天然气能量计量逆向混合建模数字孪生系统

刘彬; 钟璐; 冯全源; 陈毅红

doi:10.11999/JEIT260289

天然气能量计量逆向混合建模数字孪生系统

doi: 10.11999/JEIT260289 cstr: 32379.14.JEIT260289

刘彬^{1, 2},
钟璐²,
冯全源^1, ,,
陈毅红¹

1.
西南交通大学信息科学与技术学院成都市 611700
2.
成都市秦川物联网科技股份有限公司成都市 610100

基金项目: 国家自然科学基金重点项目(项目编号：62531021)，四川省中央高校及科研院所科技成果转化项目(项目编号：2025ZHCG0008)

详细信息

通讯作者:
冯全源　fengquanyuan@swjtu.edu.cn

中图分类号: TP391.1
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- 文章访问数: 8
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- 被引次数: 0
出版历程
- 修回日期: 2026-04-23
- 录用日期: 2026-04-23
- 网络出版日期: 2026-05-23

An Inverse-Hybrid-Modeling Digital Twin System for Natural Gas Energy Metrology

LIU Bin^{1, 2},
ZHONG Lu²,
FENG Quanyuan^{1
, ,},
CHEN Yihong¹

1.
College of Information Science and Technology, Southwest Jiaotong University, Chengdu 611700, China
2.
Chengdu Qinchuan IoT Technology Co., Ltd., Chengdu 610100, China

