Multivariate State Estimation Technique Combined with Modiﬁed Information Entropy Weight Method for Steam Turbine Energy Efﬁciency Monitoring Study

: An energy efﬁciency monitoring method of the steam turbine system is studied in this paper. Multivariate state estimation technique (MSET) is utilized to compare the actual monitoring parameters and the healthy data of the equipment in normal working condition with a multi parameter estimation model. Due to the limitation of a single heat rate index in evaluating energy efﬁciency variation, the energy efﬁciency deviation degree combined with improved information entropy weight is proposed to judge the steam turbine’s operation condition levels. The index value in the modiﬁed weight method has been searched for more steady weight values calculated by information entropy values with small variation. Taking a 600 MW unit as an example, the energy efﬁciency levels of the unit under a 550 MW normal working condition are clustered into four groups, testifying the MSET model correctness and calculating the deviation degree value. Then, the energy efﬁciency status monitoring model is utilized to record residuals of actual data and estimated data during abnormal energy efﬁciency period. The residuals over deviation degree are then marked and judged as related with the abnormal data. The results show that the MSET model can timely and accurately judge the change of unit operation state, and the deviation degree calculated by the modiﬁed information entropy weight method can provide earlier warnings for the abnormal energy efﬁciency working conditions.


Introduction
Common energy consumption diagnosis methods are usually divided into two categories: the heat method and exergy method, based on the first law of thermodynamics and the second law of thermodynamics, respectively [1]. The heat method is used to evaluate the unit economy efficiency from the perspective of energy conversion quantity, such as the equivalent enthalpy drop method [2], heat balance method, cyclic function method [3], matrix method, and so on. An improved equivalent enthalpy drop method was proposed to determine the equivalent enthalpy drop through the isentropic expansion process of steam [4]. The involved method can further diagnose the causes of the economic reduction in the steam turbine system, compared with the traditional one. The matrix theory was used to deduce the main flow relations of the cyclic function method from the generalized mathematical model of the thermal system [5], which made the thermal economic analysis and diagnosis of the thermal system more effective. However, these methods cannot evaluate the system efficiency from energy quality aspect. The second law of thermodynamics analysis method, such as entropy method and exergy method, can analyze the unit economy from both quantity and quality aspects [6]. A general loss matrix equation [7] of steam water distribution in a thermal system has been established based on the loss balance theory, which can accurately calculate the loss distribution of

Preliminaries and Problems Statement
The research object in the paper is a 600 MW supercritical intermediate reheat condensing steam turbine. The thermodynamic process of this condensing steam turbine can be seen in Figure 1. The feedwater is heated by the boiler and turns into superheated steam. The main steam at a certain pressure and temperature enters the high pressure (HP) cylinder of steam turbine, flows through the nozzle and expands in the nozzle to obtain a high speed. Two extraction stages in HP cylinder feed the #1 and #2 HP heaters. Then, the HP cylinder exhaust goes back into the boiler and then works the reheated steam into the intermediate pressure (IP) cylinder after being reheated. In the IP cylinder, there are two extraction stages supplying the #3 HP heater and the deaerator. Exhaust steam of IP cylinder goes into the low pressure (LP) cylinder with four extraction stages for #5, #6, #7 and #8 LP heaters. The LP cylinder exhaust goes into the condenser as the cold end of Rankine cycle. The heat energy of steam is converted into steam kinetic energy in the nozzle, and the kinetic energy is converted into mechanical energy by the moving blade. The rotors of the steam turbine and generator are connected by coupling. When the rotor of the steam turbine rotates at a certain speed, the generator rotor also rotates, generating electricity.
Energies 2021, 14,6795 3 of 20 under a 550 MW normal working condition are clustered by selected operation data, testifying the MSET model correctness and calculating the deviation degree value. Then, the energy efficiency status monitoring model is utilized to record residuals of actual data and estimated data during abnormal energy efficiency period. The residuals over deviation degree are then marked and can provide an earlier diagnosing, compared with the actual operation state. The results of MSET combined with the modified information entropy method can provide the earlier warning information for the energy efficiency abnormal variation.

