A Novel Method for the Estimation of Higher Heating Value of Municipal Solid Wastes

Abstract

The measurement of the higher heating value (HHV) of municipal solid wastes (MSWs) plays a key role in the disposal process, especially via thermochemical approaches. An optimized multi-variate grey model (OBGM (1, N)) is introduced to forecast the MSWs’ HHV to high accuracy with sparse data. A total of 15 cities and MSW from the respective city were considered to develop and verify the multi-variant models. Results show that the most accurate model was POBGM (1, 5) of which the least error measured 5.41% MAPE (mean absolute percentage error). Ash, being a major component in MSW, is the most important factor affecting HHV, followed by volatiles, fixed carbon and water contents. Most data can be included by using the prediction interval (PI) method with 95% confidence intervals. In addition, the estimations indicated that the MAPE from estimating the HHV for various MSW samples, collected from various cities, were in the range of 3.06–34.50%, depending on the MSW sample.

1. Introduction

Municipal solid wastes (MSWs) are amalgamations of different solid waste streams produced as a result of domestic, commercial and institutional activities [1]. Owing to rapid economic growth, changing lifestyles and rapid urbanization, China’s MSWs production witnessed an exponential growth in the last decade. The annual generation of MSWs in Chinese urban areas reached ~0.23 billion tons in 2020 [2] and is expected to reach more than ~0.25 billion tons in 2022. The traditional methods for MSWs treatment include landfilling, anaerobic digestion and composting [3]. Landfilling was once commonly adopted due to its technological maturity and feasibility. However, in recent years, due to the unavailability of land and other associated environmental issues, landfilling has not been considered as a sustainable solution for the disposal of solid wastes [4]. While anaerobic digestion and composting, both being considered as technologically advanced and environmentally friendly techniques to recover energy and other valuable resources from MSWs, there are drawbacks, such as applicability only to biodegradable fractions of MSWs, long processing time and requirement of post-treatment for residues, which further hinder their adoptability [5].

Thermochemical treatments, such as incineration, pyrolysis and gasification, offer superiority over the biochemical conversion platforms, which include complete elimination of pathogens, very short processing time and complete conversion of carbon to fuels and other valuable chemicals [6]. Incineration is considered as the most reliable and economic route of waste to energy conversion in developing countries for reasons such as significant reduction in the volume of wastes to be disposed, processing time and better process control. For instance, the volume and mass of waste can be reduced by 80–90% and 70–80%, respectively [7]. The drawbacks, such as high capital cost and release of dioxins can cause detrimental effects on human and animal health, resulting in the development of emerging technologies, such as pyrolysis and gasification, to meet the various energy needs and carbon reduction strategy [8,9]. It is to be noted that the stable operation of biomass thermal conversion relies mostly on the characteristics of biomass, namely, the heating value, ash and water content and elemental composition [10]. ‘Higher’ heating value (HHV) or ‘lower’ heating value (LHV) is the common property of solid waste and fuel and can be experimentally determined by using an adiabatic oxygen bomb calorimeter. The availability and accessibility of the equipment may limit its usage, and, under such circumstances, researchers can use the proximate and ultimate analysis data to predict the HHV of fuel material by employing more practical models [11,12]. The HHV of feedstocks is the key parameter that determines the stability and efficiency of the thermochemical reactions [13].

The correlation between HHV and proximate/ultimate analysis is not new and dates back to late 1800s, with the first correlation proposed by Dulong based on ultimate analysis of coal [14]. Developing models for estimating the HHVs of MSW has attracted much attention owing to the benefits that are offered to handlers during solid waste management handling and disposal. Regression methods and back propagation neural network methods are the most common techniques used to predict the HHV of MSWs with simple composition based on known mechanisms or a mass of data [12,15]. As the thermal decomposition mechanism of the MSWs remains unclear, the HHV cannot be accurately predicted by the aforementioned models.

Grey model (GM), introduced by Deng in 1982, can be used to accurately predict the values of HHV under ambiguous conditions such as unknown mechanisms and scant data [16]. GM (1, 1) is a univariate model with high accuracy, which is most widely used to predict production under complicated situations, such as wind power generation [17] and MSW production forecasting [18]. The GM (1, 1) is used to characterize an unfamiliar system by employing a first-order differential equation. MSWs are characterized with complicated physical and chemical composition, which cannot be described by univariable model [19]. Therefore, to predict the HHV of the MSWs accurately, grey model with multivariable is more suitable [20]. Zhang et al. [21], Intharathirat et al. [22], and Duman et al. [23] studied the prediction of MSW production using the multi-variable GM (1, N) model.

To our knowledge, little or no studies have reported on the prediction of HHV for MSWs. To this end, proximate and elemental analysis could be used [12,15], and the newly developed GM (1, N) (N ≥ 2) model might offer a better method for HHV prediction as more variables are involved.