摘要

摘要: 针对传统天然气能量计量方法在复杂工况下存在动态响应延迟、参数辨识困难、物理简化过度及抗干扰能力弱等问题，本文提出一种基于逆向混合建模的天然气能量计量数字孪生系统。系统以“算法-系统-场景”三层级架构为核心，整合热值、流量、能量机理模型与多源实时数据流，引入变分自编码器(VAE)实现无监督工况特征挖掘，结合动态贝叶斯网络与变分期望最大化(VEM)系统校准构建参数自矫正循环，解决传统模型对海量标注数据的依赖。集成超声波流量计、气相色谱仪等工业级设备，保障数据实时传输与闭环控制。覆盖动态压力波动、含氢混合气、多气源切换等核心工况，确保模型与实际应用高度适配。通过全尺寸工业级实验平台25周连续验证，结果表明，系统运行延迟≤3.8 s，数据传输抖动≤0.5 s，单设备日均能耗≤1.2 kW·h，平均无故障工作时间(MTBF)≥4 100 h，能量计量误差≤0.15%，热值误差≤0.12%，流量示值误差≤0.2%。同时，系统通过工业以太网加密与权限分级控制满足安全需求，为智能管网优化与标准化集成提供工程支撑。
- 天然气能量计量 /
- 数字孪生 /
- 逆向混合建模 /
- 变分自编码器 /
- 动态贝叶斯网络
Abstract: Objective Global natural gas consumption continues to rise at an average annual growth rate of 3.2%. A 0.1% reduction in energy measurement error can reduce trade disputes by approximately $750 million per year. Traditional research primarily adopts indirect methods for energy measurement, with the mainstream chromatographic analysis and acoustic velocity correlation methods facing multiple bottlenecks and certain application limitations. Chromatographic analysis exhibits low anti-interference error but suffers from high flow dynamic response delay and insufficient dynamic calibration capability. Additionally, it has poor adaptability to multi-gas source switching, requires manual calibration, and has high usage and maintenance costs. The lack of interoperability standards for energy networks further exacerbates system integration difficulties. Although acoustic velocity correlation demonstrates low-latency flow dynamic response, it has high anti-interference error, which may increase significantly with single-component content changes (e.g., hydrogen content from 5% to 10%) and even fail under complex operating conditions (e.g., multi-gas source mixing, dynamic pressure fluctuations). To address these challenges, new mechanism modeling-oriented methods have emerged, with the two most representative research directions being “mechanism modeling-driven” and “hybrid modeling”. Both approaches integrate multi-source data fusion with virtual-real interaction to establish mechanism models between flow rate, other parameters, and energy, providing a new paradigm for accurate energy measurement, but new challenges have also arisen. The “mechanism modeling-driven” approach is based on static flow modeling using Computational Fluid Dynamics (CFD), but it has low dynamic parameter update efficiency (delay > 30 seconds), struggles to adapt to real-time operating condition changes, and relies on massive labeled data with insufficient interpretability. The “hybrid modeling” approach has unresolved issues in multi-module collaborative optimization. Furthermore, the core challenge of existing research lies in the lack of industrial-grade verification platform support, addressing the problems of dynamic response delay, parameter identification difficulties, excessive physical simplification, and weak anti-interference capability in traditional natural gas energy measurement methods under complex conditions. Building on the latest research findings of the “mechanism modeling-driven” and “hybrid modeling” approaches, this study innovatively introduces a variational autoencoder (VAE)-based operating condition feature extraction algorithm and a dynamic Bayesian network parameter calibration mechanism, combined with variational expectation-maximization (VEM) algorithm for offline calibration. It proposes a reverse hybrid modeling-driven digital twin system, which effectively solves the aforementioned problems in traditional natural gas energy measurement processes. Methods This study proposes a natural gas energy measurement digital twin system based on reverse hybrid modeling, which centers on a three-tier architecture of “algorithm-system-scenario”. It integrates calorific value, flow rate, and energy mechanism models with multi-source real-time data streams. A variational autoencoder (VAE) is introduced to achieve unsupervised operating condition feature mining, and a parameter self-correction loop is constructed by combining dynamic Bayesian network with variational expectation-maximization (VEM) system calibration. Industrial-grade devices such as ultrasonic flowmeters and gas chromatographs are integrated to ensure real-time data transmission and closed-loop control. The system covers core operating conditions such as dynamic pressure fluctuations, hydrogen-containing gas mixtures, and multi-gas source switching, ensuring a high degree of adaptability between the model and practical applications. Through 25 weeks of continuous verification on a full-scale industrial-grade experimental platform, the results show that the system has an operational delay ≤ 3.8 s, data transmission jitter ≤ 0.5 s, average daily energy consumption per device ≤ 1.2 kW·h, mean time between failures (MTBF) ≥ 4 100 h, energy measurement error ≤ 0.15%, calorific value error ≤ 0.12%, and flow rate indication error ≤ 0.2%. Meanwhile, the system meets security requirements through industrial Ethernet encryption and hierarchical access control, providing engineering support for intelligent pipeline network optimization and standardized integration. Results and Discussions First, a multi-level hybrid modeling framework is established: modular hybrid modeling is achieved through an algorithm-system-scenario three-tier architecture. Numerical methods combined with data are generally more flexible than purely analytical models and can be used to represent complex multi-physical systems because they employ fewer physical lumped parameterizations; moreover, under mechanical, energy, and hydrodynamic effects, these parameters may change during the energy measurement process. Deep integration of mechanism models and real-time data is realized through variational autoencoder (VAE) and dynamic Bayesian network, reducing parameter synchronization delay to 3.8 seconds, which effectively supports fluid-acoustic co-simulation and rapid response for complex working conditions such as hydrogen-containing natural gas. Second, an integrated algorithm for reverse hybrid modeling and system calibration is proposed: by introducing variational autoencoder (VAE), dynamic Bayesian network, and variational expectation-maximization (VEM) algorithm, a reverse hybrid modeling algorithm is constructed to form a self-supervised and adaptive intelligent system with inner closed-loop operation. The VAE encoder compresses high-dimensional operating condition data into low-dimensional feature vectors, enabling unsupervised feature extraction without massive labeled data. It can also automatically generate perturbed data similar to the input data based on learned internal distribution laws of the data, simulating abnormal operating conditions to verify anti-interference capability. Combined with dynamic Bayesian network, a continuous iterative cycle of "prior → evidence → posterior" is constructed to realize system self-correction and adaptive response to operating condition changes. The VEM algorithm specifically compensates for systematic errors that are difficult to cover by dynamic Bayesian networks, overcoming the limitations of traditional static models. Conclusions This study describes and gradually validates a hybrid digital twin system that combines experimental data-driven approaches with physical models, successfully simulating the physical characteristics of natural gas energy measurement. A full-scale test platform was constructed, and all major parameters of the system were rigorously validated through experimental measurement data and compared with industry benchmark data. Each independent module within the "algorithm-system-scenario" three-tier hybrid modeling architecture (including calorific value measurement, flow calculation, and energy conversion) underwent 25 weeks of continuous experimental validation, confirming a high degree of consistency between model predictions and actual measurements.On the established digital twin experimental platform for natural gas energy measurement, systematic validation was conducted on the three core functions: flow measurement under dynamic conditions, multi-component calorific value determination, and energy accumulation. The results demonstrated that the output of the digital twin model matched the physical device measurement data with over 99.5% accuracy. Notably, under complex operating conditions such as pressure pulsations and hydrogen-containing gas mixtures, the system maintained measurement accuracy within 0.5%, significantly outperforming traditional methods and meeting the Class A accuracy requirements for natural gas measurement.By introducing a multi-tier hybrid modeling framework, this study successfully addressed the challenges of parameter identification difficulties and excessive physical simplifications inherent in traditional natural gas energy measurement methods. The integration of Variational Autoencoders (VAEs), dynamic Bayesian networks, and Variational Expectation- Maximization (VEM) algorithms enabled unsupervised feature extraction for complex operating conditions and adaptive model parameter calibration, reducing reliance on prior physical knowledge and massive labeled datasets. Experimental evidence demonstrates that the proposed method maintains high precision and robustness even in complex scenarios, such as pressure pulsations and hydrogen-containing gas mixtures, where traditional models struggle to provide accurate descriptions.
- Natural gas energy metrology /
- Digital twin /
- Inverse hybrid modeling /
- Variational autoencoder (VAE) /
- Dynamic bayesian network

