Peak Carbon Dioxide Emissions Strategy Based on the Gray Model between Carbon Emissions and Urban Spatial Expansion for a Built-Up Area

Abstract

Urban spatial expansion affects almost every dimension of sustainable urban development. A good grasp of the relationship between urban spatial evolution and carbon emissions can be the key to urban spatial governance. As a central city in the central region and a national low-carbon pilot city, Changsha has experienced a rapid expansion of construction land and growing carbon emissions. In this paper, four variable factors and five variable factors of carbon emission were used for the case city Changsha in 1979, 1996, 2003, 2008, 2013 and 2016. Based on the “double carbon” constraint target, the total carbon emissions, carbon-emission intensity and per capita carbon emission constraint indices were forecasted until 2030. They are 87.29 million t-CO₂, 0.45 t-CO₂/CNY10⁴ and 8.73 t-CO₂/person, respectively. The scale of urban land is controlled at 889.61; the constraint indices of residential, commercial service land, industrial land and road square land scales are 231.3 km², 143.88 km², 150.17 km² and 135.83 km², respectively. The land expansion intensity, urban compactness and shortest travel distance constraint indices are 6.19, 0.236 and 96,086.76 km, respectively. The results of this analysis can provide scientific guidance for the next step in territorial spatial master planning and low-carbon governance.

1. Introduction

Low-carbon governance has become an important element of global carbon emission reduction. Urban expansion has a certain locking effect on the good operation of cities and the release of urban carbon emissions. Because the main strategy in low-carbon urban planning is urban spatial planning, the relationship between carbon emissions and urban spatial expansion has been the focus of relevant research. The existing academic literature mainly focuses on this relationship, particularly studying land use, urban spatial form, urban compactness, urban transportation and construction control. The research on land use and carbon emissions mainly includes three aspects: land-use carbon-emission effect, land-use carbon emissions accounting and low-carbon land-use policy. Dong [1] and Houghton [2] studied the impact of land-use change on carbon emissions; Wei [3], Niu [4], Xu [5] and Zhang [6] considered the spatiotemporal pattern, spatial differentiation characteristics, spatial correlation and influencing factors, focused on the impact of landout mode on carbon emissions; Chang [7], Yan [8], Yang [9] and Zhang [10] analyzed the relationship between construction land expansion and carbon emissions or carbon-emission intensity by building a Tapio decoupling model, a gray correlation model, a Kuznets curve model and a coupling coordination model. The calculation of land-use carbon emissions is conducted by using the model method [11,12], sample plot inventory method [13,14], remote sensing estimation method [15,16,17] and other calculation methods to study the change in carbon emissions with the dynamic change in land use. From the perspective of low-carbon land-use policy, Glaeser [18], Mathy [19], Xi [20] and Yang [21] and Zhou [22] considered the policy aspect of the subject and proposed a study of the relationship between land-use control regulations and carbon emissions at the policy level. On urban morphology and carbon emissions, Ishii, S. [23] analyzed the potential for greenhouse gas (GHG) emission reduction under different scenarios for three potential urban forms in Utsunomiya, Japan; Ou [24], Yi [25] and Teng [26] used time-series data to analyze the impact of carbon-emission evolution on the urban spatial morphology of four cities, including Beijing, Shanghai, Tianjin and Guangzhou, and 108 cities in China, including the Yangtze River Delta, and agglomeration, as research objects. Wang [27] studied the coupling relationship between the shape index of 35 urban built-up areas and carbon-emission intensity in China. Shi [28] analyzed the relationship between urban morphology and carbon dioxide emissions of 264 Chinese cities; Wang [29] discussed the empirical relationship between urban morphology and the CO₂ emissions of 104 prefecture-level cities in China; Ding [30] collected data from 295 cities in China and quantified the urban form compactness (UFC) from 2000 to 2015. The STIRPAT model is used to estimate the impact of UFC on CO₂ emissions, and the panel threshold regression model is used to estimate the threshold effect that limits the impact of compact urban morphology on carbon emissions. Li [31] used extreme gradient enhancement and a multiscale geographically weighted regression model to study the different importance, spatial multiscale change and spatial heterogeneity of the socioeconomic and urban form of the CO₂ emissions of 275 cities in China in 2005, 2012 and 2015. Urban compactness and carbon emissions: Ye [32], Li [33] and Yang [34] studied the interaction and driving factors between urban compactness and carbon-emission intensity of Xiamen and Changsha Zhuzhou Xiangtan urban agglomeration, respectively. Urban transport and carbon emissions: Research on the relationship between transport and carbon emissions can be divided into two schools: centralism and decentralization. The representative scholars of centralism, Kenworthy [35], Ewing [36] and Andong [37], attributed the causal chain of the growth of carbon emissions in the transport sector to “urban sprawl growth-private car use-carbon emissions” and believed that a compact, high-density urban form can reduce urban-transport carbon emissions. Pan [38], Waygood [39], Andrea [40], Sukarno [41], Zhao [42] and Wang [43] analyzed the relationship between urban scale, spatial structure, public transport participation, public transport facility saturation, land-use structure and carbon-emission intensity from the perspective of urban transport and travel. This view was questioned by decentralized scholars. Representative figures such as Alexander [44], William [45], and Christian [46], proposed that a compact development model may not necessarily reduce the carbon emissions of transport travel, and they carried out empirical research in some German cities.