Preliminaries and Problems Statement
The research object in the paper is a 600 MW supercritical intermediate reheat condensing steam turbine. The thermodynamic process of this condensing steam turbine can be seen in Figure 1. The feedwater is heated by the boiler and turns into superheated steam. The main steam at a certain pressure and temperature enters the high pressure (HP) cylinder of steam turbine, flows through the nozzle and expands in the nozzle to obtain a high speed. Two extraction stages in HP cylinder feed the #1 and #2 HP heaters. Then, the HP cylinder exhaust goes back into the boiler and then works the reheated steam into the intermediate pressure (IP) cylinder after being reheated. In the IP cylinder, there are two extraction stages supplying the #3 HP heater and the deaerator. Exhaust steam of IP cylinder goes into the low pressure (LP) cylinder with four extraction stages for #5, #6, #7 and #8 LP heaters. The LP cylinder exhaust goes into the condenser as the cold end of Rankine cycle. The heat energy of steam is converted into steam kinetic energy in the nozzle, and the kinetic energy is converted into mechanical energy by the moving blade. The rotors of the steam turbine and generator are connected by coupling. When the rotor of the steam turbine rotates at a certain speed, the generator rotor also rotates, generating electricity.  The heat rate is usually taken as an important index to study and measure the power plant economic performance. Thus, the assessment of the heat rate index has been widely focused on in power plants, and has become one of the important means to monitor the performance of steam turbines [18]. The heat rate value indicates the energy consumed when the turbine generates 1 KWh electricity, calculated as Equation (  The heat rate is usually taken as an important index to study and measure the power plant economic performance. Thus, the assessment of the heat rate index has been widely focused on in power plants, and has become one of the important means to monitor the performance of steam turbines [18]. The heat rate value indicates the energy consumed when the turbine generates 1 KWh electricity, calculated as Equation (1) where P is the power output; MW: F g , F f , F r , F c and F sp stand for the main steam flow, feed water flow, reheated steam flow, cold reheated steam flow and spray flow respectively; t/h: and analysis on Rankine cycle, the main steam parameters, reheat steam parameters, feedwater parameters, condenser vacuum and other parameters can be taken as main characteristic parameters for thermal economy analysis in the condensing steam turbine.

Methods
Multivariable state estimation technique (MSET) algorithm is a nonlinear multivariate predictive diagnosis technology proposed by Singer [19]. The algorithm is based on the unit's normal operation data, and estimates the normal operation data under the current condition according to the similarity principle. The actual operation condition of the unit can be judged by comparing the residual between the actual data of the system and the estimated data by MSET algorithm. With the characteristics in dealing with industrial operation data, the MSET algorithm has been widely used in auxiliary equipment fault diagnosis [20] and equipment monitoring [21].

MSET Method for Parameter Estimation
MSET, a kind of advanced pattern recognition technology, uses the similarity between monitoring parameters under normal operation conditions to realize the state estimation [22]. The algorithm reveals the relationship between parameters based on historical data, and estimates the real state of the object according to the obtained knowledge.
 has m data samples and n interrelated parameters. The main steps of MSET algorithm can be listed as follows: Step 1: Construct memory matrix D, shown in Equation (2). The construction of the memory matrix is the core content of the MSET algorithm, and tends to cover the normal data of all operating conditions, as possible. The purpose to cover the whole normal data is to ensure the accuracy of prediction results. Usually, isometric interpolation method is taken for memory matrix construction.
Step 2: Calculate estimated vector X est according to the actual observed vector X obs as Equation (3) where W is the corresponding vector weight in D and X est is a linear combination of historical observation vectors. W can be calculated by residual minimization constraint by X obs and X est , as Equation (4) where ε is the minimum residual value between X obs and X est . Then, search for the partial derivative of w 1 , w 2 , . . . w m respectively as shown in Equation (5) Equation (6) can be modified as Equation (7) by using nonlinear operator instead of dot multiplication of matrix, since the matrix D is irreversible.
where ⊗ is a nonlinear operation symbol for Euclidean distance calculation in this paper as Equation (8).
Then, the estimated vector can be obtained by bringing Equation (7) into Equation (3), as the following Equation (9)