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图 1 天然气超声波流量计计量原理图

Figure 1. Measurement principle of ultrasonic flowmeters for natural gas

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图 2 时差式超声波流量计原理图

Figure 2. Schematic diagram of time-difference ultrasonic flowmeter

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图 3 VAE网络结构示图

Figure 3. Schematic diagram of VAE network structure

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图 4 数字孪生架构

Figure 4. Digital twin architecture version

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图 5 数字孪生实施流程图

Figure 5. Implementation flow chart of digital twin

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图 6 “算法-系统-场景”三层级混合建模架构图

Figure 6. Algorithm-system-scenario" three-level hybrid modeling architecture diagram

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图 7 天然气能量计量实验平台原理

Figure 7. Principle of the experimental platform for natural gas energy measurement

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图 8 能量计量实验平台

1.一体化能量计量装置 2.管道本体 3.标气 4.氢气 6.数据采集与控制系统 7.色谱分析仪 8.电源 9.采样器10.流量计11.上位机系统

Figure 8. Experimental platform for energy measurement

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图 9 天然气能量计量数字孪生平台

Figure 9. Natural gas energy measurement digital twin platform

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图 10 压缩因子计算的不确定度范围

Figure 10. Uncertainty range of compressibility factor calculation

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图 11 天然气流量测量示值误差

Figure 11. Indicated value error of natural gas flow measurement

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图 12 天然气周发热量

Figure 12. Weekly calorific value of natural gas

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图 13 天然气周发热量标准偏差

Figure 13. Standard deviation of weekly calorific value of natural gas

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表 1 基础参数与符号定义

Table 1. Definitions of Basic Parameters and Symbols

符号	定义	单位
Z	天然气混合气体压缩因子(无量纲)	-
Z_j	组分j的压缩因子	-
b_j	组分j的求和因子(b_j=1-Z_j)	-
ρ_m	摩尔密度	kmol/m³
H_s	计量参比条件下高位发热量，30MJ/m³≤H_s≤45MJ/m³	MJ/m³
d	相对密度(空气=1，20℃、101.325kPa)，0.55≤d≤0.80	-
x_co2	气体混合物中不可燃的非烃组分含量，即CO2的摩尔分数，0≤x_co2≤0.20	-
x_H2	气体混合物中可燃的非烃组分含量即H2的摩尔分数，0≤x_H2≤0.10	-
p	绝对压力，0MPa≤p≤12MPa	MPa
T	热力学温度，263K≤T≤338K	K
B	第二维利系数，为H_s、d、x_co2、x_H2、T的函数	m³/kmol
C	第三维利系数，为H_s、d、x_co2、x_H2、T的函数	m⁶/kmol²
R	摩尔气体常数，R=0.008314510MJ/(kmol·K)	MJ/(kmol·K)

