point
Constructing temperature entropy of single measured value
The degree of order for a single temperature monitoring point, a dimensionless standardized value, is introduced before calculating the singlepoint temperature entropy. This value measures the degree of temperature order at single point. During the operation period, a concrete dam body is mainly influenced by the dead load and temperature load; consequently, the temperature variation can be viewed as a probability event (Wu 2003). For those complicated mechanical properties caused by materials, load, and environment and so on, they are reflected comprehensively in the measured temperature data from thermometers placed in the different part of dam. Therefore, when we proposed the temperature entropy expression that can describe degree of order of the temperature field in concrete dams, we didn’t consider various mechanical and thermal properties. The small probability method for calculating the monitoring index discloses that the smaller the probability of temperature reoccurrence is, the more dangerous the dam will be. Therefore, the probability of temperature occurrence at monitoring points represents the safety degree of the temperature load and can be used as an important index to measure the degree of order of the temperature field.
 1.When temperature increases, u _{ ij } is$$u_{ij} = F(x_{ij} ) = \int_{  \infty }^{{x_{ij} }} {f_{i} (\delta )} d\delta$$(11)
 2.When temperature decreases, u _{ ij } is$$u_{ij} = 1  F(x_{ij} ) = \int_{{x_{ij} }}^{ + \infty } {f_{i} (\delta )} d\delta$$(12)
Formulas (11) and (12) state that larger deviations of the measured temperature of a concrete dam from the initial temperature would lead to larger u _{ ij } values; otherwise, the u _{ ij } will be smaller.
Obviously, Formulas (11) and (12) indicate that \(0 < u_{ij}^{1} < 1\) and \(0 < u_{ij}^{2} < 1\). Meanwhile, \(u_{ij}^{1} + u_{ij}^{2} = 1\), which implies that the sum of the degree of order and the degree of disorder is 1. The degree of disorder decreases while the degree of order increases, and vice versa.
Weight entropy
The overall temperature entropy includes two levels: overall temperature entropy at the top and singlepoint temperature entropy at the bottom. The monitoring points interact with each other and influence the evolution of overall temperature entropy together.
The temperature monitoring point with greater contributions to the overall temperature entropy possesses the higher weight. According to the principle of synergetics, the synergic evolution equation involves both stable and unstable modals. Structural evolution is mainly determined by unstable modals. Therefore, only unstable modes dominate the orderly variation of the dam temperature when the overall temperature field change is under critical phase change. The stable modals slightly influence the orderly variation of the dam temperature.
Weight optimization based on projection pursuit
Construction of the temperature entropy of the spatial temperature field
The degree of order sequence (\(\{ u_{ij}^{1} \}\)), the degree of disorder sequence (\(\{ u_{ij}^{2} \}\)), the deformation entropy sequence (\(\{ S_{i}^{j} \}\)), and the weight distribution entropy of the spatial temperature field were calculated by using Formula (11)–(14). Based on these equations, the formula to calculate the multipoint temperature entropy was deduced as follows. The degree of order and the weights of the monitoring points were combined. For the ith monitoring point, the contribution of its degree of order to the overall temperature entropy is \(\omega_{i} u_{ij}^{1}\), whereas the contribution of its degree of disorder to the overall deformation entropy is \(\omega_{i} u_{ij}^{2}\). According to definition of generalized information entropy, the orderly entropy of the whole temperature field is:
Formula (22) reveals that the overall temperature entropy involves the weight distribution entropy (\(S_{\omega }^{j}\)) and the weighted average of the temperature entropy of different monitoring points (\(\sum\nolimits_{i = 1}^{n} {\omega_{i} S_{i}^{j} }\)).
Establishment of an earlywarning index based on the temperature entropy
Subsequently, the distribution of this sample space was tested by smallsample statistical testing methods (e.g., the A–D method or the K–S method) to determine the distribution function F(X) of its probability density f(x). K–S (Kolmogorov–Smirnov) method bases on empirical distribution function, which is used to determine whether a sample is from a specific distribution. A–D (Anderson–Darling) method is a correction of A–D method and gives weight to the distribution of the tail. Besides, K–S inspection has nothing to do with the specific distribution, that’s to say, its critical value doesn’t depend to the tested specific distribution.
S _{ m } is the earlywarning index value under the failure probability of α. Dam security can be evaluated by comparing the temperature entropy and S _{ m }. If temperature entropy is smaller or equal to S _{ m }, attention should be paid to determine probable causes, to strengthen the monitoring system, and to analyze whether other monitoring items have abnormalities. An appropriate α should be chosen to calculate S _{ m }. The value of α is determined by various factors, such as engineering grade, engineering scale, operation progress and so on. Meanwhile, several confidence values of α shall be set for the dam risk management: α _{1}, α _{2}, …, α _{ n }(α _{1} > α _{2} > ···α _{ n }).The corresponding S _{ m } (\(S_{{m_{1} }} ,S_{{m_{2} }} , \ldots ,S_{{m_{n} }}\)) should be calculated to get the multilevel early warning index of the overall temperature field in dams.
Result
Project overview
To study the temperature field changes during dam operation and the safe impounding period, temperature monitoring data from August 1, 2007 to January 20, 2014 were used as the sample data in this paper. Temperature data of all monitoring points within the 6th dam block were used to calculate temperature entropy and earlywarning index value.
Partitioning calculation results
Measuring point clustering partition table of the spatial temperature field
Partitions  Measuring points  