This study attempts to predict the HHV value of MSW collected from different locations, using correlation models (including 2 traditional regression methods and, GM (1, N) with different N) based on the proximate or ultimate analysis data. In addition, the practicality of multivariate grey prediction model GM (1, N), which employ grey rational analysis to estimate the HHV of MSW is studied in this work.

2. Data and Basic Methods

The flow path of the modelling program is shown in Figure 1 and is carried out as follows: input of the key factors and HHVs, determination of grey factors with Grey Relational Analysis (GRA) method, finding the best prediction model among the alternative 12 methods, and discussing the model influencing factors.

2.1. Data Sources

Shanxi Province, with a total area of 156,579 km² has a population over 37.18 million as per 2018 census. The generation of MSWs in Shanxi Province has increased rapidly to 5 million tons in 2019 from 3.54 million tons since 2008. The characteristic data related to MSW from 15 cities of Shanxi Province were collected and averaged in time (3–6 points) within three months. The MSW in Shanxi Province is characterized by high moisture and ash content, and a relatively low content of volatile and fixed carbon, as shown in

Table 1. The HHVs and proximate analysis of MSWs of 15 cities in Shanxi Province.

Num.	City	HHV	Proximate Analysis
		kcal·kg−1	Mad/%	Vad/%	Aad/%	FCad/%
1	Taiyuan	764	34.41	14.86	48.04	2.71
2	Datong	1258	23.04	20.08	50.74	6.14
3	Yangquan	980.5	39.46	12.55	41.84	6.17
4	Changzhi	1353	15.91	12.26	61.04	10.8
5	Jincheng	1393	22.5	12.64	54.3	10.59
6	Shuozhou	868	35.38	20.99	40.08	3.55
7	Xinzhou	1004	25.36	14.28	56.48	3.89
8	Jinzhong	1212	23.52	13.52	54.94	8.01
9	Linfen	1046	25.13	14.92	54.91	5.05
10	Yuncheng	919	18.62	8.7	65.72	6.96
11	Lvlinag	947.5	24.01	11.43	59.59	4.98
12	Yuanping	968	21.41	13.33	60.56	4.72
13	Dingxiang	947	23.68	12.8	58.83	4.7
14	Fenyang	1259	14.79	11.43	64.25	9.53
15	Xiaoyi	1267	25.3	14.95	52.09	7.67

HHV—Higher heating value, air dried basis, kcal/kg; M_ad—Water content, air dried basis, wt.%; V_ad—Volatile content, air dried basis, wt.%; A_ad—Ash content, air dried basis, wt.%; FC_ad—Fixed carbon, air dried basis, wt.%.

. The contents of moisture, volatile, ash and fixed carbon ranged from 14.79–39.46%, 8.70–20.99%, 40.08–65.72 and 2.71–10.80%, respectively.

The contents of C and O were higher than other elements (

Table 2. The ultimate analysis of MSWs of 15 cities.

City Num.	Ultimate Analysis (wt.%, Air Dried Basis)
	C	H	O	N	S	Cl
1	8.61	1.09	7.00	0.29	0.59	0.16
2	15.34	1.30	8.89	0.44	0.25	0.20
3	10.61	0.91	6.39	0.31	0.51	0.18
4	13.28	1.31	7.52	0.45	0.51	0.13
5	15.26	1.21	5.91	0.39	0.45	0.18
6	14.47	1.16	8.28	0.34	0.29	0.14
7	9.80	1.16	5.98	0.37	0.87	0.17
8	12.08	1.17	7.03	0.36	0.83	0.19
9	10.86	1.12	6.97	0.31	0.71	0.18
10	8.28	1.23	5.06	0.40	0.69	0.17
11	8.08	1.12	6.60	0.36	0.25	0.18
12	9.60	1.25	6.11	0.39	0.70	0.16
13	9.15	1.20	6.36	0.41	0.39	0.13
14	12.01	1.27	6.46	0.35	0.90	0.16
15	12.95	1.23	6.62	0.39	1.43	0.17

). The HHVs for all the samples were much lower than coal, and ranged between 764–1267 kcal·kg⁻¹ [24]. Research on the HHV analysis and predictions prior to thermal disposal were critically important to avoid damage to equipment and ensure the operation under stable conditions [25,26]. In this work, the proximate and ultimate analysis results were used to develop the models for HHV prediction.

2.2. Traditional Forcasting Method

The performance of the established models is evaluated by the mean absolute percentage error (MAPE, %) as following [22].

$$MAPE(\%)=\left({\displaystyle \sum _{k=1}^m\frac{\left|A_k-P_k\right|}{A_k\times m}}\right)\times 100$$

(1)

MAPE can be divided into four categories as follows: Excellent (<10%), Good (10–20%), Reasonable (20–50%) and Incorrect (>50%) [27].

In this work, two traditional, regression prediction models based on proximate and ultimate analysis are introduced and compared to the proposed model, which is discussed in detail in Section 4.2.

