Under the volumetric pricing model, assume that the gas demand function of gas users is D v = a − b 1 p + b 2 H v , where a is the total potential market demand, b 1 is the price elasticity coefficient, and b 2 is the coefficient of influence of the calorific value of natural gas per unit of gas on the expansion of market demand. The profit function of gas supplier S is as follows: π v H v = p − g D v − 1 2 θ H v 2 (1)

The above equation shows that the profit function for gas merchant S is the difference between revenues from gas sales and costs, with costs including not only the cost of unit sales but also the cost of regulating the calorific value of the gas. Denote the lower limit value of the GB17820-2012 standard on the unit calorific value of natural gas as H s ( H s = 31.4 M J / m 3 ), and if the gas merchant S seeks to maximize the profit, the following optimization model in Eq. 2 needs to be solved: Max H v   π v H v   s t .   H v ≥ H s (2)

According to the extreme value theory and the KKT condition, the optimal unit calorific value decision H v * and the maximized profit level π v * of the gas trader S under the volumetric pricing model are as follows, respectively: H v * = p − g b 2 θ , if  θ < p − g b 2 H s   H s   , if  θ ≥ p − g b 2 H s   (3) π v * = p − g a − b 1 p θ + 1 2 p − g b 2 2 θ , if  θ < p − g b 2 H s   p − g a − b 1 p + b 2 H s − 1 2 θ H s 2   , if  θ ≥ p − g b 2 H s   (4)

Proof: the second-order derivative of π v H v with respect to H v in Eq. 1 yields ∂ 2 π v H v ∂ H v 2 = − θ . Then π v H v is a convex function with respect to H v , and there exists an optimal per-unit calorific value decision H v * and a maximized level of profit π v * for gas trader S .

The above results show that the optimal unit calorific value decision of gas supplier S in volumetric pricing model H v * is related to the value of its natural gas unit calorific value moderating cost coefficient θ . When the value of unit calorific value control cost coefficient θ is lower, the gas supplier S will set the unit calorific value higher than GB17820-2012 standard, i.e., H v * > H s ; on the contrary, when the value of unit calorific value control cost coefficient θ is higher, the optimal unit calorific value of the gas supplier S is equal to the unit calorific value of GB17820-2012 standard, i.e., H v * > H s . The optimal unit calorific value of gas supplier S is equal to the unit calorific value of GB17820-2012 standard, i.e., H v * = H s .

This section considers city gas supplier S supplying gas to gas users (residential or non-residential) and using energy metering pricing to settle with gas users. Energy metering pricing is closely related to the unit calorific value of natural gas, and the sales price of natural gas is generally based on the standard unit calorific value of natural gas prescribed by the NDRC converted from the sales price under the volumetric metering pricing model. The standard unit calorific value of natural gas specified by the NDRC under the energy-metering pricing model is denoted as H 0 (unit MJ / m 3 ), and if the natural gas sales price under the volumetric pricing model is p (unit y u a n / m 3 ), the If the sales price of natural gas under the volumetric pricing model is p , then the sales price of natural gas under the volumetric pricing model is p e = p H o (unit yuan / MJ ), and the cost of sales per unit for gas supplier S is g e = g H o (unit yuan / MJ ).

To ensure that the model is meaningful, it is assumed that the standard unit calorific value of natural gas specified by the NDRC satisfies H 0 > 2 p − g b 2 θ . Lu et al. (2019) emphasize the importance of governmental departments such as the NDRC to regulate the standard unit calorific value of natural gas under the energy metering and pricing model, so in addition to exploring the optimal unit calorific value decision of city gas dealers under the energy metering and pricing model, this paper further analyzes the impacts of the standard unit calorific value of natural gas, H 0 , on the optimal decision-making of the gas dealers, the maximization of profits, and the surplus of gas users’ consumers. Assuming that gas supplier S sets the unit calorific value of natural gas to be H e under the energy metering pricing model, the cost to be borne by S when the unit calorific value is H e is 1 2 θ H e 2 , where θ is the cost coefficient of gas unit calorific value regulation of gas producer S . Following the parameter notation of the previous section, the gas demand of a gas customer is known to be a − b 1 p + b 2 H e in the volumetric pricing model, which can be transformed into D e = a − b 1 p + b 2 H e H e in the energy metering model. The profit function of the gas merchant S can be obtained as follows: π e H e = p e − g e D e − 1 2 θ H e 2 = p − g H o a − b 1 p + b 2 H e H e − 1 2 θ H e 2 (5)

