Robust Data Reconciliation in Supercritical Carbon Dioxide Thermal Systems: From Framework Design to Performance Evaluation

Abstract

A number of studies have been carried out to analyze the theoretical performance of super critical carbon dioxide (S-CO₂) systems, but monitoring its actual performance when gross errors appear in measurements can be quite challenging. This paper proposes a robust data reconciliation framework to cope with the gross errors happening in S-CO₂ systems. A schematic S-CO₂ recompression cycle was constructed with different types of measurement sensors, and various estimators with tuned parameters were evaluated to compare their performance. A hybrid strategy with two optimization solvers was designed to ensure the convergence of the solution. Results demonstrated the effectiveness of the proposed robust data reconciliation framework, where the mean relative error (MRE) of all measurements can be reduced from 1.02% to 0.39%, and the MRE of the gross errors can even be reduced from 4.79% down to only 1.11%. Statistics indicated that the Welsch estimator offered the best overall performance, while the Cauchy estimator proved to be more stable. The methods and conclusions provided in this paper can inspire subsequent research on data processing and the operation optimization of real S-CO₂ systems.

1. Introduction

As a new type of energy medium, supercritical carbon dioxide (S-CO₂) has gained the prevailing interests of researchers. It has excellent fluidity and thermal conductivity [1], and it is easy to produce and use due to its relatively low critical parameters [2]. Researchers have regarded it as an ideal working fluid for thermodynamic cycles, especially suitable for Brayton cycles. Currently, various thermodynamic cycle designs using S-CO₂ as the working fluid have been proposed by the academic community and applied to nuclear power systems [3], solar power systems [4], gas turbine systems [5], and energy storage systems [6]. Among the common S-CO₂ thermodynamic cycle designs, the recompression cycle has a higher thermal efficiency and is widely used as the subject of various theoretical and applied studies. For example, Al-Sulaiman and Atif [4] designed multiple S-CO₂ thermodynamic cycles for solar systems, and the recompression cycle gained the highest round-trip thermal efficiency (RTE) of 52%. Luu et al. [7] demonstrated that an S-CO₂ recompression cycle with both inter-cooling and reheating structures can achieve a maximum RTE of 60%, proving the superiority of the recompression design. You and Metghalchi [8] came up with an optimization analysis on the supercritical carbon dioxide recompression cycle, which focused on the effect of the pressure ratio on the thermal efficiency and exergetic efficiency of the system. Similarly, Li et al. [9] carried out the optimization of a S-CO₂ recompression Brayton cycle as well, though they were more concerned about the condensation margin for the system in order to apply the S-CO₂ cycle in shipboard usage. Ding et al. [10] carried out a comparative study on the control strategies of a supercritical carbon dioxide recompression Brayton cycle, which aimed for actual usage in GEN-IV nuclear reactors. Apart from the classical optimal analysis, Du et al. [11] applied deep neural network and data mining techniques in the S-CO₂ recompression cycle to achieve higher efficiency, which resulted in a high-efficiency and compact design of S-CO₂ systems for shipboard usage. As summarized by Kulhanek and Dostal [12], recompression design (with flow diving) offered better results for S-CO₂ Brayton cycles.

Existing research mainly focuses on the thermodynamic design and optimization of S-CO₂ thermal systems, which are mostly confined to the design stage. As for a constructed S-CO₂ power plant, however, it would be more crucial to capture its actual operation status, with various measurement sensors deployed around the system. Due to precision limitations, errors always exist in the measurement results. When the error of measurements exceeds the normal range, evaluation of the system would be unreliable. To overcome the problem, it is necessary to utilize the data reconciliation method in S-CO₂ systems. It is a technique for measurement correction in the real-world industrial system, based on the physical model of systems and statistical principles. It was introduced by Kuehn and Davidson [13] in 1961 for data correction in chemical engineering control systems, and later, it was accepted in various industrial sectors, including the chemical industry [14], mining industry [15], and energy industry [16]. Specific to the energy and power engineering sector, several studies have been carried out to apply data reconciliation to various types of systems. Langenstein [17] implemented data reconciliation in nuclear power plants for power recapture and power uprate for the reactors, based on VDI 2048 [18], a technical standard on data reconciliation proposed by the Association of German Engineers (VDI). Yu et al. [19] developed a data reconciliation framework for a double reheat power plant, significantly reducing the uncertainty of mass flow measurements and the calculation result of the heat rate. As for energy storage systems, Wu et al. [20] designed a compressed carbon dioxide energy storage (CCES) system test rig and applied a data reconciliation algorithm for it to improve the precision of exergy analysis. Such research has proved the practicability of the data reconciliation method in different types of energy systems.