Jigisha Parikh et al. [15] introduced a method with a MAPE of 3.74% to calculate the HHV (MJ·kg⁻¹) of solid feedstock, and the correlation is given below:

$$HHV=0.3636\times FC+0.1559\times VM-0.0078\times ASH$$

(2)

where FC, VM, ASH denote fixed carbon, volatile content and ash content, respectively, in wt.% on a dry basis.

Boumanchar et al. [12] established a multiple regression and programming formalism method for HHV calculation based on ultimate analysis. The equation can be expressed as follows:

$$\begin{array}{c}HHV=2.775+H+0.004027\times C+0.004027\times C^2+\frac{0.05706}{H-12.97}\\ +\frac{0.02323}{H-6.661}+\frac{0.009398}{H-5.961}+\frac{12.97-H}{H^3-5.922\times C}\end{array}$$

(3)

2.3. Multivairant Grey Model, GM (1, N)

In this section, the multivariable grey model GM (1, N) is introduced based on the works of Zeng and Li [20].

GM (1, N) is a first-order grey model with N variables, including N-1 independent variables and one dependent variable. Let $X_1{}^{(0)}$ be the dependent variable that has m initial sequences, as presented in Equation (4).

$$X_1{}^{(0)}=(x_1{}^{(0)}(1),x_1{}^{(0)}(2),\dots ,x_1{}^{(0)}(m))$$

(4)

N-1 independent variables denote $X_i{{}^{(}}^{0)}(i=2,3,\dots ,N)$ with m initial sequences are of strong correlation with $X_1{}^{(0)}$, as presented in Equation (5).

$$X_i{{}^{(}}^{0)}=(x_i{}^{(0)}(1),x_i{}^{(0)}(2),\dots ,x_i{}^{(0)}(m))$$

(5)

Then, the first order accumulated generation operation (1-AGO) of the initial series, which is $X_j{{}^{(}}^{1)}$, can be denoted as following Equation (6).

$$X_j{}^{(1)}=(x_j{}^{(0)}(1),x_j{}^{(0)}(2),\dots ,x_j{}^{(0)}(m))(j=1,2,\dots ,N)$$

(6)

where $X_j{}^{\left(1\right)}$ can be calculated by $X_j{}^{\left(1\right)}\left(k\right)={\displaystyle \sum }_{g=1}^k{x}_i{}^{\left(0\right)}\left(g\right)$, $k=1,2\cdots ,m.$

$Z_1{}^{\left(1\right)}$ is the mean sequence that is calculated by consecutive neighbour of the above $X_1{}^{\left(1\right)}$,

$$Z_1{}^{(1)}=(z_1{}^{(1)}(2),z_1{}^{(1)}(3),\dots ,z_1{}^{(1)}(m))$$

(7)

where $z_1{}^{\left(1\right)}\left(k\right)=0.5\times \left(x_1{}^{\left(1\right)}\left(k\right)+x_1{}^{\left(1\right)}\left(k-1\right)\right)$, $k=2,3\cdots ,m$.

Then,

$$x_1{}^{(0)}(k)+a{z}_1{}^{(1)}(k)={\displaystyle \sum _{i=2}^N{b}_i{x}_i{}^{(1)}(k)}$$

(8)

is the GM (1, N) model.

The parameters column can be presented as $\widehat{a}={[a,b_1,b_2,b_3,\dots ,b_N]}^T$, which satisfies the least square estimate equation given below.

$$P={(B^{T}B)}^{-1}{B}^{T}Y$$

(9)

where,

$$B=\left[\begin{array}{cccc}-{\mathrm{z}}_1^{\left(1\right)}\left(2\right)& x_2^{\left(1\right)}\left(2\right)& \cdots & x_N^{\left(1\right)}\left(2\right)\\ -{\mathrm{z}}_1^{\left(1\right)}\left(3\right)& x_2^{\left(1\right)}\left(3\right)& \cdots & x_N^{\left(1\right)}\left(3\right)\\ ⋮& ⋮& ⋮& ⋮\\ -{\mathrm{z}}_1^{\left(1\right)}\left(m\right)& x_2^{\left(1\right)}\left(m\right)& \cdots & x_N^{\left(1\right)}\left(m\right)\end{array}\right]$$

(10)

$$Y=\left[\begin{array}{c}{x}_1^{\left(0\right)}\left(2\right)\\ x_1^{\left(0\right)}\left(3\right)\\ ⋮\\ x_1^{\left(0\right)}\left(m\right)\end{array}\right]$$

(11)

Then, the desired grey parameters obtained in the last step can be substituted into Equation (12) to obtain the (k + 1) th 1-AGO result as following:

$${\widehat{x}}_1{}^{(1)}(k+1)=[x_1{}^{(0)}-\frac{1}{a}{\displaystyle \sum _{i=2}^N{b}_{{}^i}{x}_{{}^i}{}^{(1)}(k+1)]}{e}^{-ak}+\frac{1}{a}{\displaystyle \sum _{i=2}^N{b}_i{x}_i{}^{(1)}(k+1)}$$

(12)

Finally, the (k + 1) th predictive value, also ${\widehat{x}}_1{}^{\left(0\right)}\left(k+1\right)$, can be obtained through the inverse accumulated generating operation using Equation (13)

$${\widehat{x}}_1{}^{(0)}(k+1)=a{\widehat{x}}_1{}^{(1)}(k+1)={\widehat{x}}_1{}^{(1)}(k+1)-{\widehat{x}}_1{}^{(1)}(k)$$

(13)

The aforementioned grey model has some drawbacks in practice, including mechanism, parameter and structure defects [20]. The optimization of current grey model is discussed in detail in the subsequent sections.