Previous studies on the correlation between carbon emissions and urban spatial expansion are abundant, but they mostly start from a single factor reflecting the spatial characteristics and study the relationship between a single indicator and total carbon emissions or carbon-emission intensity. There is still insufficient evidence to establish the relationship between multiple factors. There are also many documents in the recent literature covering the relationship between land-use changes and carbon emissions. When implemented at the planning level, most of them propose carbon reduction constraint control policies in terms of population size and land-use scale, but there is no classified restrictive control on various types of urban construction land. In this paper, on the one hand, a nonequally spaced approach is applied, which serves to gradually optimize the Gray relational model and carry out the correlation analysis between carbon emissions and urban spatial expansion. On the other hand, the future trend of urban construction land and carbon emissions is predicted. Considering the target of the “double carbon” constraint in China, we propose a carbon emission constraint index, an urban land scale constraint index and an urban spatial expansion constraint index. This proposition provides a scientific guidance for future overall territorial planning and low-carbon policy.

2. Areas and Research Methods

2.1. Background of the Study Area

Located in the middle of China, Changsha is an important central city in the middle reaches of the Yangtze River, one of the pilot areas for the comprehensive reform of the “two-oriented society”, an important grain production base in China and an important node city in the middle reaches of the Yangtze River City cluster and the Yangtze River Economic Belt, playing an increasingly prominent role in the national development strategy. In recent years, Changsha’s vigorous economic development has brought about the rapid outwards expansion of the city. From 1949 to 2016, the urban population grew from 383,500 to 3,362,500, the GDP grew from CNY 2.87 million to CNY 851,013 million, and the built-up area increased from 6.7 km² to 476.34 km². During this period, the city’s master plan went through six revisions, in 1979, 1996, 2003, 2008, 2013 and 2016. As of 2021, Changsha’s GDP has reached CNY 1,214,250 million, jumping to 15th place in the country. The population, economic and built-up area indices are shown in

Table 1. Population, economic and built-up area indices.