Despite its successful usage, the standard data reconciliation method is facing an important challenge: gross errors of the measurements. Usually, a measurement with gross error means its error exceeds its supposed random distribution, which means the measurement does not follow the supposed distribution (e.g., a standard normal distribution). This may happen when instrument failure or leakage exists in the system. Though the data reconciliation algorithm may still try to correct the errored measurements (as we have explored in the previous study [21]), its excessive error may spread to relating measurements and make them unreliable as well, which is referred to as the “smearing effect” [22]. To overcome the challenge, the concept of the robust data reconciliation method was proposed by Tjoa and Biegler [23], with a contaminated normal distribution for the measurements with gross errors. Under the robust data reconciliation framework, different forms of objective functions, rather than the standard least-square function, is applied in the data reconciliation optimization problem. These objective functions are called robust estimators, and they represent the supposed error distribution other than standard normal distribution. It can be regarded as the direct application of robust regression theory in the data reconciliation method. Johnston and Kramer [24] proved that robust data reconciliation can effectively reduce both random errors and gross errors, without the categorization of the measurements into normal ones and errored ones. This also means that the smearing effect can be reduced under robust data reconciliation. Successively, various robust estimators have been proposed to improve the correction effect, as well as the augmented algorithms, to cope with these nonlinear objective functions. Prata et al. [25] carried out a comparative analysis for eight robust estimators, and they pointed out the superiority of the Welsch [26] and Lorentz estimators. Besides theoretical studies, several works focused on the application of robust data reconciliation in industrial systems. For example, Xie et al. [27] utilized their newly proposed robust estimator in the evaporation process of alumina production, showing its effectiveness in gross error correction. Zhang et al. [28] developed a robust data reconciliation analysis framework for a double-flash smelter system and eventually programmed their algorithm into a metal balance software system. These cases prove the feasibility of applying robust data reconciliation in the industrial scene.

As for energy systems, however, the application of robust data reconciliation is still rare. Such works include the one by Valdetaro and Schirru [29], where they combined the robust data reconciliation method with the particle swarm optimization algorithm and applied it to a simple turbine plant model, as well as a reactor. However, the systems analyzed in their work were rather simple, where most sensors were for mass flow, so the system was almost linear. Zhang et al. [30] constructed a performance estimation model for an entrained-flow pulverized coal gasification system, based on the data reconciliation method enhanced with several machine learning techniques. In their model, the standard least-square target function was applied, which limited its performance facing gross errors. As for the S-CO₂ system, no existing studies have implemented robust data reconciliation. This is more serious than conventional energy systems, as the measurement techniques for the S-CO₂ system are still not reliably verified, and there is a lack of a standard measurement and operation process for such systems, which implies larger and less controllable errors of the measurement results. Therefore, it is necessary to consider robust data reconciliation specifically for S-CO₂ energy systems and evaluate its effect in enhancing their reliability.

2. Methodology

This paper aimed to explore the application of robust data reconciliation in a S-CO₂ recompression cycle. To cope with the possible excessive measurement errors, it was necessary to apply robust data reconciliation. Several theoretical challenges needed to be considered, including the definition of the robust data reconciliation problem, selection among various robust estimators, and appropriate application of optimization solvers. These topics are discussed in detail in the following sections.

2.1. Robust Data Reconciliation Framework

In a thermal system, all known parameters (including the sensors’ measurement values and assumed empirical parameters) can be arranged into the vector $x$, and the unknown parameters can be arranged into the vector $u$. In a standard heat balance computation, several equations should be constructed to calculate the values of $u$ from $x$, where the components of $x$ are mostly the design parameters of the equipment deployed in the system. In the data reconciliation method, however, most of the elements in $x$ are the measurements from various types of sensors. Suppose all equations can be expressed as a residual vector $f$, that is:

$$f=f(x,u)$$

(1)

When all parameters are precise, there should be $f=0$. The length of $x$ is denoted as $n$, the length of $u$ as $r$, and the length of $f$ as $m$. As a necessary condition for data reconciliation, one should ensure the following:

$$m>n-r$$

(2)

This condition ensures that there are more equations (or correspondingly, more known parameters) than needed in order to solve the system so that redundancy exists for data correction. Otherwise, when both sides in Equation (2) are equal to each other, it implies that one has to suppose all known parameters are precise to solve the unknown parameters, just like in the heat balance calculation.

Under the robust data reconciliation framework, the following optimization problem was constructed to acquire the best correction results for known and unknown parameters:

$$\begin{array}{l}\underset{v}{\min }{\displaystyle \sum _{i=1}^n\rho \left({\xi }_i\right)},{\xi }_i={\displaystyle \frac{v_i}{{\sigma }_i}}\\ s.t.f(x-v,u)=0\end{array}$$

(3)

In Equation (3), the vector $v$ stands for the correction vector of known variables $x$ (which should have the same length), which should be subtracted from the original measurements as an offset to acquire the corrected results $x-v$. The target function of the optimization problem is the summation of the normalized correction ${\xi }_i=v_i/{\sigma }_i$ transformed by the estimator $\rho$, where ${\sigma }_i$ is the standard deviation of the ith parameter and can be transformed from its measurement uncertainty $a_i$ when it obeys a normal distribution [18]:

$${\sigma }_i={\displaystyle \frac{a_i}{1.96}}$$

(4)