3. Optimized Grey Forecasting Model

3.1. OGM (1, N)

In this section, GM (1, N) optimized to obtain POGM (1, N) based on proximate analysis and EOGM (1, N) based on ultimate analysis was obtained.

The optimized grey model, OGM (1, N) is presented as follows:

$$x_1{}^{\left(0\right)}\left(k\right)+a{z}_1{}^{\left(1\right)}\left(k\right)={\displaystyle \sum }_{i=2}^N{b}_i{x}_i{}^{\left(1\right)}\left(k\right)+h_1\times \left(k-1\right)+h_2$$

(14)

where k = 2, 3, …, m. $h_1\times \left(k-1\right)$ and $h_2$ are linear correlation term and grey action quantity, respectively.

Then, the parameter series $\widehat{p}={[b_1,b_2,b_3,\dots ,b_N,a,h_1,h_2]}^T$ satisfies:

$$\widehat{p}={(B^{T}B)}^{-1}{B}^{T}Y$$

(15)

where,

$$B=\left[\begin{array}{ccccccc}-x_2^{\left(1\right)}\left(2\right)& x_3^{\left(1\right)}\left(2\right)& \cdots & x_N^{\left(1\right)}\left(2\right)& {-\mathrm{z}}_1^{\left(1\right)}\left(2\right)& 1& 1\\ -x_2^{\left(1\right)}\left(3\right)& x_3^{\left(1\right)}\left(3\right)& \cdots & x_N^{\left(1\right)}\left(3\right)& {-\mathrm{z}}_1^{\left(1\right)}\left(3\right)& 2& 1\\ ⋮& ⋮& ⋮& ⋮& ⋮& ⋮& ⋮\\ -x_2^{\left(1\right)}\left(m\right)& x_3^{\left(1\right)}\left(m\right)& \cdots & x_N^{\left(1\right)}\left(m\right)& {-\mathrm{z}}_1^{\left(1\right)}\left(m\right)& m-1& 1\end{array}\right]$$

(16)

$$Y=\left[\begin{array}{c}{x}_1^{\left(0\right)}\left(2\right)\\ x_1^{\left(0\right)}\left(3\right)\\ ⋮\\ x_1^{\left(0\right)}\left(m\right)\end{array}\right]$$

(17)

Then, the required grey parameters obtained in the last step can be substituted in Equation (18), and the result can be presented as,

$${\widehat{x}}_1{}^{(1)}(k)={\displaystyle \sum _{t=1}^{k-1}[{\mu }_1{\displaystyle \sum _{i=2}^N{\mu }_2{}^{t-1}{b}_i{x}_i{}^{(1)}(k-t+1)}]+{\mu }_2{}^{k-1}}{\widehat{x}}_1{}^{(1)}(1)+{\displaystyle \sum _{j=0}^{k-2}{\mu }_2{}^j[(k-j){\mu }_3+{\mu }_4]}$$

(18)

where, $k=2,3,\dots ,m$, ${\mu }_1=\frac{1}{1+0.5a}$, ${\mu }_2=\frac{1-0.5a}{1+0.5a}$, ${\mu }_3=\frac{h_1}{1+0.5a}$, ${\mu }_4=\frac{h_2-h_1}{1+0.5a}$.

Finally, the inverse accumulated generating operation is introduced to calculate the (k + 1)th predictive value.

$${\widehat{x}}_1{}^{\left(0\right)}\left(k\right)={\widehat{x}}_1{}^{\left(1\right)}\left(k\right)-{\widehat{x}}_1{}^{\left(1\right)}\left(k-1\right)k=2,3,\dots ,m$$

(19)

3.2. OBGM (1, N)

In the grey system, the background-value coefficient, $\xi (0<\xi <1)$, strangely equates to 0.5. In this study, the background-value coefficient will be further studied after the best grey model has been determined.

With parameter series $\widehat{p}={[b_2,b_3,\dots ,b_N,a,c,d]}^T$, the OGM (1, N) is presented as Equation (20).

$$x_i{}^{(0)}(k)+a\xi x_1{}^{(1)}(k-1)={\displaystyle \sum _{i=2}^N{b}_i{x}_i{}^{(1)}(k)+kc+d},k=2,3,\dots ,m$$

(20)

However, many works showed that dynamic $\xi$ can reduce the mean absolute prediction error (MAPE %). Therefore, the background-value coefficient ($\xi$) of the above OGM (1, N) is optimized using the particle swarm optimization algorithm (PSO) method [28]. With the optimized $\xi$, OBGM (1, N) is denoted. The introduction of this optimization method including the detailed MATLAB code is given in the work of Zeng and Li [20].