Modified Information Entropy Weight Method
C. E. Shannon introduced the concept of entropy from the second law of thermodynamics into the field of informatics in 1948, taking entropy to measure the amount of information [23] widely used in various fields. Generally speaking, if the information entropy of a parameter is smaller, it indicates the parameter value has a bigger variation degree. Then, this parameter may contain more information, and can play a more important role in data analysis. On the contrary, a parameter with a larger entropy value means a less important role in index evaluation. Therefore, parameter information entropy values can be calculated to quantize different parameters' weight in comprehensive evaluation.

Traditional Information Entropy Weight Method
. . . . . . x m1 , x m2 , . . . , x mn     as the evaluated object, and X j (j = 1, 2, . . . , n) as the parameter set. m, n represent data number and parameter number, respectively. The parameter values set x ij are then taken to calculate the characteristic weight value set p ij as Equation (10).
Then, the information entropy value of parameter X j can be expressed as Equation (11) where ln p ij = 0 when p ij = 0. 0 ≤ H j ≤ 1, and if x ij = x sj , i = s, then H j = 1. The traditional information entropy weight method uses parameter entropy value for the jth parameter information entropy weight w 0 j calculation as Equation (12) [24]  = 0.3333 0.5000 = 0.6667. In fact, four groups contain different information deviation, thus, the traditional information entropy weight method leads to an unreasonable result, especially for the small information entropy difference values in groups 1-3 in Table 1.

Modified Information Entropy Weight Method
In order to avoid the shortcomings by traditional information entropy weight method, a weight index 1 w is then calculated as Equation (13)

Modified Information Entropy Weight Method
In order to avoid the shortcomings by traditional information entropy weight method, a weight index w 1 is then calculated as Equation (13) where H represents the average value of parameters' information entropy values unequal to 1. Assume a group of information entropy values as H j , including H p and H q . Then, the compound information entropy weight is modified as Equation (14) where the information entropy weight value ratio of H p and H q is then expressed as The modified information entropy weight method is taken to calculate with the four group information entropy values as Table 1. The results are shown in Figure 3 with the index n varying from 0 to 50. The modified information entropy weight method is taken to calculate w w with the four group information entropy values as Table 1. The results are shown in Figure 3 with the index n varying from 0 to 50.  The fourth group in Table 1 shows relatively bigger differences between information entropy values. Thus, curves in Figure 3d can be compared with Figure 2b for the suitable value of n . Table 2 lists   The fourth group in Table 1 shows relatively bigger differences between information entropy values. Thus, curves in Figure 3d can be compared with Figure 2b for the suitable value of n. Table 2 lists    . Thus, the modified information entropy weight method can be clearly expressed as Equation (16) Therefore, information entropy weight values calculated by Equation (16) can reflect the relationship between information entropy differences and weight values.