2.2. Selection of Robust Estimator

Under the standard data reconciliation process, the weighted least-squares (WLS) function is applied as the estimator. If the normalized correction for a known parameter is denoted as $\xi$, WLS estimator should give the following:

$${\rho }_{\mathrm{WLS}}(\xi )={\displaystyle \frac{1}{2}}{\xi }^2$$

(5)

Under the WLS estimator, one can see the surge of ${\rho }_{\mathrm{WLS}}$ when the correction scale $\xi$ increases, which would cause excessive influence on the objective function. For robust data reconciliation purposes, the estimators should be applied to imper the scale of the objective function when specific elements (the measurements with gross errors) of $v$ become too large.

Over the decades, a number of robust estimators have been proposed by different researchers. In this paper, the most representative robust estimators were compared for evaluation in S-CO₂ systems, which are listed below.

Fair estimator: It was proposed by Albunquerque and Biegler [31] and is convex and has continuous first and second derivatives. It is shown in the following equation, where $c_{\mathrm{F}}$ is a constant parameter:

$${\rho }_{\mathrm{fair}}(\xi )=2c_{\mathrm{F}}^2\left[{\displaystyle \frac{\left|\xi \right|}{c_{\mathrm{F}}}}-\ln \left(1+{\displaystyle \frac{\left|\xi \right|}{c_{\mathrm{F}}}}\right)\right]$$

(6)

Logistic estimator: The definition of the logistic estimator is given below, where $c_{\mathrm{L}}$ is a constant parameter:

$${\rho }_{\mathrm{logistic}}(\xi )=2\ln \left[1+\exp \left({\displaystyle \frac{\xi }{c_{\mathrm{L}}}}\right)\right]-{\displaystyle \frac{\xi }{c_{\mathrm{L}}}}$$

(7)

Cauchy estimator: It can be derived from a Cauchy distribution, where $c_{\mathrm{C}}$ is a constant parameter:

$${\rho }_{\mathrm{cauchy}}(\xi )=c_{\mathrm{C}}^2\ln \left(1+{\displaystyle \frac{{\xi }^2}{c_{\mathrm{C}}^2}}\right)$$

(8)

Welsch estimator: This is an estimator suitable for data with a large deviation, where $c_{\mathrm{W}}$ is a constant parameter:

$${\rho }_{\mathrm{welsch}}(\xi )={\displaystyle \frac{c_{\mathrm{W}}^2}{2}}\left[1-\exp \left(-{\displaystyle \frac{{\xi }^2}{c_{\mathrm{W}}^2}}\right)\right]$$

(9)

To compare these different estimators, their coefficient should be decided “impartially”, which means they should have the same asymptotical efficiency when $\xi$ is under the standard normal distribution. In this work, the tuned parameters given by Özyurt and Pike [32], as well as Prata et al. [25], were applied to decide the coefficient of each estimator, which made the efficiency of every estimator 95%. The tuned coefficients for each estimator are listed in

Table 1. Tuned robust estimator coefficient to achieve 95% efficiency.

Robust Estimator	Coefficient	Value
Fair	$c_{\mathrm{F}}$	1.3998
Logistic	$c_{\mathrm{L}}$	0.602
Cauchy	$c_{\mathrm{C}}$	2.3849
Welsch	$c_{\mathrm{W}}$	2.9846

.

The relations between the forementioned estimators and the correction value are shown in Figure 1. It was clear that the WLS estimator increased significantly as the input magnitude grew, and it was not robust compared with the other estimators. As for fair and logistic estimators, the slope was approaching a constant when $\xi$ increased. For Cauchy and Welsch estimators, they tended to show a constant value after receiving a large $\xi$. This implied their robustness when facing gross error measurements. This implied a relatively large (and effective) correction for the errored sensors under those estimators compared with the WLS estimator, as the rapid growth of ${\rho }_{\mathrm{WLS}}$ can inhibit the meaningful correction of these sensors. On the other hand, when the errors were relatively small, Cauchy and Welsch estimators may be less competitive than the WLS estimator, as excessive corrections may be generated due to the slow growth of these estimators.

To summarize their characteristics, the influence function (IF) can be introduced to show the global robustness of different estimators. As for most estimators, IF may simply be defined as the derivative of the estimator function $\rho$ towards the correction $\xi$. The IF curves of all estimators are shown in Figure 2. From the figure, we can see three different patterns. For the classical WLS estimator, its IF increased linearly towards the correction. For fair and logistic estimators, their IF approximated to a constant. For Cauchy and Welsch estimators, their IF even descended when $\xi$ became larger. Especially for the Welsch estimator, the IF immediately dropped to almost zero when $\xi$ exceeded about 3~4, the acknowledged bound of a standard normal distribution with 99% confidence. Such differences can influence their effectiveness to deal with the gross errors in S-CO₂ systems, which will be studied in the following part of this paper.