The final time response function and restored expression of the OBGM (1, N) model can be expressed as follows:

$$\begin{array}{c}{\widehat{x}}_1{}^{(1)}(k)={\displaystyle \sum _{u=1}^{k-1}[{\upsilon }_1{\displaystyle \sum _{i=2}^N{\upsilon }_2{}^{u-1}{b}_i{x}_i{}^{(1)}(k-u+1)}]+{\displaystyle \sum _{\upsilon =0}^{k-2}{\upsilon }_2{}^v}[(k-v){\upsilon }_3+{\upsilon }_4]+{\upsilon }_2{}^{k-1}}{\widehat{x}}_1{}^{(1)}(1)\\ k=2,3,\dots ,m\end{array}$$

(21)

and

$$\begin{array}{c}{\widehat{x}}_1{}^{(0)}(k)={\upsilon }_1{\displaystyle \sum _{i=2}^k\left[{\displaystyle \sum _{i=2}^N{({\upsilon }_2-1)}^{\left[\frac{d-2}{d}\right]}{\upsilon }_2{}^{\left[\frac{d-2}{d}\right]\left[\frac{d-3}{d}\right](d-3)}{b}_i{x}_i{}^{(1)}(k-d+2)}\right]}\\ +{\displaystyle \sum _{v=0}^{k-3}\left[{\upsilon }_2{}^{k-2}{(2{\upsilon }_3+{\upsilon }_4+1)}^{\left[\frac{v}{k-3}\right]}+{\upsilon }_2{}^v{\upsilon }_3\right]}\end{array}$$

(22)

where $k=2,3,\dots ,m$, ${\upsilon }_1=\frac{1}{1+a\xi }$, ${\upsilon }_2=1-\frac{a}{1+a\xi }$, ${\upsilon }_3=\frac{c}{1+a\xi }$, ${\upsilon }_4=\frac{d}{1+a\xi }$.

The best multivariable grey model with optimized background-value coefficient will be denoted as POBGM (1, N) or EOBGM (1, N).

3.3. Grey Relational Analysis (GRA)

The method of grey relational analysis (GRA) was used to recognize the relational grade of selected factors in the aforementioned models and carried out on MATLAB software. Thus, the relationship can be determined between the reference series and compared series, which were denoted as x₀ = (x₀(1), x₀(2), …, x₀(k)) and x_i = (x_i(1), x_i(2), …, x_i(k)) (i = 1, 2,…, n), respectively. GRA was used to rank the selected factors, of which the one with the highest GRA grade (0~1.0) was the most important factor affecting the HHVs of MSWs. The GRA grades can be obtained using the equations listed in the work of Intharathirat et al. [22].

4. Results and Discussions

The relational grades of selected factors were firstly introduced. To identify the best prediction model, POGM (1, N) and EOGM (1, N) were studied with different N values and compared against the two traditional regression prediction methods mentioned in Section 2.2. Moreover, the coefficients ($\xi$) of the aforementioned best model were discussed. The results of using the best model to predict the HHVs of MSW is presented in the last section.

4.1. Identification of the Facotors

Two kinds of multivariable grey models are introduced in this study on the basis of the proximate and ultimate data of MSWs. In the GRA process, firstly, a series of HHVs and selected factors should be normalized to reduce the inaccuracy. In this work, the raw data are normalized by the mean value.

The grey relational grade (

Table 3. Grey relational grade of selected factors.

Proximate Analysis
Water Content		Volatile Content	Ash	Fixed Carbon
0.5759		0.6361	0.7247	0.6157
Elemental Analysis
C	H	O	N	S	Cl
0.8049	0.7502	0.7218	0.7814	0.5292	0.7617

) shows that, in the proximate analysis related model, ash was the most influencing factor (0.7247), followed by volatile content (0.6361), fixed carbon (0.6157) and water content (0.5759). Soponpongpipat et al., [29] reported a prediction equation without the parameters of ash and mentioned that the ash content did not have a significant impact on the HHV of torrefied biomass. However, Qian et al. [11] reported that ash content had a significant impact on the HHV of biochars. Therefore, in view of the high ash content in the selected MSW feedstocks, the HHV of the MSW samples is expected to be greatly influenced by the ash content.

In the elemental analysis related model, C is the most influencing factor (0.8049), followed by N (0.7814), Cl (0.7617), H (0.7502), O (0.7218) and S (0.5292). The factors with values > 0.5 have a remarkable influence on HHVs [23]. Boumanchar et al. [12] reported that the change in HHV of MSW followed two trends: an increasing trend can be witnessed with the increase of carbon and hydrogen while a decreasing trend can be noticed with the increase in nitrogen and sulphur rates.