Item		1949	1979	1996	2003	2008	2013	2016
Population (104 person)		38.35	99.28	160.38	196.38	237.06	299.25	336.25
GDP (CNY106)		2.87	21.38	415.77	1077.22	3300.98	7153.13	8510.13
The built-up area (H, unit: sq. km)		6.7	51.8	104.93	96.26	209.63	393.78	476.34
development land (unit: sq. km)	Residential land (R)	1.92	11.02	25.38	30.95	69.95	134.79	155.32
	Commercial service land (A and B)	1.26	13.33	26.97	37.02	43.98	73.4	66.58
	Industrial land (M)	1.52	12.5	21.9	27.1	30.6	66.5	94.56
	Road and transportation land (S)	0.8	2.97	9.59	20.58	31.76	37.67	52.87
	Public green space (G)	0.6	4.8	7.94	12.29	21.42	29.13	34.13

Note: the built-up area was meant to express the total area constructed. It contains both the developed land and the non-developed land. The developed land area contains eight categories: R, A, B, M, W, S, U and G, which refer to the GB50137-2011 standard [47] classification of urban land. W (warehouse land) and U (utility land) have little to do with urban carbon emissions. Therefore, classes W and U are not described in the table.

. Nevertheless, Changsha has always had the problems of a single energy structure and tight energy consumption, and its energy consumption and carbon emissions rank in the forefront of Hunan Province. According to the energy statistics yearbook data of Hunan Province, from 2005 to 2015, the total energy consumption of Changsha increased from 1851.785 t-ce to 9709.27 t-ce. Compared with the total energy consumption of Hunan Province increasing from 4455.207 t-ce to 15468.61 t-ce, the growth rate of Changsha is far higher than that of Hunan Province.

The National Development and Reform Commission announced nearly 100 low-carbon pilot cities three times in 2010, 2014 and 2017. Changsha was one of the third batches of pilot cities announced. However, Changsha’s booming economy has brought several drawbacks, such as the rapid outwards expansion of urban land and excessive energy consumption. Energy problems restrict the development of Changsha city to some extent. In the new historical development stage, effective ways to reduce energy consumption and reduce carbon emissions is an important issue to be solved. To achieve the two types of societal goals and accelerate the low-carbon city-pilot projects, it is necessary to optimize Changsha’s territorial planning and reach peak carbon emissions by 2030. The intensity of carbon emissions needs to be reduced by 60% to 65% from the 2005 level.

2.2. Gray Relational Model

Gray system theory was founded by Professor Deng, J.L. [48] in the 1980s. It has its roots in the three major theories of systems theory, information theory and cybernetics, which were developed in the 1940s. Gray system theory allows us to deeply analyze a given set of data in a “small sample” and “bad information” system to produce an accurate perception of the whole system and formulate predictions. Since the relationship between carbon emissions and the urban spatial expansion index in this study is a typical Gray system, the nonequally spaced stepwise optimization GM (1,1) model method is chosen [49]. This model accurately fits urban carbon emissions. It is an easy-to-model method that has a saturation curve. It is a relatively accurate model for urban carbon emissions prediction with good generalizability.

2.3. Urban Space Indicators

2.3.1. Land Expansion Intensity Index

The land expansion intensity index [50] reflects the change in land use and the expansion speed in the study area, which is monitored by the urbanization intensity index model. The calculation is shown in Formula (1):

$$I_{{}^H}=\frac{I_a-I_b}{H_i}\times \frac{1}{T}$$

(1)

where I is the land expansion intensity index; I_b and I_a are the land-use areas at the end and the beginning, respectively; H is the built-up land area; i is the area of each type of land, which refers to residential land (R), commercial service land (A and B), industrial land (M), road and transportation land (S) and public green space (G); T is the study period.

2.3.2. Urban Morphology Indices

The urban morphology analyses incorporate the Boyce and Clark shape index [51] and Moellering’s harmonic analysis method included in the double Fourier axis shape analysis [52]. This paper, inspired by the radius shape index algorithm, compares the shape of the urban boundary to a standard circle (Figure 1). The calculation is shown in Formula (2):

$$SBC={\displaystyle \sum _{i=1}^n\left|\left(\frac{R_i}{{\displaystyle \sum R_i}}\right)\times 100-\frac{100}{n}\right|}$$

(2)

where SBC is the urban morphology index; n is the number of radii with the same angle in the study figure, and R_i is the radius from the dominant point to the edge of the figure. The dominant point can be the centroid of the urban shape or the CBD center. The calculation results of the urban spatial shape index are shown in

Table 2. Results of urban land expansion intensity.