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表 2 计量参比条件(20℃，101.325kPa)下天然气中常见组分的摩尔质量及物性参数

Table 2. Molar Mass and Physical Property Parameters of Common Components in Natural Gas Under Metering Reference Conditions (20 °C, 101.325 kPa)

序号	组分	摩尔质量 M_j(kg/kmol)	理想体积发热量 $ \tilde{H}_{j}^{0} $(MJ/m³)	理想摩尔发热量 $ \overline{H}_{j}^{0} $(MJ/kmol)	理想质量发热量 $ \hat{H}_{j}^{0} $(MJ/kg)	压缩因子 Z_j	求和因子 $ \sqrt{{b}_{j}} $
1	C₁甲烷	16.043	37.044	891.09	55.545	0.9981	0.0436
2	C₂乙烷	30.070	64.91	1561.41	51.93	0.9920	0.0894
3	C₃丙烷	44.097	92.29	2220.13	50.35	0.9834	0.1288
4	n-C₄正丁烷	58.123	119.66	2878.57	49.53	0.9682	0.1783
5	i-C₄异丁烷	58.123	119.28	2869.38	49.37	0.9710	0.1703
6	n-C₅正戊烷	72.150	147.04	3537.17	49.03	0.9450	0.2345
7	i-C₅异戊烷	72.150	146.76	3530.24	48.93	0.9530	0.2168
8	Neo-C₅新戊烷	72.150	146.16	3516.01	48.73	0.9590	0.2025
9	C₆己烷	86.177	174.46	4196.58	48.70	0.9190	0.2846
10	N₂氮气	28.0135	0	0	0	0.9997	0.0173
11	He氦气	4.0026	0	0	0	1.0005	0.0000
12	CO₂二氧化碳	44.010	0	0	0	0.9650	0.1871
13	Ar氩气	2.0159	11.889	285.99	141.87	1.0006	-0.0051
14	H₂S硫化氢	34.082	23.37	562.19	16.50	0.9900	0.1000
15	H₂O水	18.0153	1.84	44.224	2.45	0.9520	0.2191

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表 3 纯气体第二维利系数和非同类交互作用维利系数温度展开式中b(0)、b(1)、b(2)的数值

Table 3. Values of b(0), b(1), and b(2) in the temperature expansion formulas for the second virial coefficients of pure gases and the cross virial coefficients for heterogeneous interactions

	ij	b(0)	b(1)	b(2)
CH	H0	–4.25468×10^–1	2.86500×10^–3	–4.62073×10^–6
CH	H1	8.77118×10^–4	–5.56281×10^–6	8.81510×10^–9
CH	H2	–8.24747×10^–7	4.31436×10^–9	–6.08319×10^–12
N2	22	–1.44600×10^–1	7.40910×10^–4	–9.11950×10^–7
CO2	33	–8.68340×10^–1	4.03760×10^–3	–5.16570×10^–6
H2	44	–1.10596×10^–3	8.13385×10^–5	–9.87220×10^–8
CO	55	–1.30820×10^–1	6.02540×10^–4	–6.44300×10^–7
CH+N₂	12	y = 0.72 + 1.875 × 10^–5 (320 – T)²
CH+CO₂	13	y = -0.865
CH+H₂	14	–5.21280×10^–2	2.71570×10^–4	–2.50000×10^–7
CH+CO	15	–6.87290×10^–2	–2.39381×10^–6	5.18195×10^–7
N₂+CO₂	23	–3.39693×10^–1	1.161176×10^–3	–2.04429×10^–6
N₂+H₂	24	1.20000×10^–2	0.00000	0.00000