4.2. Comparative Analysis

The HHV series, including 10 data sets for establishing the model and five datasets for verification, were used to develop the alternative models, which include two multiple regression methods [12,15], and 10 multivariable models (including POGM (1, N) (N = 2, 3, 4, 5) which are based on the proximate analysis of MSWs, and EOGM (1, N) (N = 2,3,4,5,6,7) as based on the elemental analysis). The results of the 12 alternative models are presented in

Table 4. Comparison of the selected models.

Models		Variable	Modelling MAPE * (%)	Accuracy	Prediction MAPE (%)
Traditional Models
1	Proximate analysis		12.634	←G G→	15.45
2	Elemental analysis		11.99	←G E→	7.45
Multivariate Grey Prediction Models
POBGM (1, N)
	(1,2)	Ash	61.04	←I I→	1555.00
	(1,3)	Ash, Vad	40.40	←R I→	887.35
	(1,4)	Ash, Vad, FC	5.42	←E E→	2.89
	(1,5)	Ash, Vad, FC, MC	5.41	←E E→	3.06
EOBGM (1, N)
	(1,2)	C	16.39	←G I→	79.16
	(1,3)	C N	11.12	←G R→	25.16
	(1,4)	C N Cl	9.27	←E G→	15.98
	(1,5)	C N Cl H	7.72	←E G→	14.64
	(1,6)	C N Cl H O	4.90	←E R→	38.33
	(1,7)	C N Cl H O S	0.0026	←E I→	123,173.23

* MAPE—Mean absolute percentage error, %. E—Excellent. G—Good. R—Reasonable. I—Incorrect. The best fit model is in bold.

.

The results show that traditional models still bear good accuracy in forecasting HHVs of MSWs. Generally, POGM (1, N) exhibits better performance than EOGM (1, N) because the proximate analysis was obtained using a greater testing quantity than elemental analysis. It can be observed that multivariable grey models with larger N using proximate analysis data not only displays high modeling accuracy but also shows excellent performance in prediction. Similar analysis was reported in studies reported on multivariable grey models with larger N values for other areas [30,31,32]. Among the alternative models, POGM (1, 4) and POGM (1, 5) had similar performances and displayed excellent accuracy in forecasting HHV with the least error of 2.89% MAPE. Water content can be seen as a vital parameter which greatly affected the HHVs. Therefore, POGM (1, 5) that included water content was chosen to be used for further studies. The model parameters of multivariable grey models are presented in

Table 5. The model parameters of multivariable grey models.

Type		Parameters
	N	a	c	d	b2	b3	b4	b5	b6	b7
POBGM	2	−0.54	−1532.21	1951.21	16.74
	3	−0.52	1348.74	1928.10	15.14	−4.6621
	4	2.23	152.52	−153.49	8.91	56.8333	152.37
	5	2.23	145,846.98	153.49	1468.46	1517.3970	1613.02	1460.57
	2	−0.47	−331.90	1659.18	−18.24
EOBGM	3	−0.11	−1210.68	1414.10	−27.48	3816.6772
	4	0.16	−3087.30	1095.48	8.86	5543.4456	6680.63
	5	0.70	−2960.29	578.08	38.62	8572.7496	9620.30	−1270.18
	6	−0.05	−7208.66	267.51	−42.33	13306.5705	16865.80	−1863.00	319.64
	7	−1.45	−2395.25	3576.35	23.90	5903.7691	−6479.40	−1631.27	15.00	1907.87

(the values in bold font are the parameters of the best model).

4.3. HHV Forecasting

The background-value coefficient is usually noted as 0.5 in grey model for the simplification of modeling [33,34]. In this section, POBGM (1, 5) models with different background-value coefficients were established and their performances was compared in

Table 6. POBGM (1,5) model results with different background-value coefficients (0.05–0.95 with 0.1 interval).