Year	IH	IR	IA&B	IM	IS	IG
1949–1979	22.44	15.8	16.33	13.2	9.04	23.33
1979–1996	6.03	7.67	6.02	4.42	13.11	3.85
1996–2003	7.55	3.14	5.32	3.39	16.37	7.83
2003–2008	6.14	25.2	3.76	2.58	10.86	14.86
2008–2013	17.57	18.54	6	23.46	3.72	7.2
2003–2016	6.99	5.08	5.49	14.07	4.6	5.72

.

2.3.3. Urban Compactness

Urban compactness adopts the quantitative model of urban spatial form developed by Thinh [53] based on Newton’s law of universal gravitation. The model calculates the construction land area in the grid according to the superposition of the GIS grid and land-use data layer (Figure 2), as shown in Formula (3):

$$U=\left[{\displaystyle \sum \frac{1}{C}\frac{Z_i{Z}_j}{D^2(i,j)}}\right]÷\left[\frac{1}{2}N(N-1)\right]$$

(3)

where U is the average gravity of the urban space, an expression of urban compactness; Z_i and Z_j are the urban construction land area within any grid i and grid j; D (i, j) is the geometric distance between grid i and grid j; C is a constant (100 m² is taken in the study to make the calculation result dimensionless); N is the total number of grids. The results of the urban compactness calculation are shown in

Table 3. Results of urban morphology indices, degree of urban compactness and shortest travel distance.

Content	1949	1979	1996	2003	2008	2013	2016
Urban morphology indices (SBC)	34.0	28.81	33.80	33.29	31.89	26.17	25.64
Urban compactness (U)	0.521	0.228	0.172	0.146	0.21	0.205	0.195
Shortest travel distance (Z)	607.12	1948.76	4740.335	15,463.21	29,804.45	53,102.61	71,254.58

.

2.3.4. Urban Traffic Accessibility

The calculation of urban traffic accessibility is based on distance metrics such as the time distance, economic distance or minimum cost distance. In this study, the minimum cost distance is used for the calculation as in Formula (4):

$${\mathrm{Z}}_i={\displaystyle \sum _{j=1}^n{T}_{ij}/(n-1)}$$

(4)

where Z_i is the accessibility of a node in the traffic network, which reflects the average value of the sum of the shortest distances from a node to all other nodes in the network. The smaller the value of the shortest spatial distance is, the better the accessibility of the node in the whole network; T_ij is the shortest spatial distance from node i to node j; n is the number of traffic network nodes.

3. Results

3.1. Urban Space Indicators Results

According to Formula (1), the expansion intensity of urban land in six stages is calculated, and the results are shown in Table 2. The expansion intensity of the built-up area remained stable, with large fluctuations between 2008 and 2013. This fluctuation is mainly due to the change in the intensity of industrial-land expansion caused by the construction of industrial parks. The expansion of construction land remained stable and fluctuated greatly from 2008 to 2013. This fluctuation is mainly due to the change in industrial-land expansion intensity caused by the construction of industrial parks and the development of a large amount of residential land. The expansion intensity of residential land is relatively the largest, especially between 2003 and 2008, as it is several times that of other types of land. The expansion intensities of commercial-service land and public green space are relatively stable.

According to Formula (4), the shortest travel distance is calculated. The calculation process is as follows: First, the traffic network “line” elements of Changsha city were loaded in Arc GIS for six periods: 1949, 1979, 1996, 2003, 2008, 2013 and 2016 (Figure 3). The number of network nodes in the six periods was 341, 554, 558, 1785, 3468, 4201 and 4988, respectively. Network Analyst is used to establish the transportation OD cost matrix, and the spatial presentation of network accessibility of each node is obtained by solving the matrix (Figure 4). The results of urban morphology indices, degree of urban compactness and shortest travel distance are shown in Table 3.