The above equation shows that unlike volumetric pricing, the profit level of gas merchant S under the energy metering pricing model is directly affected by the standard unit calorific value of natural gas, H 0 , set by the NDRC. If the calorific value of natural gas sold by gas supplier S is equal to the standardized calorific value of natural gas in the weather set by NDRC, i.e., H e = H 0 , then the profit of gas supplier S will be equal under the two metering and pricing models. If gas trader S seeks to maximize its profit, it needs to solve the following optimization model in Eq. 6: Max H e   π e H e   s t . H e ≥ H 0 (6)

According to the extreme value theory and KKT condition, the optimal unit calorific value H e * and the maximized profit π e * of the gas supplier S under the energy metering and pricing model can be obtained as follows, respectively, H 0 ^ = p − g ( − b 1 p + a ) θ + p − g b 2 2 + b 2 p − g θ : H e * = p − g ( a − b 1 p H 0 θ − 2 b 2 p + 2 b 2 g , if  2 p − g b 2 θ < H 0 < H 0 ^   H 0   , if  H 0 ≥ H 0 ^   (7) π e * = p − g 2 − b 1 p + a 2 2 H 0 θ − 4 b 2 p − g H 0 ,  if  2 p − g b 2 θ < H 0 < H 0 ^ p − g a − b 1 p + b 2 H 0 − 1 2 θ H 0 2   , if  H 0 ≥ H 0 ^ (8)

Proof: the second order derivative of π e H e with respect to H e in Eq. 5 yields ∂ 2 π e H e ∂ H e 2 = 2 p − g b 2 H 0 − θ . The purpose of finding the maximum value of the profit of the gas trader S requires that ∂ 2 π e H e ∂ H e 2 < 0 . It is then required that H 0 > 2 p − g b 2 θ .

The above results indicate that the optimal unit calorific value decision H e * for gas supplier S under the energy metering pricing model is related to both the value of its regulation cost coefficient θ per unit calorific value and the standard unit calorific value of natural gas, H 0 , as specified by the NDRC. The optimal unit calorific value decision H e * for S is monotonically decreasing with respect to its unit calorific value regulation cost coefficient θ as well as the standard unit calorific value of natural gas H 0 , when all else is equal. This suggests that a higher standard unit calorific value of natural gas H 0 set by the NDRC may cause gas suppliers to reduce their unit calorific value H e * to a level exactly equal to the standard unit calorific value H 0 .

First, we analyze the impact of the reform of the metering and pricing model on gas suppliers’ optimal unit calorific value decisions. By comparing the optimal calorific value per unit of gas merchant S in the volumetric pricing mode H v * Eq. 3 and the optimal calorific value per unit of gas merchant S in the energy metering pricing mode H e * Eq. 7. It can be obtained that under the energy metering pricing model, if the standard calorific value H 0 set by NDRC is lower than the threshold H 0 ∼ i.e., it satisfies 2 p − g b 2 θ < H 0 < H 0 ∼ , then the optimal unit calorific value of the gas merchant is higher than its optimal unit calorific value under the volumetric metering pricing model i.e., H e * > H v * . Conversely, if H 0 ≥ H 0 ∼ , then H e * ≤ H v * , where H 0 ∼ = p − g 2 H s b 2 − b 1 p + a H s θ .

The above results firstly show that government departments such as the NDRC can guide gas suppliers to adjust the unit calorific value H e * of the natural gas they sell by regulating the standard unit calorific value H 0 of natural gas. Under the energy metering pricing model, if the NDRC controls the standard unit calorific value of natural gas, H 0 , within a reasonable range, it can effectively guide gas suppliers to increase the unit calorific value of the natural gas they sell to a level that exceeds the unit calorific value of the natural gas under the volumetric metering pricing model.