X1(0)	(a): ξ = 0.05			(b): ξ = 0.15			(c): ξ = 0.25			(d): ξ=0.35
	x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)
764.00.	764.00	0.00	0.00	764.00	0.00	0.00	764.00	0.00	0.00	764.00	0.00	0.00
1258.00	1377.97	119.97	9.54	1235.77	−22.23	1.77	1139.48	−118.52	9.42	3877.55	2619.55	208.23
980.50	983.65	3.15	0.32	959.05	−21.45	2.19	927.54	−52.96	5.40	1915.93	935.43	95.40
1353.00	1499.66	146.66	10.84	1414.45	61.45	4.54	1363.20	10.20	0.75	2737.34	1384.34	102.32
1393.00	1333.11	−59.89	4.30	1276.75	−116.25	8.35	1246.21	−146.79	10.54	2593.84	1200.84	86.21
868.00	1026.27	158.27	18.23	972.35	104.35	12.02	944.78	76.78	8.85	2080.16	1212.16	139.65
1004.00	988.13	−15.87	1.58	933.73	−70.27	7.00	874.37	−129.63	12.91	1982.58	978.58	97.47
1212.00	1298.84	86.84	7.17	1230.78	18.78	1.55	1184.37	−27.63	2.28	2457.40	1245.40	102.76
1046.00	1018.73	−27.27	2.61	968.94	−77.06	7.37	926.16	−119.85	11.46	2069.77	1023.77	97.88
919.00	1097.05	178.05	19.37	1038.42	119.42	12.99	971.21	52.21	5.68	2159.22	1240.22	134.95
			8.22			6.42			7.48			118.32
(e): ξ = 0.45			(f): ξ = 0.5			(i): ξ = 0.55			(j): ξ = 0.65
x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)
764.00.	0.00	0.00	764.00	0.00	0.00	764.00	0.00	0.00	764.00	0.00	0.00
1674.55	416.55	33.11	1267.45	9.45	0.75	1438.32	180.32	14.33	1893.00	635.00	50.48
1131.13	150.63	15.36	969.79	−10.71	1.09	1030.43	49.93	5.09	1523.77	543.27	55.41
1595.18	242.18	17.90	1383.89	30.89	2.28	1463.41	110.41	8.16	1810.90	457.90	33.84
1512.60	119.60	8.59	1330.65	−62.35	4.48	1424.15	31.15	2.24	1855.74	462.74	33.22
1166.78	298.78	34.42	998.30	130.30	15.01	1080.36	212.36	24.47	1526.14	658.14	75.82
1119.62	115.62	11.52	921.37	−82.63	8.23	1012.70	8.70	0.87	1415.38	411.38	40.98
1401.58	189.58	15.64	1196.81	−15.19	1.25	1264.86	52.86	4.36	1585.71	373.71	30.83
1189.52	143.52	13.72	1005.70	−40.30	3.85	1105.63	59.63	5.70	1505.64	459.64	43.94
1234.31	315.31	34.31	1026.91	107.91	11.74	1128.04	209.04	22.75	1542.45	623.45	67.84
		20.51			5.41			9.77			48.04
(k): ξ = 0.75			(l): ξ = 0.85			(m): ξ = 0.95			(n): ξ= 0.500667307066868
x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)	x1(0)(k)	△(k)	re(k)
764.00	0.00	0.00	764.00	0.00	0.00	764.00	0.00	0.00	764.00	0.00	0.00
1977.68	719.68	57.21	1071.99	−186.01	14.79	944.89	−313.11	24.89	1258.21	0.21	0.02
3822.85	2842.35	289.89	1154.71	174.21	17.77	1113.45	132.95	13.56	980.95	0.45	0.05
9909.00	8556.00	632.37	1376.25	23.25	1.72	1281.88	−71.12	5.26	1371.56	18.56	1.37
28,801.36	27,408.36	1967.58	1025.45	−367.55	26.39	1152.02	−240.98	17.30	1326.27	−66.73	4.79
87,582.77	86,714.77	9990.18	1284.83	416.83	48.02	966.79	98.79	11.38	997.92	129.92	14.97
272,022.56	271,018.56	26,993.88	431.28	−572.72	57.04	818.97	−185.03	18.43	915.38	−88.62	8.83
849,023.53	847,811.53	69,951.45	2457.82	1245.82	102.79	1376.55	164.55	13.58	1191.19	−20.81	1.72
2,654,524.81	2,653,478.81	253,678.66	−1512.57	−2558.57	244.61	565.81	−480.19	45.91	999.86	−46.14	4.41
8,304,352.45	8,303,433.45	903,529.21	5555.04	4636.04	504.47	1262.34	343.34	37.36	1019.57	100.57	10.94
		140,787.83			113.06			20.85			5.23

. The mathematical definitions in Table 6 are as follows: X₁(0)—the actual HHV, x₁⁽⁰⁾(k)—the predicted HHV, △(k)—the absolute error, re(k)—the relative error. It can be concluded that grey prediction models with background-value coefficient of 0.5 is rather accurate. Similar results were reported in literature [17]. For the following study, ξ = 0.5 and the model parameters are shown in Section 4.2.

The accuracies of both modeling and prediction steps using different sampling scales were determined in this part. As shown in Figure 2 and

Table 7. Comparison of POBGM (1, 5) model prediction accuracy when using a different sample scale.