MSET Model Validation
Energy consumption characteristic data are then selected from steady state database from May 1 to May 30 in 2017, under a 550 MW working condition (with the load in the range (548, 552) MW), and the environment temperature in the range (27, 30) • C. According to Equation (1), unit characteristic parameters of energy consumption containing 13 attributes are seen in Table 3. The modeling process is shown in Figure 4 for data mining guidance. The 325 data are then clustered by Kmeans method into four clusters, and clustering results based on heat rate can be seen in Table 4. The table shows that #1 cluster contains data samples with the best heat rate, indicating the best working condition. Furthermore, #4 cluster contains those with the highest heat rate, meaning the highest energy consumption, while #2 and #3 clusters are those in the middle level. A total of 50 data from the #1 cluster are then      In Figure 7a       Moreover, the MSET model established by partial data in cluster #1 is utilized to test the rule between the change in unit operation level and the estimation accuracy. The data  Moreover, the MSET model established by partial data in cluster #1 is utilized to test the rule between the change in unit operation level and the estimation accuracy. The data from cluster #1, cluster #2, cluster #3 and cluster #4 are taken as test data. Error percentage values are then calculated by parameters as main steam temperature, main Moreover, the MSET model established by partial data in cluster #1 is utilized to test the rule between the change in unit operation level and the estimation accuracy. The data from cluster #1, cluster #2, cluster #3 and cluster #4 are taken as test data. Error percentage values are then calculated by parameters as main steam temperature, main steam pressure, reheated steam temperature, reheated steam pressure, feedwater temperature, condenser vacuum and heat rate, seen in Figure 9. Cluster #1 obtains the lowest error percentage values as 0.09%, 0.33%, 0.03%, 0.02%, 0.01%, 0.08% and 0.07%. Further, with the increase in heat rate in cluster #2 to cluster #4, the accuracy of the MSET multi parameter estimation model based on cluster #1 decreases gradually.
Energies 2021, 14,6795 14 of 20 steam pressure, reheated steam temperature, reheated steam pressure, feedwater temperature, condenser vacuum and heat rate, seen in Figure 9. Cluster #1 obtains the lowest error percentage values as 0.09%, 0.33%, 0.03%, 0.02%, 0.01%, 0.08% and 0.07%. Further, with the increase in heat rate in cluster #2 to cluster #4, the accuracy of the MSET multi parameter estimation model based on cluster #1 decreases gradually. Thus, the MSET multi parameter estimation model can be established through the samples in the lowest heat rate group (cluster #1 in the paper), and the residual changes between actual values of the parameters and the estimated ones in the model can be used to monitor unit working condition variation.

Case Analysis of Unit Energy Efficiency Monitoring
Unit energy efficiency status can be reflected by differences between actual observed and model estimated values, seen in Section 4.1. Therefore, the deviation degree index calculated by errors of energy consumption characteristic parameters as Equation (17) can measure the energy efficiency.
where k w represents the weight value of kth characteristic parameter, calculated by the modified information entropy weight method in Section 3.2.2. In Section 4.1, four classes have been clustered by heat rate value with Kmeans method. The deviation degree average value of MSET model with the data from cluster Thus, the MSET multi parameter estimation model can be established through the samples in the lowest heat rate group (cluster #1 in the paper), and the residual changes between actual values of the parameters and the estimated ones in the model can be used to monitor unit working condition variation.