Further analyze the impact of the two metering and pricing models on the profit level of gas suppliers. By comparing the maximized profit of gas trader S in the volumetric pricing model π v * Eq. 4 with the maximized profit in the energy metering pricing model π e * Eq. 8. It can be obtained that under the energy metering pricing model, if the standard unit calorific value H 0 set by the NDRC and other governmental departments is lower to satisfy 2 p − g b 2 θ < H 0 ≤ H α , then the gas supplier’s maximized profit is higher than its maximized profit level under the volumetric pricing model i.e., π e * ≥ π v * . Otherwise π e * < π v * , where H α = B 2 − 4 A C − B 2 A , A = 2 θ p − g a − b 1 p + H s b 2 − θ 2 H s 2 , B = 2 p − g b 2 ( θ H s 2 − 2 p − g a − b 1 p + H s b 2 , C = − p − g 2 a − b 1 p 2 .

The above results indicate that under the energy metering pricing model, the standard unit calorific value of natural gas, which is set by the NDRC, directly affects the profit level of gas suppliers. If the standard unit calorific value of natural gas under the energy-metered pricing model is low (below the threshold H α ), the energy-metered pricing model is able to generate higher profits for the gas supplier than the volumetric pricing model. Therefore, government departments such as the NDRC can regulate the standard heat unit value of natural gas, thereby affecting the incentives of gas suppliers to produce and operate under the energy metering and pricing model.

This section analyzes the consumer surplus of gas customers under two metered pricing models and compares the relationship between their magnitudes. Under the volumetric pricing model, the consumer surplus function the consumer surplus function in Eq. 9 that a unit of natural gas with a calorific value of H v can generate for a gas customer is: C S v = ∫ 0 a − b 1 p + b 2 H v H v b 2 − D v + a b 1 d D v − p a − b 1 p + b 2 H v (9)

Substituting into H v * Eq. 3, the consumer surplus of a gas customer under the volumetric pricing model can be obtained as: C S v * = b 2 2 p − b 1 p θ − b 2 2 g + a θ 2 2 b 1 θ 2 , if  θ < p − g b 2 H s   b 2 H s − b 1 p + a 2 2 b 1   , if  θ ≥ p − g b 2 H s   (10)

Under the energy pricing model, the consumer surplus function in Eq. 11 that a unit of natural gas with a calorific value of H e can generate for a non-residential gas customer is: C S e = ∫ 0 a − b 1 p + b 2 H e H e H e 2 b 2 − D e + H e a b 1   H e d D e − p a − b 1 p + b 2 H e H e (11)

Substituting into H e * Eq. 7, the consumer surplus under the energy metering pricing model can be obtained as: C S e * = p − g − b 1 p + a 3 − b 2 p + b 2 g + H 0 θ 2 2 H 0 θ − 2 b 2 p + 2 b 2 g 3 b 1 (12)

By comparing the relationship between consumer surplus C S v * in Eq. 10 and C S e * in Eq. 12 under the two metering models, it can be seen that if the standard calorific value H 0 set by the government department is lower to satisfy 2 p − g b 2 θ < H 0 ≤ H β , the consumer surplus under the energy metering model is higher than that under the volumetric model, i.e., C S e * ≥ C S v * . Otherwise C S e * < C S v * , where H β is the unique real root of the difference between the two types of consumer surplus.

The above results indicate that the change of natural gas metering and pricing model will directly affect the consumer surplus of city gas users. When the standard unit calorific value of natural gas in the energy-metered pricing model is low (below the threshold H β ), the energy-metered pricing model is able to generate a higher consumer surplus for city gas users than the volumetric pricing model. Otherwise, the energy metering pricing model will reduce consumer surplus. The finding also suggests that the NDRC can regulate the standard unit calorific value of natural gas under the energy metering pricing model, thereby guiding the preference of city gas users for the natural gas metering pricing model.