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表 4 基于验证集的不同扰动幅度下校准前后模型预测精度对比

Table 4. Comparison of Model Prediction Accuracy Before and After Calibration Under Different Disturbance Amplitudes Based on the Validation Set

VAE工况模式	扰动幅度	评估指标	校准前值	校准后值	提升率 (%)
稳态(C₀)	1%	NRMSE	0.0150	0.0080	46.67
		MAPE	0.0200	0.0120	40.00
		R²	0.9850	0.9920	0.71 *
		鲁棒性	82%	98%	19.51
2Hz流速波动(C₂)	2%	NRMSE	0.0030	0.0150	50.00
		MAPE	0.0350	0.0180	48.57
		R²	0.9750	0.9880	1.33 *
		鲁棒性	75%	96%	28.00
含氢 10%(C₄)	5%	NRMSE	0.0500	0.0200	60.00
		MAPE	0.0650	0.0300	53.85
		R²	0.9500	0.9800	3.16 *
		鲁棒性	60%	92%	53.33
注：R²提升率计算方式为 $ \frac{\text{校准后}{R}^{2}\text{-校准前}{R}^{2}}{\text{1-校准前}{R}^{2}}×100\% $，更能反映对未解释方差的改善程度，结果分别为 46.67%， 52.00%， 60.00%。NRMSE和MAPE的提升率计算方式为 $ \frac{\text{校准前值-校准后值}}{\text{校准前值}}×100\% $。

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表 5 工业现场对比测试

Table 5. Industrial Field Comparative Testing

指标	色谱分析法	声速关联法	本系统
热值误差	±0.35%	±1.8%*	±0.12%
流量示值误差	±0.45%	±0.8%	±0.2%
能量计量误差	±0.5%	±1.9%	±0.15%
流量动态响应速度	180 s	40 s	3.8 s
多气源切换适应性	需人工校准	失效	自动切换(响应≤0.3 s)
含氢工况适配性	需手动调整	含氢＞5%	VAE 自动识别 + 参数自适应
维护频次	2次/周	1次/周	1次/月
鲁棒性	±0.5%	±1.9%	±0.15%
资源消耗	1.8 kW·h	1.9 kW·h	1.2 kW·h
数据传输稳定性	抖动≤2 s	抖动≤5 s	抖动≤0.5 s
安全性	基础加密	无防护	工业以太网加密 + 权限分级(上位机)
*注：含氢＞5%时，声速关联法的热值误差急剧增大。

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参考文献(28)