2.3. Application of Optimization Solvers

As one can see in Equation (3), the robust data reconciliation problem is an equation-constrained optimization problem, where both the target function and the constraints were nonlinear. For such a problem, only a few algorithms (i.e., optimization solvers) are capable of quickly and accurately solve it. These candidates include the following:

Sequential quadratic program (SQP): It is a classical algorithm for a nonlinear optimization problem, which converts the target function into a sequence of quadratic approximations in iterations. For each of the quadratic problems, a high-accuracy solution can be acquired based on the Lagrange multiplier and Newton method, which linearize the constraints and turn the problem into a non-constrained one. Thanks to its simplicity, SQP has very high efficiency and is suitable under most situations. However, a large step size or simplification of constraints may sometimes cause trouble, especially for S-CO₂ system calculations, as wrong values of physical properties can interrupt the calculation.

Interior point method (IPM): It is also called the barrier method, whose practical form was proposed by Karmarkar [33] in 1984. It is an algorithm for the constrained convex optimization problem, featured for its high efficiency (polynomial runtime) and strictly satisfying the constraints, as it always searches for a solution in the feasible region, which is indicated by its name. This means it can better cope with the rigorous requirements of the thermal system calculation than SQP.

To compromise between efficiency and convergence, both algorithms were utilized to solve the robust data reconciliation problem. SQP was chosen as the default solver and applied first. When the SQP solver failed or did not converge for constraints, IPM was used instead to acquire a reasonable solution. The optimization solver for robust data reconciliation was implemented through the MATLAB R2024a platform in this work, which had good convergence, compared with other open-source solvers. It should be noted that equivalent or better results may still be possible with those solvers under carefully tuned parameters, which was not the focus of this paper.

3. Configuration of the System and Conditions

3.1. Recompressed S-CO₂ System Layout

To study robust data reconciliation for S-CO₂ systems, we set up a schematic recompressed S-CO₂ cycle for measurement simulations. The layout of the recompressed S-CO₂ system is shown in Figure 3, mainly based on the original design proposed by Dostal et al. [3].

The system was supposed to have already been constructed, and various types of measurement sensors were deployed all over the system. The cycle included two compressors (C1, C2), two regenerators (R1, R2), one turbine (T), a high-temperature heat source (H), and a condenser (COND). In the recompression cycle, only a part of the CO₂ passed through the condenser to be cooled down to the inlet temperature of C1, while the other part was directly entering C2. The two streams merged at the outlet of the low-temperature regenerator, R1. Then, they entered the heat source H to be heated up to a predetermined temperature, after which they flowed into the turbine T to expand and generate power. Subsequently, they entered the two regenerators sequentially to preheat the incoming CO₂. The cooling water (13, 14) entering and exiting COND was also included in the scope of robust data reconciliation analysis in order to provide more redundancy of measurements. The shaft power generated by T was supplied to drive C1 and C2 and then transformed into electricity by G.

To simplify the thermal model and the simulation process, several assumptions were proposed for the schematic S-CO₂ system studied in this work:

• For all the heat exchangers in the system (including COND, R1, R2, and H), they were assumed to have a “perfect” design, which meant their heat transfer loss and pressure drop were not considered.
• For C1, C2, and T, it was assumed that there was no leakage of S-CO₂, and their mechanical loss during the energy conversion process was also ignored.
• As for the mixture point a, the pressure of outflow No. 6 took the minimum of inflows No. 4 and No. 5. For the split point b, the pressure of outflow No. 3 and No. 12 took the value of inflow No. 11. No extra pressure drop and heat loss were considered for them.

Despite these assumptions, the robust data reconciliation framework studied in this paper could still be verified and be applied to the actual S-CO₂ systems, with estimations about the various losses over the conversion process. They could be easily added to the analysis process. For the current system, removal of these factors helped to give prominence to the most important parameters. Under the assumptions above, the state parameters of each point under the standard condition are shown in

Table 2. Principal parameters of the schematic system under the standard condition.

Substance	State Point	Mass Flow (kg/s)	Pressure (MPa)	Temperature (°C)	Enthalpy (kJ/kg)
CO2	1	2.64	7.6	32	315.09
	2	2.64	30	81.66	354.02
	3	1.36	7.6	104.76	529.05
	4	1.36	30	251.02	644.00
	5	2.64	30	251.02	644.00
	6	4	30	251.02	644.00
	7	4	30	352.73	778.79
	8	4	30	550	1028.86
	9	4	7.6	388.89	855.14
	10	4	7.6	270.52	720.43
	11	4	7.6	104.76	529.05
	12	2.64	7.6	104.76	529.05
Water	13	27.01	0.101	25	104.91
	14	27.01	0.101	30	125.83

, slightly different from the original design of Dostal et al. [3]. To acquire the enthalpy of each point, the open-source property library CoolProp [34] was applied.