Verify how the gas supplier sets the optimal unit calorific value and the effect of unit calorific value on the gas supplier’s profit under the volumetric pricing model. By assigning values to the parameters p = 3.1 , g = 0.1 , b 1 = 3 , b 2 = 4 , a = 100 , Figures 1, 2 can be obtained based on the findings in Section 2. Figure 1 shows the optimal unit calorific value decision of the gas supplier under the volumetric pricing model, indicating that when the gas supplier’s natural gas unit calorific value regulation cost coefficient θ is low, i.e., does not exceed the threshold p − g b 2 H s , then the optimal unit calorific value θ of the gas supplier will decrease with increasing θ . Until the unit calorific value regulation cost coefficient θ increases beyond the threshold p − g b 2 H s , the gas supplier reduces its unit calorific value to the lower limit of the unit calorific value in the GB17820-2012 standard, H s ( H s = 31.4 MJ / m 3 ). Figure 2 shows the profit level of the gas supplier in the volumetric pricing model, which shows that the profit level of the gas supplier increases and then decreases with respect to the calorific value of the natural gas it sells, H v , and maximizes its profit at H v = H v * .

Under the energy metering and pricing model, verify how gas suppliers set the optimal unit calorific value and the impact of the unit calorific value on the profitability of gas suppliers. By assigning values to the parameters p = 3.1 , g = 0.1 , H 0 = 37 , b 1 = 3 , b 2 = 4 , θ = 0.87 , a = 100 , Figure 3 can be obtained based on the findings in Section 3. Figure 3 shows the profit level of the gas supplier under the energy metering pricing model, and it can be seen that the profit function of the gas supplier increases and then decreases with respect to the calorific value of the natural gas sold by the supplier H e per unit, and maximizes the profit at H e = H e * .

By assigning values to the parameters p = 3.1 , g = 0.1 , b 1 = 3 , b 2 = 4 , a = 100 , and comparing the magnitude of the relationship between the standard unit calorific value H 0 , which is set by the NDRC and other governmental departments, and the optimal unit calorific value of the natural gas sold by gas merchants, H e * , under the energy metering pricing model, Figure 4 can be obtained. Figure 4 shows that under the energy metering and pricing model, if the standard unit calorific value H 0 set by the NDRC and other governmental agencies is low enough not to exceed the threshold H 0 ^ i.e., the value of H 0 falls into region I, then the gas supplier S is encouraged to increase the optimal unit calorific value of the natural gas it sells beyond the standard unit calorific value level, i.e., H e * > H 0 . If a higher value of H 0 falls into region II, it may result in the unit calorific value of natural gas sold by gas trader S being lower than the standard unit calorific value, i.e., H e * ≤ H 0 .

The following verifies the optimal unit calorific value decision and the change in profit level of the gas supplier under the two metering and pricing models. By assigning values to the parameters p = 3.1 , g = 0.1 , b 1 = 3 , b 2 = 4 , θ = 0.87 , a = 100 , Figures 5, 6 can be obtained based on the findings of Section 4.

Figure 5 shows the optimal unit calorific value decision of the gas supplier under the two metered pricing models. Firstly, it is shown that the optimal unit calorific value of natural gas sold by gas suppliers under the energy metering pricing model decreases as the standard unit calorific value H 0 set by the NDRC and other governmental departments increases. Secondly, when the standard unit calorific value H 0 is below the threshold H 0 ∼ , the optimal unit calorific value of the gas merchant under energy metering is higher than in the volumetric pricing model. Conversely if the standard unit calorific value H 0 exceeds the threshold H 0 ∼ , the gas merchant’s optimal unit calorific value under energy metering is lower than in the volumetric pricing model.

Figure 6 shows the maximized profit levels for gas suppliers under the two metered pricing models. Firstly, it shows that the profit level of gas suppliers in the energy metering and pricing mode decreases with the increase of the standard unit calorific value H 0 stipulated by the NDRC and other governmental departments, while the profit level of gas suppliers in the volumetric metering and pricing mode is independent of the H 0 due to the implementation of the GB17820-2012 standard. Secondly when the standard unit calorific value H 0 is lower than the threshold H α , the gas supplier maximizes a higher level of profit under energy metering than under volumetric pricing models. Conversely if the standard unit calorific value H 0 exceeds the threshold H α , the gas merchant maximizes a lower level of profit under energy-metered pricing than under the volumetric pricing model.