[1]	SUN Qie, LI Hailong, MA Zhanyu, et al. A comprehensive review of smart energy meters in intelligent energy networks[J]. IEEE Internet of Things Journal, 2016, 3(4): 464–479. doi: 10.1109/JIOT.2015.2512325.
[2]	MONTUORI L, ALCÁZAR-ORTEGA M, VARGAS-SALGADO C, et al. Enabling the natural gas system as smart infrastructure: Metering technologies for customer applications[C]. 2020 Global Congress on Electrical Engineering (GC-ElecEng), Valencia, Spain, 2020: 96–100. doi: 10.23919/GC-ElecEng48342.2020.9286291.
[3]	ULBIG P and HOBURG D. Determination of the calorific value of natural gas by different methods[J]. Thermochimica Acta, 2002, 382(1/2): 27–35. doi: 10.1016/S0040-6031(01)00732-8.
[4]	LANG Xianming, LI Ping, GUO Ying, et al. A multiple leaks’ localization method in a pipeline based on change in the sound velocity[J]. IEEE Transactions on Instrumentation and Measurement, 2020, 69(7): 5010–5017. doi: 10.1109/TIM.2019.2955774.
[5]	SELEZNEV V. Computational fluid dynamics methods for gas pipeline system control[M]. Computational Fluid Dynamics. IntechOpen, 2010: 335–362. doi: 10.5772/7110. (查阅网上资料,未找到本条文献出版地,请确认).
[6]	KUNZ O and WAGNER W. The GERG-2008 wide-range equation of state for natural gases and other mixtures: An expansion of GERG-2004[J]. Journal of Chemical & Engineering Data, 2012, 57(11): 3032–3091. doi: 10.1021/je300655b.
[7]	连远锋, 田天, 陈晓禾, 等. 数字孪生辅助强化学习的燃气站场巡检任务分配算法[J]. 电子与信息学报, 2025, 47(7): 2285–2297. doi: 10.11999/JEIT241027. LIAN Yuanfeng, TIAN Tian, CHEN Xiaohe, et al. Gas station inspection task allocation algorithm in digital twin-assisted reinforcement learning[J]. Journal of Electronics & Information Technology, 2025, 47(7): 2285–2297. doi: 10.11999/JEIT241027.
[8]	刘宁波, 张子豪, 陈宝欣, 等. 海上目标多源数据特征提取与特征相关性分析[J]. 电子与信息学报, 2025, 47(10): 3745–3758. doi: 10.11999/JEIT250200. LIU Ningbo, ZHANG Zihao, CHEN Baoxin, et al. Features extraction and correlation analysis of multi-source data for maritime targets[J]. Journal of Electronics & Information Technology, 2025, 47(10): 3745–3758. doi: 10.11999/JEIT250200.
[9]	胡小方, 杨涛. 基于忆阻循环神经网络的层次化状态正则变分自编码器[J]. 电子与信息学报, 2023, 45(2): 689–697. doi: 10.11999/JEIT211431. HU Xiaofang and YANG Tao. Hierarchical state regularization variational autoencoder based on memristor recurrent neural network[J]. Journal of Electronics & Information Technology, 2023, 45(2): 689–697. doi: 10.11999/JEIT211431.
[10]	杨旗, 薛定宇. 基于双尺度动态贝叶斯网络及多信息融合的步态识别[J]. 电子与信息学报, 2012, 34(5): 1148–1153. doi: 10.3724/SP.J.1146.2011.01012. YANG Qi and XUE Dingyu. Gait recognition based on two-scale dynamic Bayesian network and more information fusion[J]. Journal of Electronics & Information Technology, 2012, 34(5): 1148–1153. doi: 10.3724/SP.J.1146.2011.01012.
[11]	ULBIG P and HOBURG D. Determination of the calorific value of natural gas by different methods[J]. Thermochimica Acta, 2002, 382(1/2): 27–35. (查阅网上资料,本条文献与第3条重复,请确认). doi: 10.1016/s0040-6031(01)00732-8.
[12]	中华人民共和国国家质量监督检验检疫总局, 中国国家标准化管理委员会. GB/T 17747.3-2011 天然气压缩因子的计算第3部分: 用物性值进行计算[S]. 北京: 中国标准出版社, 2012. General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China and Standardization Administration of the People's Republic of China. GB/T 17747.3-2011 Natural gas—calculation of compression factor—Part 3: Calculation using physical properties[S]. Beijing: Standards Press of China, 2012.
[13]	国家市场监督管理总局. JJF 1993-2022 天然气能量计量技术规范[S]. 北京: 中国标准出版社, 2022. State Administration for Market Regulation. JJF 1993-2022 Metrological specification for the energy measurement of natural gas[S]. Beijing: Standards Press of China, 2022. <b>(查阅网上资料, 未找到本条文献出版年, 请确认)</b>.