3.2. Measurement and Trend Prediction of Urban Carbon Emissions

3.2.1. Urban Carbon Emission Measurement

Urban carbon emissions are generally measured using indicators, such as total carbon emissions, carbon intensity, carbon emissions per capita and carbon emissions per unit of land. In some previous studies, Fong [54], Feng [55] and Liu [56] proposed measuring urban carbon emissions using system dynamics. The system dynamics model defines five corresponding su-models, namely, a residential sub-model (F_Res), a commercial services sub-model (F_Com), an industrial sub-model (F_Ind), a transportation sub-model (F_Tra) and a carbon-sequestration sub-model (F_G). Total carbon emissions are obtained as the difference between emissions and absorption. Based on this, Liu [57] optimized the model. The carbon emission of the city was calculated by taking Changsha city as an example. This paper analyzes the interactions between carbon emissions and several indicators, namely urban population, land expansion and economic growth, as well as the complex relationships covered by the results. The data of three indicators in the Statistical Yearbook in 1979, 1996, 2003, 2008, 2023 and 2016 were extracted to verify the SD model outputs. The comparison between the actual values and the model outputs is shown in Figure 5. The results show that the error between them is small, so the model is feasible for urban-carbon-emission assessment. The results calculated using the system dynamics are shown in

Table 4. Total carbon emissions, carbon-emission intensity, per capita emissions and emissions per unit of land.

Content	Units	1949	1979	1996	2003	2008	2013	2016
Total carbon emissions (TCE)	t-CO2/104	1.73	5.88	5.21	6.96	12.32	17.88	18.13
Per capita emissions (PCE)	104 t-CO2/per.	66.24	601.73	851.07	1392.75	2963.87	5404.60	6095.21
Emission growth rate (EG)	%	/	26.95	2.44	9.09	22.56	16.47	4.26
Residential sub-model (FRes)	104 t-CO2	6.42	25.32	50.95	132.37	208.77	347.48	410.2
Commercial-service sub-model (FCom)		5.02	52.24	65.89	240.39	471.61	784.93	1125.6
Industrial sub-model (FInd)		42.35	472.27	587.47	641.26	1813.72	2147.59	2333.6
Transportation sub-model (FTra)		12.52	51.96	146.81	378.79	469.81	2124.64	2225.8
Sequestration sub-model (FGa)		−695.95	−609.3	−592.44	−552.02	−427.69	−343.41	−231.54
Carbon-emission intensity (EI)	t-CO2/CNY104	23.08	28.14	2.05	1.29	0.9	0.76	0.72
Emission per unit land (EPL)	106 t-CO2/Km2	9.89	11.62	8.11	6.72	14.14	13.72	12.80

.

3.2.2. Prediction of Carbon Emission Trend

The average relative error calculated in MATLAB R2019a software is 1.4669%, and the expression Y = −670.18787 − 332125.83042X^1 + 504.71453X^2 − 0.25566X^3 + 4.31658E − 5X^4 of carbon emissions and years. The total carbon emissions in future years are predicted. The results are shown in

Table 5. Carbon emission prediction of Changsha.

Year	Carbon Emissions
	Base Scenario	COVID-19 Scenario
2017	6431.31	6824.57
2018	6903.27	7773.32
2019	7398.59	8393.766
2020	7917.81	9084.231
2021	8461.51	9571.036
2022	9030.24	10,070.21
2023	9624.56	10,581.52
2024	10,245.02	11,104.73
2025	10,892.20	11,550.77
2026	11,566.66	12,000.61
2027	12,268.95	12,453.83
2028	12,999.66	12,910.04
2029	13,759.34	13,375.11
2030	14,548.55	13,836.29

. The overall carbon emissions in Changsha City show an upwards trend. By 2030, there will be a large increase in urban carbon emissions.