Once the state parameters were decided, the performance of the schematic S-CO₂ system and its key devices could be calculated through heat balance calculation, which is listed in

Table 3. Performance of the cycle and key device parameters under the standard condition.

Performance Parameter	Value
RTE	43.55%
C1 isentropic efficiency	85%
C2 isentropic efficiency	85%
T isentropic efficiency	90%
Generator power	431.27 kW

. It should be noted that the total efficiency of the generator was supposed to be 99% when transforming the shaft power output. As the outstanding index of the cycle, the round-trip efficiency (RTE) of the system can be decided, which is defined as:

$$\mathrm{RTE}={\displaystyle \frac{W}{Q}}={\displaystyle \frac{W_{\mathrm{T}}-W_{\mathrm{C}1}-W_{\mathrm{C}2}}{Q}}$$

(10)

where $W$ is the shaft power output of the system and can be computed through the subtraction of the shaft power generated by the turbine, T, and the power consumed by two compressors, C1 and C2. $Q$ is the heat power input into the system, and in the current S-CO₂ recompression cycle, it was simply the heat generated by H:

$$Q={\dot{m}}_7\cdot (h_8-h_7)$$

(11)

where ${\dot{m}}_7$ is the mass flow into H, and $h_7$ and $h_8$ are the enthalpies corresponding to the inflow and outflow S-CO₂ of H (see Figure 3). Based on the current design values, the RTE of the system turned out to be 43.55%, slightly higher than the design given by Dostal et al. [3], who used a lower estimation for efficiencies for the key devices in their system. As for a real S-CO₂ power plant, however, the actual value of RTE should be lower.

3.2. Configuration of the Measuring Sensors

As the schematic system was supposed to be constructed, a number of measurement sensors were necessary to be deployed all over the system. Their types and positions are shown in Figure 3, and their uncertainties are listed in

Table 4. Sensors deployed in the S-CO₂ recompression cycle and their uncertainties.

State Point	Pressure	Temperature	Mass Flow Rate
1	1%	1 °C	2.5%
2	1%	-	-
3	-	-	2.5%
4	1%	-	-
5	-	-	-
6	-	1 °C	-
7	-	1 °C	2.5%
8	1%	1 °C	2.5%
9	1%	1 °C	-
10	-	1 °C	-
11	-	1 °C	-
12	1%	-	-
13	const.	1 °C	1%
14	const.	1 °C	1%

. Temperature sensors had the largest number, as they were the most crucial to deciding the thermodynamic state of S-CO₂. Pressure sensors were fewer than temperature sensors, and there were only four mass flow sensors for the S-CO₂ part of the system. As for S-CO₂ measurements, the pressure sensors (supposed to be pressure transmitters [35]) took 1% of uncertainty, and temperature sensors (supposed to be thermocouples [36]) used an absolute uncertainty of 1 °C. For mass flow measurement, they were usually low-precision for gas [37], so the uncertainty was supposed to be 2.5%. As for cooling water, the pressure measurements were supposed to be constants, as their values almost caused no impact on the enthalpy of the water under normal conditions. The uncertainty of temperature was set as the same value as S-CO₂ (1 °C), and the uncertainty of the mass flow rate was supposed to be a little bit lower [38], configured as 1%.

Apart from these three types of sensors, the power generated by G (converted from the shaft power generated by T; part of them were also consumed by C1 and C2 for compression) was also supposed to be measured through a dynamometer. It was a high-precision meter, compared with the thermal sensors, so its uncertainty was set as 0.25%, much lower than any of the sensors deployed in the system.

3.3. Simulation Conditions

To evaluate the effect of the selected estimators for robust data reconciliations, a number of conditions, including sensors with gross errors, were generated. In this work, 2000 conditions were generated in total in order to cover various combinations of the gross errors happening at different sensors. Two categories of errors were generated based on the standard condition:

Gross errors: For each condition, the number of sensors with gross errors was decided by random at first, from only one sensor up to eight sensors. The sampling ratio for the number of errored sensors is shown in Figure 4. After deciding the number, corresponding sensor(s) with gross error were sampled among all sensors randomly. It should be highlighted that the generator power measurement was never considered when adding gross errors, as it was assumed to be more reliable than the thermal sensors. As for the selected sensors, their measurements were attached to gross errors, ranging between 3 to 10 times their uncertainties, whose values were sampled from a uniform random distribution. The directions of errors were also decided by random. In other words, for a measured parameter $x_i$ with gross error, if its uncertainty was $a_i$, then the gross error $\delta x_i$ should satisfy the following random distribution:

$$\delta x_i=\mathrm{sgn}(S_i)\cdot \left|X_i\right|,S_i~U(-1,1),X_i~U(3a_i,10a_i)$$

(12)

where $S_i$ and $X_i$ are two random variables, $\mathrm{sgn}$ is the sign function, and $U$ denotes a uniform distribution. Once $\delta x_i$ was decided, it was then added to the value of $x_i$ under the standard condition as the actual measurement result under such a condition.