In order to verify the effect of the two metering and pricing models of natural gas on the consumer surplus of gas users, the parameters are assigned the values p = 3.1 , g = 0.1 , b 1 = 3 , b 2 = 4 , θ = 2.5 , a = 100 , and according to the results of the study in Section 4; Figure 7 can be obtained.

Figure 7 first shows that under the energy metering pricing model, the consumer surplus of gas users will be affected by the standard unit calorific value of natural gas H 0 set by the NDRC and other governmental departments, i.e., it decreases as the standard unit calorific value H 0 rises. In the volumetric pricing model, consumer surplus is not related to H 0 because gas suppliers implement the GB17820-2012 standard. Second, when the standard unit calorific value H 0 is lower than the threshold H β , the consumer surplus of gas users under energy metering is higher than in the volumetric pricing model. Conversely, if the standard unit calorific value H 0 exceeds the threshold H β , the consumer surplus of a gas customer under energy metering is lower than in the volumetric pricing model.

Finally, based on the conditions related to the energy metering and pricing model to increase the profit level of gas merchants as well as the consumer surplus of gas users in Section 4, the strategy-oriented diagram for setting the standard unit calorific value of natural gas under the energy metering and pricing model of the Development and Reform Commission (DRC) and other governmental departments is plotted by assigning the values of the parameters p = 3.1 , g = 0.1 , b 1 = 3 , b 2 = 4 , a = 100 in Figure 8.

As can be seen from Figure 8, the government should set the standard unit calorific value of natural gas, H 0 , between the thresholds 2 p ‐ g b 2 θ and H β under the energy metering model, as shown in region I of the figure, so that the gas supplier and the gas user can obtain a higher level of profit and consumer surplus than that under the volumetric pricing model, respectively. If the government sector sets the standard unit calorific value H 0 between the thresholds H β and H α , as shown in region II of the figure, only gas users can obtain a higher consumer surplus under the energy metering tariff model than under the volumetric tariff model, whereas the gas supplier can obtain a lower level of profit under the energy metering tariff model than under the volumetric tariff model. If government departments set the standard unit calorific value H 0 above the threshold H α , as shown in region III of the figure, this would result in gas suppliers as well as gas consumers having lower profits and consumer surplus under the energy metering tariff model than under the volumetric tariff model.

This paper establishes a mathematical model based on optimization theory to explore the operational decisions of city gas suppliers under volumetric and energy metering pricing models. In the volumetric pricing mode, the gas supplier only needs to set the unit calorific value of the natural gas sold in accordance with the national standard GB17820-2012, and the study found that due to the cost of regulating the unit calorific value of natural gas, the gas supplier lacks the incentive to increase the calorific value of the natural gas sold, and only ensures that the high level of heat generation is not lower than that of the GB17820-2012 standard, which is 31.4 MJ / m 3 . This is all that is required. In contrast, under the energy metering pricing model, the sales price of natural gas is generally based on the standard unit calorific value of natural gas prescribed by the NDRC and other government departments, which is converted from the sales price under the volumetric pricing model. The demand for natural gas under this model is not only price-dependent, as in the volumetric pricing model, but is also affected by the calorific value of the gas Thus under the energy metering pricing model, gas suppliers have an incentive to voluntarily adjust the calorific value of natural gas in order to increase sales and improve profits, even if doing so increases costs. The study also found that by setting a reasonable standard unit calorific value for natural gas, the government can effectively influence the decision of gas suppliers to adjust the calorific value of natural gas. In short, different pricing models affect the strategic choices of gas suppliers regarding the calorific value of natural gas.

In addition, the reform of the gas metering and pricing model may have both positive and negative impacts on the profits of gas suppliers as well as on the consumer surplus of gas users. This paper shows that compared to the volumetric pricing model, the energy metering pricing model can increase both the gas provider’s profit and the consumer’s surplus for the user. This finding correlates with the government-mandated standard unit calorific value of natural gas. Theoretically, it has been demonstrated that the NDRC can influence the operations of gas suppliers by adjusting the standard unit calorific value of natural gas and induce customers to prefer the energy metering pricing model. Meanwhile, a formula for deriving a reasonable range of standard unit calorific values for natural gas is proposed.