3.3. Gray Correlation Analysis Results

3.3.1. Correlation between Carbon Emissions and Urban Land Area

The GM (1,1) model was run in MATLAB software, and correlation indices were calculated between the carbon-emission-increase rate and the urban-land expansion intensity on the one hand and between the carbon-emission intensity and the land-expansion intensity per unit of urban land area on the other hand, as shown in

Table 6. Relational grades of the carbon emissions index and urban land use area.

Relational Grades	Correlation with Land Expansion Intensity
	R	A&B	M	S	G
Total carbon emissions (TCE)	0.999	0.938	0.967	0.962	−0.984
Per capita emissions (PCE)	0.978	0.946	0.932	0.95	−0.961
Emission growth rate (EG)	0.788	0.549	0.168	−0.148	0.906
Carbon-emission intensity (EI)	−0.621	−0.844	−0.634	−0.723	−0.613
Emission per unit land (EPL)	0.678	0.554	0.538	0.634	0.667

. The correlations between both the total carbon emissions and the per capita carbon emissions on the one hand and the area of each type of land on the other hand are very high, all above 0.9. The correlation between industrial-land expansion, which is the largest source of carbon emissions, and the carbon emission growth rate is the lowest, only 0.168, reflecting the decoupling trend between industrial lands and urban carbon emissions in Changsha. The correlation between the increase rate of carbon emissions and the expansion intensity of road lands is weak (−0.148). Road lands themselves do not produce significant carbon emissions, but the increase in residential lands and commercial lands is caused by the increase in road lands. Indeed, the correlation of residential and commercial lands with the increase rate of carbon emissions is considerably strong (0.788 and 0.549, respectively).

There was a negative correlation between carbon-emission intensity and all types of land area. It is strongly correlated with the amount of commercial service land, followed by roads and public-square lands. The GDP output from the expansion of commercial-service land, as well as transportation and logistics lands, accounts for a larger proportion of the urban GDP growth. The correlation of carbon-emission intensity is low for residential lands and industrial lands (0.621 and 0.634, respectively). One can observe that the energy consumption per unit of GDP for urban activities carried out in residential areas is lower than that of industrial areas. Therefore, its effect on the intensity of urban carbon emissions is less significant. The correlation between carbon emissions per unit of land and the area of each type of land (H) is generally positive and is above 0.5. It has a negative correlation with the area of green space. The correlation between carbon emission per unit land use and residential land use is the highest, with a value of 0.678, followed by road and public-square land use and commercial-service land use, with 0.634 and 0.554, respectively, and the lowest correlation was with industrial land use, with a value of 0.538.

3.3.2. Carbon Emissions and the Correlation of Urban Spatial Extension

Similarly, the correlations between the five indicators of carbon emissions and the intensity of land expansion (I), urban morphology indices (SBC), urban compactness (U) and shortest travel distance (Z) were calculated. The results of the calculation are shown in

Table 7. Relational grades between carbon emission indices and the urban sprawl index.

Content	Urban Expansion Intensity(I)	Urban Morphology Indices (SBC)	Urban Compactness (U)	Shortest Travel Distance (Z)
TCE	/	−0.827	−0.392	0.992
PCE	/	−0.834	−0.482	0.963
EG	0.669	/	/	/
EI	/	0.278	0.858	−0.598
EPL	/	0.707	−0.094	0.666

. All five indicators of carbon emissions have some correlation with the shortest travel distance. The correlation between the shortest travel distance and total carbon emissions and per capita emissions is the highest, 0.992 and 0.963, respectively, which is close to 1. The correlation between shortest travel distance and carbon-emission intensity and carbon emission per unit of land is the second highest, −0.613 and 0.666, respectively. The carbon emission growth rate and urban land expansion intensity also have a high correlation, with an index of 0.669.