Random errors: As for the sensors not selected for gross error attachment in the previous process, their values were still added with random errors sampled within their uncertainties, based on their values under the standard condition (listed in Table 2). In other words, for a measured parameter $x_i$, if its uncertainty was $a_i$, then a random error $\delta x_i~U(-a_i,a_i)$, which was decided through a uniform distribution to avoid excessive errors, was generated and added to its value under the standard condition.

3.4. Computation Process

Based on previous sections, we established a robust data reconciliation framework for the proposed schematic S-CO₂ system. The computation process, from condition generation to data correction, is shown in Figure 5. It should be noted that solution failure may still happen under the IPM solver, and to ensure that 2000 conditions could be acquired in the final results, backup conditions were generated for those failed cases. All computations were carried out on a personal computer with Intel Core i7-10700 CPU. MATLAB parallel pools [39] were enabled to accelerate the computations.

3.5. Statistics on Computation Results

For the purpose of result analysis, two types of statistics were utilized in this study.

Mean relative error (MRE): To evaluate the overall performance of estimators under all conditions, it was useful to take the average of all the results by relative error format. For a measured parameter with a standard value $x_i$, if its reconciled values under $N$ conditions were denoted as a vector ${\widehat{x}}_i$:

$${\widehat{x}}_i=\left({\widehat{x}}_{i,1},{\widehat{x}}_{i,2},\dots ,{\widehat{x}}_{i,N}\right)$$

(13)

then the MRE of the parameter $x_i$ under all conditions can be computed as:

$$\mathrm{MRE}\left(x_i\right)={\displaystyle \frac{1}{N}}{\displaystyle \sum _{k=1}^N\left|\frac{{\widehat{x}}_{i,k}-x_i}{x_i}\right|}$$

(14)

Similarly, the concept of MRE can be extended to the overall results. One can acquire the MRE of all $n$ measured parameters through the following equation:

$$\mathrm{MRE}\left(x\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\mathrm{MRE}(x_i)}$$

(15)

Root mean-square error (RMSE): The disadvantage of MRE is that it may ignore the extreme values hidden in individual conditions or parameters. To reveal such problems for estimator comparison, it was necessary to compute the RMSE of the results as well. For a measured parameter with a standard value $x_i$ and with its reconciled values’ vector under $N$ conditions ${\widehat{x}}_i$, their RMSE was defined as:

$$\mathrm{RMSE}(x_i)=\sqrt{{\displaystyle \frac{1}{N}}{\displaystyle \sum _k^N{\left(\frac{{\widehat{x}}_{i,k}-x_i}{x_i}\right)}^2}}$$

(16)

Compared with the direct average, the RMSE can magnify the influence of extreme data, which is useful for detecting fault in robust data reconciliation. As for overall results, it was also possible to define the overall RMSE of all measurements like MRE, simply through the average among all the measurements’ RMSE, which can be represented by the following equation:

$$\mathrm{RMSE}\left(x\right)={\displaystyle \frac{1}{n}}{\displaystyle \sum _{i=1}^n\mathrm{RMSE}(x_i)}$$

(17)

4. Results

4.1. Overview of Solutions

For all generated conditions, five kinds of estimators were evaluated to carry out data reconciliation for the schematic S-CO₂ system. Over the solution process, only 26 conditions were terminated due to solver failure, and 2026 conditions were generated in total to eventually acquire 2000 valid conditions. The computation cost box chart of the five estimators is shown in Figure 6. The duration shown in the figure corresponded to the time cost of a single condition, including the total cost of both SQP and IPM solvers if the SQP solver failed to provide a valid solution. Among all estimators, the WLS estimator had the lowest time cost, as its target function had the simplest form. In fact, when the constraints were linearized in the optimization problem (3), the WLS estimator turned the problem into the standard form of SQP, which ensured the convergence of the solver. The logistic estimator also had a relatively low computation cost, while the average computation times of the other three estimators were over 30 s per condition. The Welsch estimator had the highest computation cost among all the estimators, due to its complex form and rather low derivative when the parameter became larger (see Figure 2), which may inhibit the convergence of the optimization solvers.

The root mean-square error (RMSE) of the constraints after data reconciliation are shown in Figure 7 with a box chart. It is clear to see that under all five estimators, the residual of the constraints, which were all less than $2\times {10}^{-8}$, could be significantly reduced. The mean values of the constraints’ RMSE under all estimators were very close to each other, while the fair estimator reached a lower bound of the constraints’ RMSE. In general, the RMSE results proved that fine convergence was achieved through the proposed robust data reconciliation framework, thanks to the hybrid strategy of the SQP and IPM solvers mentioned in Section 2.3.