Case Analysis of Unit Energy Efficiency Monitoring
Unit energy efficiency status can be reflected by differences between actual observed and model estimated values, seen in Section 4.1. Therefore, the deviation degree index calculated by errors of energy consumption characteristic parameters as Equation (17) can measure the energy efficiency.
where X i_obs = [x i1_obs , x i2_obs , . . . , x id_obs ] and X i_est = [x i1_est , x i2_est , . . . , x id_est ] are the actual observed and model estimated values of the ith characteristic parameters, respectively. d is the energy consumption characteristic parameter number. Since parameters have different influences on heat rate, weight values should be added in the deviation degree calculation, as Equation (18).
where w k represents the weight value of kth characteristic parameter, calculated by the modified information entropy weight method in Section 3.2.2.
In Section 4.1, four classes have been clustered by heat rate value with Kmeans method. The deviation degree average value of MSET model with the data from cluster #4 (group with the highest heat rate) as the observation vector can then be taken to determine the energy efficiency warning threshold δ.
Random factors such as measurement errors and noises in the actual operation may lead to a large fluctuation of deviation degree, and even error warnings. Thus, a sliding window method should be used to deal with the deviation calculation.
Assume deviation degree sets ∆D = [∆D 1 , ∆D 2 , . . . , ∆D n ] in time sequence, and time window width is m, seen in Figure 10. The ith deviation degree ∆D i can be calculated as Equation (19).
Energies 2021, 14, 6795 15 of 20 #4 (group with the highest heat rate) as the observation vector can then be taken to determine the energy efficiency warning threshold  .
Random factors such as measurement errors and noises in the actual operation may lead to a large fluctuation of deviation degree, and even error warnings. Thus, a sliding window method should be used to deal with the deviation calculation.
Assume deviation degree sets Therefore, the deviation degree index and energy efficiency warning threshold, combined with MSET model under different conditions can be used to handle energy efficiency monitoring in the steam turbine system, and the method's flow chart can be seen as Figure 11. The steps can be described as follows: Step 1: based on the heat consumption characteristic parameter condition library, the samples with the highest heat rate under each condition are selected.
Step 2: the energy efficiency deviation warning threshold  under each condition is determined by the multi parameter estimation model.
Step 3: according to the estimated vector of characteristic parameters under MSET multi parameter estimation model, the deviation degrees of energy efficiency are calculated, and are then processed by the sliding window method for the final deviation degree D  as Equation (19).
Step 4: operation data indicate a normal working condition, unless D  exceeds  . If D  exceeds  , there is the necessity for energy efficiency diagnosis and abnormal characteristic parameter location. Therefore, the deviation degree index and energy efficiency warning threshold, combined with MSET model under different conditions can be used to handle energy efficiency monitoring in the steam turbine system, and the method's flow chart can be seen as Figure 11. The steps can be described as follows: Step 1: based on the heat consumption characteristic parameter condition library, the samples with the highest heat rate under each condition are selected.
Step 2: the energy efficiency deviation warning threshold δ under each condition is determined by the multi parameter estimation model.
Step 3: according to the estimated vector of characteristic parameters under MSET multi parameter estimation model, the deviation degrees of energy efficiency are calculated, and are then processed by the sliding window method for the final deviation degree ∆D as Equation (19).
Step 4: operation data indicate a normal working condition, unless ∆D exceeds δ. If ∆D exceeds δ, there is the necessity for energy efficiency diagnosis and abnormal characteristic parameter location.
Historical data from 2020/05/17 09:34 to 2020/05/17 13:49 with 1 min as the time interval are then taken for the 600 MW steam turbine monitoring, partly listed in Table 5 Figure 11. Flow chart of energy efficiency monitoring method.
Historical data from 2020/05/17 09:34 to 2020/05/17 13:49 with 1 minute as the time interval are then taken for the 600 MW steam turbine monitoring, partly listed in Table 5. The environment temperature and power output are (27, 30) °C and (548, 552) MW, respectively.   Characteristic parameters of heat rate in cluster #1 (from Section 4.1) are taken to establish the MSET multi parameter estimation model, and energy efficiency warning threshold δ is 1.916, calculated by data in cluster #4.
The deviation degree values during the period are then shown in Figure 12a, with window width m = 5. The red line in the figure represents the energy efficiency warning threshold value. The warning happens when the deviation degree value is larger than δ. A partial view is enlarged to clearly tag the 37th deviation degree value, which is the first one larger than δ in the time series. Then, the time stamp of the 37th deviation degree value, 2020/05/17 10:10, is tagged in the figure as the warning time.
Characteristic parameters of heat rate in cluster #1 (from Section 4.1) are taken to establish the MSET multi parameter estimation model, and energy efficiency warning threshold  is 1.916, calculated by data in cluster #4.