There was a negative correlation between total carbon emissions and carbon emissions per capita with the urban form index and urban compactness. It has a stronger correlation with the urban form index at −0.827 and −0.834, respectively. Its correlation with urban compactness is −0.392 and 0.482, respectively, which is weaker than the former. The correlation between carbon-emission intensity and urban form compactness is stronger, with a correlation coefficient of 0.858. The correlation between carbon emissions per unit of land and the urban form index is stronger, with a correlation coefficient of 0.707. The smaller the urban form index and the more compact the city, the lower the carbon-emission intensity.

3.3.3. The CO₂ Constraint Value

Changsha should accelerate the reduction of its carbon emissions and strive to meet the national requirements by 2030. Estimated from 2005, a reduction of 60% to 65% in carbon-emission intensity should be achieved by 2030. According to the six-stage revision timeline of the city’s master plan, we take 2003 as the initial year. According to the 65% requirement for controlled reduction, the indicators of construction land scale are controlled within the corresponding upper limits. The constraint values for the carbon-emission intensity and the per capita carbon emission were determined to be 0.45 and 8.73, respectively. The new urban masterplan (now the municipal spatial masterplan) predicts the population and the urban area of Changsha’s city center to be 10 million people and 1000 km², respectively. The GDP will reach CNY 200 billion. The per capita construction land will be maintained at 100 km² per million people. The carbon-emission intensity will be controlled at approximately 0.45. By 2030, the total carbon emissions should be reduced by 40% from the projected value of 14548.55. In other words, it should be no more than 87,291,300 tons, and the carbon emission constraint value per unit of land is 9.12 (as in

Table 8. The CO₂ constraint value of Changsha city in 2030.

Content	2003 (Base Year)	2016 (Current Situation)	2030 (Target Year)	2030 (COVID-19 Scenario)
GDP (CNY104)	1077.22	8510.13	≥20,000	≥19,000
Population (ten thousand people)	196.38	336.25	≥889.61	≥830.44
H (Km2)	160.41	476.34	≤889.61	≤830.44
Total carbon emissions (104 t-CO2)	1392.75	6095.21	≤8729.13	≤8301.77
Carbon emissions per capita (t-CO2/per)	6.96	18.13	≤8.73	≤8.3
Carbon-emission intensity (t-CO2/CNY104)	1.29	0.72	≤0.45	≤0.48
Carbon emission per unit land (t-CO2/hm)	8.68	12.8	≤9.12	≤9.33

).

Using the G (1,1) model established in the previous paper, with the total carbon emissions as the binding variable, the inverse projection is used to control the various construction-land indicators. The predicted results for 2030 are tested for correlation with carbon-emission intensity (EI). The maximum areas for residence, public-service industrial facilities, road and transportation and green space are estimated to be 143.88 km, 150.17 km, 135.83 km and 61.33 km, respectively, by 2030, while the percentage of residential and industrial land will decrease. By 2030, the percentage of public service, road and transportation land is expected to increase. The specific indicators are shown in

Table 9. The land constraint value of Changsha city in 2030.

Content	Built-Up Area	R	A&B	M	S	G
2016 (current situation)	476.34	155.32	66.58	94.56	52.87	34.13
Percentage of land %	100	32.61%	13.98%	19.85%	11.10%	7.17%
The relevance of carbon emissions	0.990	0.999	0.938	0.967	0.962	−0.984
2030 (target year)	889.61	231.3	143.88	150.17	135.83	61.33
Percentage of land %	100	26.0%	16.17%	16.88%	15.27%	6.89%
EI correlation test results	−0.873	−0.902	−0.855	−0.817	−0.849	−0.904

. Similarly, the carbon emission growth rate, carbon emissions per capita, carbon-emission intensity and total carbon emissions, which have the strongest correlations, are used as the binding variables. The constraint indices of urban land expansion intensity, urban morphology indices, urban compactness and shortest travel distance are calculated. The results are shown in

Table 10. The setting value of the urban morphology indices of Changsha city in 2030.