Partition I  A6T15  A6T16  A6T21  A6T22  A6T26  
Partition II  A6T09  A6T10  A6T15  A6T16  A6T20  A6T23  A6T25  A6T29  
Partition III  A6T06  A6T07  A6T08  A6T11  A6T12  A6T13  A6T14  A6T17  A6T18  A6T19  A6T24  A6T27 
A6T28  A6T35  A6T36  A6T37  A6T38 
Calculation results of the weight ω _{ i }
Weight table of the observation points of the 6th dam block in the rising and declining air temperature phases
Partitions  Measuring points  Temperature rise phase  Temperature decline phase 

Partition I  A6T15  0.13  0.23 
A6T16  0.14  0.09  
A6T21  0.34  0.39  
A6T22  0.32  0.15  
A6T26  0.07  0.14  
Partition II  A6T09  0.07  0.05 
A6T10  0.09  0.11  
A6T15  0.12  0.14  
A6T16  0.13  0.11  
A6T20  0.26  0.27  
A6T23  0.18  0.14  
A6T25  0.04  0.05  
A6T29  0.06  0.07  
Partition III  A6T06  0.06  0.04 
A6T07  0.04  0.07  
A6T08  0.03  0.07  
A6T11  0.06  0.05  
A6T12  0.04  0.05  
A6T13  0.05  0.04  
A6T14  0.07  0.08  
A6T17  0.06  0.06  
A6T18  0.08  0.04  
A6T19  0.09  0.08  
A6T24  0.06  0.07  
A6T27  0.07  0.05  
A6T28  0.03  0.07  
A6T35  0.06  0.04  
A6T36  0.05  0.05  
A6T37  0.07  0.06  
A6T38  0.08  0.08 
Calculation of spatial temperature entropy
Annual extreme temperature entropies
Temperature entropy  Year  

2008  2009  2010  2011  2012  2013  
Minimum value of partition I  0.763  0.632  1.235  1.032  1.124  1.342 
Minimum value of partition II  1.142  1.208  1.126  1.034  1.171  1.164 
Minimum value of partition III  1.132  1.418  1.324  1.053  1.132  1.145 
Calculation of earlywarning index of concrete dam
K–S test results
Probability distribution  Partition I  Partition II  Partition III 

Lognormal distribution  0.32  0.17  0.25 
Normal distribution  0.41  0.21  0.31 
Uniform distribution  0.78  0.95  0.86 
Triangular distribution  0.32  0.44  0.43 
Exponential distribution  0.44  0.46  0.65 
γ distribution  0.34  0.53  0.21 
β distribution  0.53  0.71  0.53 
The most reasonable probability distribution  Normal distribution  Normal distribution  Normal distribution 
Parameter values of the probability density function
Partition  Parameter values  

μ  σ ^{2}  
Partition I  1.021333  0.07549 
Partition II  1.140833  0.003519 
Partition III  1.200667  0.019356 
Earlywarning index values of all partitions
Earlywarning level  α = 5%  α = 1% 

Earlywarning index values  
PartitionI  0.569363  0.381156 
Partition II  1.04325  1.002615 
Partition III  0.971805  0.876503 
In this paper, the earlywarning index values of all partitions was analyzed through the theoretical method. If the temperature entropy value of partition I reaches 0.569363, the dam is in the state of primary warning. If the temperature entropy value of partition I reaches 0.381156, the dam is in the state of secondary warning. If the temperature entropy value of partition II reaches 1.04325, the dam is in the state of primary warning. If the temperature entropy value of partition II reaches 1.002615, the dam is in the state of secondary warning. If the temperature entropy value of partition III reaches 0.971805, the dam is in the state of primary warning. If the temperature entropy value of partition III reaches 0.876503, the dam is in the state of secondary warning.
Conclusions
 1.
Based on the Ward spatial clustering, partitioning of the spatial temperature field is conducted according to the intimacy degree among all the observed values. The partitioning principle attempts to make the change rule similarity degree within partitions as high as possible while make the change rule similarity between partitions as low as possible.
 2.
The degree of order and degree of disorder of the temperature monitoring points were defined by the probability method. The weight of each temperature monitoring points was explored via projection pursuit.
 3.
According to the coordinated and orderly evolution characteristics of the spatial temperature field, a temperature entropy expression that can describe degree of order of the spatial temperature field in concrete dams was established.
 4.
The earlywarning index of temperature entropy was set up via the small probability method according to the calculated sequential value of temperature entropy.
Declarations
Authors’ contributions
GY: Conceived and designed the study, calculations, analysis of the data, and wrote the manuscript. CSG and TFB: Participated in design of the study, and conducted analysis. ZMC and KK: carried out the manuscript editing and manuscript review. All authors read and approved the final manuscript.
Acknowledgements
Supported by National Natural Science Foundation of China (Grant Nos. 51139001, 41323001, 51479054, 51579086, 51379068, 51579083, 51279052, 51579085), Jiangsu Natural Science Foundation (Grant No. BK20140039), Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130094110010), Project Funded by the Priority Academic Program Development of Jiangsu Higher Education Institutions (Grant No. YS11001), Jiangsu Province “Six Talent Peaks” Project (Grant Nos. JY008, JY003), Central University Basic Research Project (Grant No. 2015B20714), Open Research Fund of State Key Laboratory of HydrologyWater Resources and Hydraulic Engineering (2015491411).
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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