4.2. Overall Correction on Measurements

To better investigate the effects of different estimators, it was necessary to evaluate the correction results on the measurements of all sensors that existed in the S-CO₂ system. The MREs of all measurements under 2000 conditions are shown in Figure 8. By average, the original error brought to the sensors was rather small, only about 1%, which meant the gross error only happened for a small portion of the measurements. After data reconciliation, the MREs of all measurements were reduced across all estimators. However, the WLS estimator yielded the highest MRE, while the Welsch estimator achieved the lowest MRE. All robust estimators performed better than the WLS estimator, in which the Welsch estimator achieved only a 0.39% MRE, almost half of that of the WLS estimator, which proved the advantage of robust data reconciliation over standard data reconciliation.

To be more specific, the MREs of each sensor before and after correction under 2000 conditions are demonstrated in Figure 9. It was obvious that for most sensors, the error of measurement was significantly reduced under any types of estimators. For example, for sensors T1, M7, and M8, the original MRE can be up to about 2%, and after data reconciliation under any robust estimator, it can be suppressed to less than 0.5%. Among all estimators, the Welsch estimator performed the best in most sensors, and it was followed by Cauchy, logistic, and fair estimators, successively. The WLS estimator was the worst for all sensors, except the sensors related to water (state points 13 and 14). For temperature sensors T6, T7, T8, and T10 and the power G, the WLS estimator even generated a larger error than the original data, which is unacceptable, as these sensors are all very important for S-CO₂ system performance monitoring. Generally, the performance order was consistent in almost every sensor, indicating the distinctive differences in the correction performances of these robust estimators.

The MRE of the measurements can reveal the general performance of robust data reconciliation, yet it may ignore the defective results happening in the change. To capture such problems, it was necessary to evaluate the RMSEs of the measurements before and after robust data reconciliation. The RMSEs of all measurements are shown in Figure 10. Compared with the original value, the RMSEs under all estimators were significantly reduced after data reconciliation, still following the successive order: WLS was the worst, followed by fair and logistic estimators, while the Cauchy estimator and Welsch estimator performed the best. It should be noted that the RMSEs of fair and logistic estimators were very close to the WLS estimator, suggesting that they may not control excessive gross errors under a number of conditions.

The RMSEs of each sensor are shown in Figure 11. Compared with Figure 9, the most obvious distinction was that the Welsch estimator had worse results for some sensors (e.g., P4), where the RMSE under the Welsch estimator was close to the WLS estimator, worse than all other robust estimators. For sensors M13 and M14, the Welsch estimator performed terribly, where the RMSE was almost the same as the original RMSE without data reconciliation. The possible reason was that these measurements were not strongly related to the S-CO₂ system, and few redundant measurements and constraints were imposed upon them. Nevertheless, the Welsch estimator still performed much worse than other robust estimators, which implies its strong dependencies on the redundancy of the measurements. Compared with the Welsch estimator, the Cauchy estimator performed more stably than the RMSEs of most sensors, including M13 and M14. Restricted to RMSE results for each parameter, the Cauchy estimator should be the preferable choice to be deployed in the robust data reconciliation framework of the S-CO₂ system.

4.3. Correction for Gross Error Measurements

On average, the overall error under each condition of the S-CO₂ system was relatively small, as only a small portion of the measurements had gross errors. To better reveal the influence of robust data reconciliation on the gross errored sensors, the best way was to separate their results from the normal sensors with random errors and only carry out statistics on them.

The MREs of all measurements with gross errors are revealed in Figure 12. Before data reconciliation, the MRE of gross errors in all conditions was almost 5%, which is generally unacceptable in practice. By data reconciliation with any of the estimators, the MRE of these gross errors can be reduced to less than 2%. The performances of the WLS, fair, and logistic estimators were very close, similar to the situation revealed in Figure 10. The Cauchy and Welsch estimators performed better in reducing the MRE of gross errors, indicating their potential to suppress excessive measurement errors.

In detail, the MRE of each sensor’s measurements when gross errors existed was enumerated through a bar chart in Figure 13. Compared with the results shown in Figure 9, it is clear that the robust data reconciliation method played an important role in reducing gross errors in the measurements of the S-CO₂ system. For example, for sensor T1, the original gross errors could be greater than 10%, and after data reconciliation, it turned out to be less than 1%. It was easy to verify that the MREs of most sensors became less than 2% after robust data reconciliation, except for only four sensors: M3, T13, T14, and M14. The last three sensors were all related to cooling water, which was analyzed in Section 4.2. As for sensor M3, one should notice that its value under the standard condition, 1.36 kg/s, was smaller than both the main flow sensor M7 (4 kg/s) and the split flow sensor M1 (2.64 kg/s), and they are always related to each other, so its MRE should always be much larger than other mass flow sensors when the absolute values of error are the same. In another way, this may also imply the lack of measurement redundancy for the split flow, corresponding to sensor M3, which should be considered for the real-world sensor deployment of the proposed S-CO₂ system.