The deviation degree values during the period are then shown in Figure 12a, with window width =5 m . The red line in the figure represents the energy efficiency warning threshold value. The warning happens when the deviation degree value is larger than  . A partial view is enlarged to clearly tag the 37th deviation degree value, which is the first one larger than  in the time series. Then, the time stamp of the 37th deviation degree value, 2020/05/17 10:10, is tagged in the figure as the warning time. Characteristic parameters are then monitored to locate the reason for the abnormal deviation degree values during operation. Figure 12b-d shows the variation of main steam pressure, condenser vacuum and heat rate in time series. In Figure 12b, main steam pressure values cannot be detected, as they contain obvious differences during the period compared with the normal operation data. Other parameters, such as main steam temperature, reheated steam temperature, feedwater temperature and reheated steam pressure have the similar performance. Figure 12c shows that the condenser vacuum suffered a significant decrease at 10:20, 10 min later than the warning time at 10:10. A lower condenser vacuum value indicates a worse working level in the steam turbine system, resulting in a higher heat rate value, seen in Figure 12d. In Figure 12d, the obvious heat rate jump can be caught at 10:31, which is 21 min later than the warning time and 11 min later than the time when the condenser vacuum suffered the significant decrease. Characteristic parameters are then monitored to locate the reason for the abnormal deviation degree values during operation. Figure 12b-d shows the variation of main steam pressure, condenser vacuum and heat rate in time series. In Figure 12b, main steam pressure values cannot be detected, as they contain obvious differences during the period compared with the normal operation data. Other parameters, such as main steam temperature, reheated steam temperature, feedwater temperature and reheated steam pressure have the similar performance. Figure 12c shows that the condenser vacuum suffered a significant decrease at 10:20, 10 min later than the warning time at 10:10. A lower condenser vacuum value indicates a worse working level in the steam turbine system, resulting in a higher heat rate value, seen in Figure 12d. In Figure 12d, the obvious heat rate jump can be caught at 10:31, which is 21 min later than the warning time and 11 min later than the time when the condenser vacuum suffered the significant decrease.
Thus, the heat rate's abnormal variety in this case can be detected to be caused by the condenser vacuum anomaly. The sudden drop of the condenser vacuum will lead to the decrease in steam energy capacity. In other words, the main steam flow will increase in order to remain at the unit load, which will result in the sudden increase in heat rate value. In this case, the misoperation of valves in the circulating water system causes a sudden decrease in circulating water flow. Thus, a lower condenser vacuum happens when the wet steam into the condenser cannot be sufficiently condensed. When the circulating water system returns to the normal working condition, the heat rate and the deviation degree will gradually fall back.
Actually, many operation parameters influence the heat rate calculation, thus, the time stamp of abnormal heat rate values can be explained later than that of the condenser vacuum in most cases. On the other hand, the heat storage in the boiler can also extend the steam turbine's heat rate response time when characteristic parameters' abnormal values exist in operation. Furthermore, while warning time by deviation calculation tagged in Figure 12a is even earlier than the time stamp of the condenser vacuum anomaly, the energy efficiency deviation index can better reflect the change of unit operation state and provide warning information in advance, when combined with the warning threshold δ.

Conclusions
In this paper, the MSET model combined with the modified information entropy weight theory is proposed as a method for the steam turbine energy efficiency monitoring study. The parameters' information entropy average value has been taken into consideration for the modified information entropy weight w 2 calculation, and the modified information entropy weight has been verified to weaken shortcomings brought by the traditional one, w 0 . Then, the MSET estimation model is established by characteristic parameters and the heat rate index under the selected steady 550 MW working condition in a 600 MW unit, with the recognition of the effectiveness in different operation levels being testified. Moreover, the deviation degree index calculated by errors in energy consumption characteristic parameters is proposed to express the current energy efficiency level of the steam turbine. The case study results with data in an abnormal energy efficiency time period show that the method proposed can accurately reflect the change in the unit operation state. Moreover, based on deviation degree values compared with δ, the method proposed in the paper can provide an earlier warning than the time provided by the abnormal heat rate index or energy consumption characteristic parameters. Thus, the combination of the multivariate state estimation technique and modified entropy weight method can be utilized as a new method for field operation guidance. Funding: This research was funded by Science Foundation of Nanjing Institute of Technology, grant number CKJA202002.

Data Availability Statement:
The data presented in this study are available on request from the corresponding author. The data are not publicly available due to the acquisition from the operation field.