The results of this study may contribute to the existing literature on natural gas pricing modeling in the following ways: On the one hand, prior to this study, there were few systematic studies in the literature on the impacts of gas energy metering and pricing reforms on the operational strategies of gas supply chain participants and the welfare of gas users. Therefore, the study fills the research gap in this field to a certain extent and provides a new perspective and theoretical framework for related research. On the other hand, related scholars can improve or extend the model based on it to provide additional support or challenge the validity of the model. Of course, this will help researchers to better understand the limitations and strengths of existing models and the extent of their applicability in different contexts. In addition, the study may provide policymakers and the industry with strategic recommendations on how to design and implement natural gas energy metering and pricing reforms. These proposals may help optimize the functioning of the gas market, improve the operational efficiency of supply chain participants and increase the welfare of gas users. Overall, this study may provide important theoretical and empirical support for the academic and practical communities in the field of natural gas market reform, and help to advance the progress of research and practice in this area.

However, the model also has some limitations. First of all, this paper is based on the condition that the natural gas sales price is relatively stable. However, in the actual market, natural gas prices may be affected by supply and demand, geopolitical factors, seasonal factors, technological advances, environmental policies and market competition. Thereby, the actual market price fluctuates and the model’s pricing results may deviate from the actual situation, leading to inaccurate operational decisions. Second, natural gas may have non-uniformities in the transmission pipeline, including temperature, pressure, and compositional inhomogeneities. Non-uniformity may lead to errors in metered billing, as gas characteristics at different times or locations may affect the accuracy of the actual energy value. On the one hand, a variety of contracts and regulations may be involved in the natural gas trading process. The complexity of these contracts and regulations may make it impossible for the energy measurement and pricing model to fully comply with the actual legal and contractual requirements, which may lead to disputes or legal problems. On the other hand, natural gas markets may have different market structures and operate differently in different regions and countries. Models may not adequately account for these geographic differences and the effects of market structure, which can affect measurement results.

(1) Accelerate the reform of natural gas energy measurement and pricing, not only to promote China’s international trade in natural gas convergence, but also to ensure the fairness of natural gas transactions. On the one hand, China’s natural gas dependence on foreign countries has reached 43%–45%, the vast majority of energy-exporting countries have energy measurement and pricing, and China’s energy companies use energy measurement and pricing for settlement when they purchase natural gas. The implementation of energy metering and pricing in China is more conducive to natural gas price diversion and international harmonization. On the other hand, since most of the natural gas sold by city gas suppliers is a mixture of gas from different pipeline sites, there are some differences in the heating value. Therefore, under an energy-metered pricing model, gas suppliers will respond more aggressively to the regulation of the calorific value of the gas they sell than they would under a volumetric pricing model. This effectively avoids gas traders from adulterating piped natural gas with nitrogen and other substandard behaviors, successfully eliminates defects in volumetric metering, better protects the interests of gas users, returns natural gas to its value attributes, and makes natural gas trading fairer.

(1) City gas operators should be supported to integrate renewable energy sources, such as biogas or hydrogen, in order to reduce dependence on non-renewable sources and promote sustainable energy development.

In the future, case studies of city gas supply companies using energy metering and pricing will be planned for further research and validation of the model. The first step is to collect the actual operational data of the company over a period of time. In the second step, the actual data are input into the energy metering pricing model to generate the optimal unit calorific value and profit of the company predicted by the model under the energy metering pricing model. In the third step, the model predictions are analyzed against the actual data to compare the company’s operational decisions in both scenarios. This is to verify whether the model can accurately reflect the actual operation situation and can provide effective support and guidance for decision-making. Finally, based on the results of the model validation, necessary adjustments and improvements are made to the energy metering and pricing model. This may include adjusting model parameters, improving model algorithms, or adding additional influences.