Sample Scale	Actual Data	Predicted Value	Absolute Deviation	Relative Deviation
	kcal·kg−1	kcal·kg−1	kcal·kg−1	%
Eight samples (Modeling MAPE % = 12.0145)
9	1046	95.2940	−950.7060	90.8897
10	919	1340.7320	421.7320	45.8903
11	947.5	1262.7320	315.2320	33.2699
12	968	1231.4700	263.4700	27.2180
13	947	1186.2230	239.2230	25.2611
14	1259	1427.3070	168.3070	13.3683
15	1267	1196.3640	−70.6360	5.5751
		Prediction MAPE %		34.4960
Nine samples (Modeling MAPE % = 2.6756)
10	919	1522.7460	603.7460	65.6960
11	947.5	1157.5170	210.0170	22.1654
12	968	1145.4470	177.4470	18.3313
13	947	1118.0910	171.0910	18.0666
14	1259	1412.0670	153.0670	12.1578
15	1267	1233.4060	33.5940	2.6515
		Prediction MAPE %		23.1781
Ten samples (Modeling MAPE % = 5.4103)
11	947.5	1000.6920	53.1920	5.6139
12	968	973.9140	5.9140	0.6110
13	947	949.9670	2.9670	0.3133
14	1259	1283.2380	24.2380	1.9252
15	1267	1180.3450	86.6550	6.8394
		Prediction MAPE %		3.0606
Eleven samples (Modeling MAPE % = 6.4045)
12	968	1565.0840	597.0840	61.6822
13	947	992.2070	45.2070	4.7737
14	1259	1333.7110	74.7110	5.9342
15	1267	1228.9080	38.0920	3.0065
		Prediction MAPE %		18.8491
Twelve samples (Modeling MAPE % = 5.6208)
13	947	683.1850	263.8150	38.6155
14	1259	1246.9930	12.0070	0.9629
15	1267	1165.4240	101.5760	8.7158
		Prediction MAPE %		16.0980

, it can be concluded that higher modeling accuracy can be obtained by using more volume of sample data.

In the predicting step, with a MAPE value of 3.0606%, using 10 samples for modeling can be considered as the most accurate scale. It is of great significance to obtain the best grey model using both modeling and prediction accuracy when establishing the grey model. A single number or point is often predicted which may not provide any necessary guidance in the actual application regarding their likely accuracy [22]. Prediction interval (PI) (95% confidence intervals) was used to forecast the upper and lower limit of HHVs of MSWs. The lower limit should meet the needs of stable operation during thermal disposal, while upper limit should fulfil the safety requirements [35]. As shown in

Table 8. Prediction interval of MSW’s HHVs at 95% PI (kcal·kg⁻¹).

No.	Actual Data	POBGM (1, 5) Model
		Prediction Data	Lower Limit	Upper Limit
11	947.50	1000.69	For Testing
12	968.00	973.91	963.80	984.03
13	947.00	949.97	927.71	972.22
14	1259.00	1283.24	1261.99	1304.48
15	1267.00	1180.35	1090.31	1270.38

and Figure 3, the POBGM (1, 5) provides good modeling accuracy of HHVs in this study. The results show that four out of five forecasted data were in the interval of 95% PI.

Data Number 11, of which the HHV is 947.5 kcal kg⁻¹, is outside the prediction curve and the prediction interval. This might be due to the measurement error or model deviation [36]. In the practical application of the established grey model, most of the data can be precisely forecasted with PI method. However, if the data bear a larger deviation, actual tests should be practiced. Normally, a specific mechanism is often proposed in advance. Callejón-Ferre et al. [37] used 20 models based on proximate or elemental analysis to predict the HHVs of crop residue, including univariate and multivariate models. These models belong to data fitting category and are adopted from other’s work. The composition and structure of organic material, especially MSW mentioned in this study, are quite complicated and cannot be simply described by fitting method. Large-scale data are often used to train the model to obtain a high prediction accuracy, e.g., neural network and other multivariate nonlinear regression methods [38]. The proposed grey model in this study can firstly filter the effect factors by using grey rational analysis, e.g., only factors with GRA larger than 0.5 will be considered in the model [17]. The larger the GRA degree, the more important the factor. Secondly, the grey model uses normalization and 1-AGO methods to reduce or even eliminate the effect of assuming mechanism [39]. More importantly, this novel method does not require large-scale data to train the model and also exhibits high accuracy even with scarce data.

5. Conclusions

The exponential growth of MSW generation in China requires highly efficient disposal on a large scale and rely significantly on thermal conversion platforms in particular. The accurate prediction of HHVs of MSWs is of great significance in the planning and management of these thermal and thermochemical operations. In this study, OBGM (1, N), on the basis of proximate or ultimate data, was developed to predict the HHV of MSWs. Thus, 12 models including two traditional regression methods, POGM (1, 2–5) and EOGM (1, 2–7) were studied. For multivariable models, GRA method is used to identify and quantify the influence of proximate and ultimate analysis. The selected best model is further studied by changing the data scale, optimizing the background-value coefficient and using prediction interval techniques.

Among the selected 12 models, POBGM (1, 5) with a background-value coefficient of 0.5 was observed as the best grey model and displayed a high accuracy, both in modeling and prediction step. According to the data scale study, less data led to higher modeling accuracy, but it cannot be concluded that less samples lead to high prediction accuracy. POBGM (1, 5) can ensure high prediction accuracy based on sparse data. With 95% PI, most of the data can be precisely forecasted. If the data bear a larger deviation, an actual test should be implemented. OBGM (1, N) models provided an effective modelling pathway for the prediction of HHV of MSWs under data scarcity conditions.