[14]	胡晶晶, 杨照明, 苏怀. 基于深度学习的天然气热值动态预测[J]. 能源, 2023, 16(2): 799. doi: 10.3390/en16020799. HU Jingjing, YANG Zhaoming, and SU Huai. Dynamic Prediction of Natural Gas Calorific Value Based on Deep Learning[J]. Energies, 2023, 16(2): 799. (查阅网上资料,未找到本条文献中文信息,请确认). doi: 10.3390/en16020799.
[15]	王池, 李春辉, 王京安, 等. 天然气能量计量系统及方法[J]. 计量学报, 2008, 29(5): 403–406. WANG Chi, LI Chunhui, WANG Jing’an, et al. The system and method for energy measurement of natural gas[J]. Acta Metrologica Sinica, 2008, 29(5): 403–406.
[16]	VASKOVSKII S and BROKAREV I. Analysis of methods and systems for natural gas composition and energy characteristics determination[C]. 2023 5th International Conference on Problems of Cybernetics and Informatics (PCI), Baku, Azerbaijan, 2023: 1–6. doi: 10.1109/PCI60110.2023.10325991.
[17]	魏伟, 鲜平, 彭刚, 等. 天然气能量计量装置研究[C]. 2011 年机电科学、电气工程与计算机国际会议, 吉林, [s. n. ], 2011: 685–688. (查阅网上资料, 未找到本条文献信息, 请确认).
[18]	ZHAO Huichao, PENG Lihui, STEPHANE S A, et al. CFD aided investigation of multipath ultrasonic gas flow meter performance under complex flow profile[J]. IEEE Sensors Journal, 2014, 14(3): 897–907. doi: 10.1109/jsen.2013.2290863.
[19]	ZHAO Yuting, WEI qiang, LI Meng, et al. Design and implementation of an external clamped ultrasonic flowmeter based on time difference method[C]. 2021 IEEE International Conference on Computer Science, Electronic Information Engineering and Intelligent Control Technology (CEI), Fuzhou, China, 2021: 685–688. doi: 10.1109/CEI52496.2021.9574588.
[20]	LI Xiaobin, XIAO Bo, CHEN Xuejiao, et al. MSDF-VAE: A cloud–edge collaborative method for fault diagnosis based on transfer learning[J]. IEEE Internet of Things Journal, 2025, 12(12): 22393–22403. doi: 10.1109/JIOT.2025.3550916.
[21]	ZHANG Haizheng, SESHADRI R, PRAKASH A A, et al. Towards dynamic Bayesian networks: State augmentation for online calibration of DTA systems[C]. 2018 21st International Conference on Intelligent Transportation Systems (ITSC), Maui, USA, 2018: 1745–1750. doi: 10.1109/ITSC.2018.8569926.
[22]	RUAH C, SIMEONE O, HOYDIS J, et al. Calibrating wireless ray tracing for digital twinning using local phase error estimates[J]. IEEE Transactions on Machine Learning in Communications and Networking, 2024, 2: 1193–1215. doi: 10.1109/TMLCN.2024.3448391.
[23]	ZHANG Yan, SUN Wen, and ALCARAZ C. Editorial CFP: IEEE transactions on industrial informatics—special section on digital twin for industrial internet of things[J]. IEEE Transactions on Industrial Informatics, 2023, 19(5): 7214–7216. doi: 10.1109/tii.2023.3261167.
[24]	TYAGUNOV M G and ANDREEV V N. The use of digital twins in the energy sector: Selection and verification of tools for modeling hybrid energy supply systems based on renewable energy sources[C]. 2025 International Russian Smart Industry Conference (SmartIndustryCon), Sochi, Russian Federation, 2025: 572–577. doi: 10.1109/SmartIndustryCon65166.2025.10986090.
[25]	KABIR R, HALDER D, and RAY S. Digital twins for IoT-driven energy systems: A survey[J]. IEEE Access, 2024, 12: 177123–177143. doi: 10.1109/ACCESS.2024.3506660.
[26]	SARGENT R G. Verification and validation of simulation models: An advanced tutorial[C]. 2020 Winter Simulation Conference (WSC), Orlando, USA, 2020: 16–29. doi: 10.1109/WSC48552.2020.9384052.
[27]	JIA Wenlong, WANG Xiujuan, WU Xia, et al. A stable solution method for natural gas density across a wide temperature range using the GERG-2008 equation of state[J]. Fluid Phase Equilibria, 2025, 593: 114328. doi: 10.1016/j.fluid.2024.114328.
[28]	李军, 于波. 天然气性质、基本状态方程与相态关系[M]. 可持续天然气藏与开采工程. 北京: 海湾专业出版社, 2022: 1–28. (查阅网上资料, 未找到本条文献信息, 请确认).