Year	Urban Expansion Intensity (I)	Urban Morphology Indices (SBC)	Urban Compactness (U)	Shortest Travel Distance (Z)
2016 (current situation)	6.99	25.64	0.195	71,254.58
2030 (target year)	6.19	32.04	0.236	96,086.76

.

4. Discussion and Conclusions

4.1. Discussion

The nonequidistant stepwise optimization GM (1,1) model is used to predict the future carbon emissions of cities and calculate the correlation between urban carbon emissions and the urban expansion index. The key findings are as follows:

(1)

The correlation of the two indicators, the total carbon emissions and the per capita carbon emissions, to the area of each type of land use is extremely high, both above 0.9. It is also further confirmed that the growth of the urban population and land use is the main source of urban carbon emissions. Road traffic land itself does not produce significant carbon emissions, but the increase in residential land and commercial land caused by the improvement in road and transportation facilities is the main reason for the increase in carbon emissions.

(2)

Three spatial morphology indicators, namely, the shortest travel distance, urban compactness and morphology indices, are highly correlated with three emission indicators: carbon emissions, carbon-emission intensity and carbon emissions per unit of land use. Among them, the most correlated indicator is carbon-emission intensity and urban compactness, with a correlation coefficient of 0.858.

(3)

Changsha city has constraint target values for total carbon emissions, carbon-emission intensity, carbon emissions per capita and carbon emissions per unit of land until 2030. They are 87,291,300 t-CO₂, 0.45 t-CO₂/CNY10⁴, 8.83 t-CO₂/10⁴ people and 9.12 million t-CO₂/km₂, respectively. The next revision of the Changsha Territorial Masterplan should control the urban construction land within 889.61 square kilometers. Residential, public-service, industrial and road lands should be controlled at approximately 231.3 km², 143.88 km², 150.17 km² and 135.83 km², respectively. The land expansion intensity, urban morphology index, urban compactness and shortest travel distance target values are 6.19, 32.04, 0.236 and 96086.76 km, respectively.

(4)

Considering the impact of COVID-19 on China’s GDP, the overall urban economy is in a downwards trend. Changsha’s GDP maintained a growth trend of 12.8% before 2019. Influenced by COVID-2019, the GDP growth rate in 2020, 2021 and 2022 decreased to 7.1%, 7.5% and 4.8%, respectively. Forecast for 2030, the GDP will decline to CNY 1900 billion. The economic downturn is likely to slow urban sprawl, so binding targets for carbon emissions will be adjusted accordingly.

4.2. Conclusions

(1)

Based on the gray correlation analysis of multiple factors, the calculation of the constraint control value of urban space expansion can provide policy guidance for the next round of spatial optimization of the city- and county-level land-space master plan and can also develop more precise constraint indicators to achieve the national 2030 carbon peak and carbon-neutral policy goals.

(2)

A coordinated development between urban land use and public transportation, particularly the promotion of rail transit to guide land development and reduce traffic trips, is the first strategy that needs to be considered for the reduction of carbon emissions.

(3)

The rapid urbanization and expansion of urban land is a necessity that continuously increases total carbon emissions and per capita carbon emissions. The goal of carbon peaking and carbon neutrality is not simply to limit urban expansion, nor is it simply to delineate urban spatial growth boundaries and control the total scale of construction lands. It should be more about the optimization of the urban space from the inside and change the urban development model from “incremental development” to “stock development” to reduce the disorderly expansion in favor of compact spatial development.

(4)

Although the gray system theoretical model introduced in this paper has high simulation accuracy, different types of gray models need to be selected for different types of data, and the quantitative relationship expression is more complex, so MATLAB software is needed to assist in analysis and research. Therefore, the correlation model of urban spatial expansion and carbon emissions is worth further deepening to select a more appropriate quantitative model to fit.