Besides the MRE results, RMSEs of each sensor under all conditions were also analyzed, the results of which are shown in Figure 14. The pattern was almost the same as the MRE, except for sensors M1, P2, and M13, where the Welsch estimator was not able to reduce the gross error under some extreme cases. Referring to Section 4.2, based on the current sensor deployment strategy, though the Welsch estimator had the optimal overall performance, the Cauchy estimator performed well for all sensors.

4.4. Correction for Unknown Parameters

Apart from measurement correction, another important role for robust data reconciliation was to better estimate the value of the unknown parameters in the S-CO₂ system. For them, original error was not able to be acquired, so the comparison was simply carried out among five kinds of estimators.

MREs of all unknown parameters in the S-CO₂ system under 2000 conditions were computed and are shown in Figure 15, where the performances of five estimators still followed the order mentioned above. The WLS estimator was the worst, followed by the fair and logistic robust estimators, while the Cauchy and Welsch estimators were significantly more outstanding. As the unknown parameters were inferred from the measurements, it was no wonder that such conclusions were acquired.

To examine the effect of robust data reconciliation on each unknown parameter, their individual results were figured out and are listed in Figure 16. For every unknown parameter studied in the S-CO₂ system, MREs under the five estimators were arranged in the same order, as shown in Figure 16. Such a high consistency, from a part of the system to the overall statistical results, proved that a more accurate correction of the measurements can effectively lessen the estimation error of any unknown parameters related to them, even if some of the measurement corrections are not satisfying (e.g., for Welsch estimator). In that way, for unknown parameter estimation purpose, the Welsch estimator seemed to be a more preferable choice than the Cauchy estimator, based on MRE results, which had more stable behavior in measurement correction.

As for RMSE results, shown in Figure 17, the pattern was consistent with MRE results. The most important unknown parameter shown in Figure 17 should be RTE, where the Welsch estimator achieved the best estimation among all estimators. Therefore, in order to apply robust data reconciliation to monitor the overall performance of the system, the Welsch estimator should be applied to acquire the most precise results.

Besides the general conclusions on the performance of the robust estimators, one can also gain insights about specific unknown parameters. The largest errors were found in two types of parameters:

Cooling water-related: This included H13 and H14, which were the enthalpies of the inflow and outflow cooling water. Their values were decided by sensors T13 and T14, and the robust estimators failed to reduce their excessive errors under some cases, as shown in Figure 13.

Compressor power: This included W(C1) and W(C2). Their errors may be induced by the error that happened in the corresponding mass flow sensors, mainly M3. Fortunately, the deviation for the compressor power estimation did not influence the total shaft power W, which was controlled by the high-precision measurements of generator power G.

5. Conclusions

In this paper, a comprehensive analysis framework was developed for data reconciliation and gross error detection in the S-CO₂ systems. The proposed approach integrated thermal calculation models and robust optimization algorithms to enhance the reliability of real-world measurements. Key findings included the following:

• This study applied robust data reconciliation into an S-CO₂ recompression cycle for the first time and explored hybrid SQP and IPM solutions for the nonlinear optimization problem. Results proved the effectiveness of the proposed framework in the convergence and correction of the measurements.
• Five estimators, including the classical WLS estimator and four robust estimators (fair, logistic, Cauchy, and Welsch), were evaluated under 2000 simulated conditions with various numbers and ranges of gross errors in the S-CO₂ system. Results demonstrated the superiority of robust estimators over the WLS estimator in the correction of both measurements and unknown parameters, where the MRE of all measurements can be reduced from 1.02% to 0.39% (Welsch estimator), and the MRE of the gross errors can be reduced from 4.79% down to only 1.11% (Welsch estimator).
• Judging by the MRE and RMSE results, the Welsch estimator achieved optimal performance for measurements and unknown parameter estimations but may fail to correct excessive errors of measurements under very few conditions. Compared with the Welsch estimator, the Cauchy estimator was able to acquire a stable performance for most conditions.

In general, as the most preferable robust estimator studied in this work, the Welsch estimator had the best MRE and RMSE results for almost all parameters, which is consistent with the conclusion drawn from the previous study for a simpler chemical system [31]. This proves the applicability of such a robust estimator in the relatively complex thermal system.

The robust data reconciliation framework constructed in this paper can lay a solid foundation for the subsequent research on data processing and operation optimization for supercritical carbon dioxide systems. Experimental validation was not carried out for the proposed study due to the lack of a comprehensive test rig of the S-CO₂ thermal cycle, which should be taken seriously at the next stage of the study. In addition, dynamic conditions were not considered in the current research, which should be necessary to apply the proposed framework to reality. Further research is expected to consider the optimal configuration of the number and position of the sensors, as well as the dynamic model of the important facility in the system under non-steady conditions, and experimental studies, as well as sensitivity or stability analysis, are necessary to verify the practical performance of the robust data reconciliation method and the optimization solvers.