Local Grey Predictor Based on Cubic Polynomial Realization for Market Clearing Price Prediction

Abstract

With the development of restructured power markets, the profit-making competitive business environment has emerged. With the help of different advanced technologies, generating companies are taking decisions regarding trading electricity with imperfect information about marketing operating conditions. The forecasting of the market clearing price (MCP) is a potential issue in these markets. Early information on the MCP can be a proven beneficial tool for accumulating profit. In this work, a local grey prediction model based on a cubic polynomial function is presented to estimate the MCP with the help of historical data. The mathematical framework of this grey model was established and evaluated for different market conditions and databases. The comparison between traditional grey models and some advanced grey models reveals that the proposed model yields accurate results.

1. Introduction

A competitive business environment enables consumer-centric policies in energy markets, while framing these policies at the generating company end, profit-making prepositions and decision-making algorithms play a vital role. In a partially known system, the fundamentals of grey mathematics are easily applicable. Equilibrium price information is a crucial parameter to determine as it depends upon several conditions, such as no. of market players, operating conditions, no. of bidding blocks and behavior of rivals. A mechanism that provides early anticipation of the market clearing price can be a potential tool for accumulating profit in energy markets. This also helps in submitting effective bids with low risks in losing revenue.

Scant information on rival behavior is a big problem in achieving a high profit accumulation and is a major hurdle when anticipating the response of the market. Hence, the energy market behaves like a grey system. A system with a partial known and partial unknown system can be treated as a grey system [1]. In an electricity market, suppliers of energy, which are also known as generating companies, offer their energy-selling propositions in terms of the energy packet and the cost of that packet. After accumulating all of the seller propositions, the system demand is calculated and the purchaser offers are also accumulated. The intersection of the demand and supply becomes the market clearing equilibrium. The clearing price related to this equilibrium point becomes the market clearing price (MCP).

Several prediction approaches have been applied to estimate the MCP through machine learning models in the literature. A neural-network-based approach along with confidence interval estimation has been applied in reference [2].

An application of a neural network combined with the Kalman filter was explored in the work due to the fact that MCP is a non stationary signal; in addition, it is the end result of the settlement of the market operations. A multi-support-vector-machine (SVM)-based approach has been proposed by Xing Yan et al. in [3]. In the work, authors divided the MCP signals into three zones as per the data characteristics, and then the SVM for each class was augmented. A fast neural network learning-based method has been employed in approach [4] for confidence interval determination and MCP prediction. From all of these reported approaches, it is evident that MCP is a significant denominator and can play a crucial role for a generating company.

Often, the neural-network-based methods, along with some statistical-analysis-based methods such as auto regressive integrating moving average (ARIMA)-based methods, require a large amount of data for generating the forecast. On the other hand, grey prediction methods take very few data and analyze the system with the processing local information. Dr. Deng suggested that grey models are developed for systems that have uncertainties and where the information is very scant [1].

Recently, local grey prediction techniques have been widely applied in energy demand forecasting [5], wind power forecasting [6] and emission forecasting [7]. In the literature, several examples are available that deal with the application of grey models for generating predictions. The work pertaining to the change in accumulation from integer order to fractional order was first introduced in reference [8]. Since then, the research on the development of new grey models can be divided into three major categories:

• Change in accumulation operator [8,9];
• Transformation of the original series if there are negative terms in the series [10];
• Change in whitening equation of the grey system [11,12].

Wu et al. [9] employed a weighted accumulation generation pattern for the better utilization of local information for accurate prediction. A recent approach based on changing the grey internal parameter through a pattern search algorithm was adopted in [13]. Considering these facts, it is empirical that the prediction of the time series through the grey model majorly depends upon the values of internal parameters [14]. A Bernoulli model has been proposed in reference and a power term has been introduced in the grey whitening differential equation [11]. A quadratic-polynomial-based realization for predicting the COVID cases has been conducted in research [12].

Recently, the integration of modern optimizers for optimizing the whitening equation has been presented in reference [15]. In the work, the polynomial power terms were made fractional in order to achieve a higher accuracy in the prediction. However, the problem with such integration is that meta optimizers give different results every time; hence, a simple-structured canonical whitening equation is not feasible. An interesting approach, however, shows the robustness of the modern optimizer in terms of time complexity in reference [16]. In addition, a rich review of wind forecasting methods has been demonstrated in reference [17]. An optimized model for forecasting nuclear energy consumption in China and America has been discussed in reference [18]. In addition, the MCP prediction can be an important part of decision making. Some approaches on decision making have been discussed in references [19,20,21].

From this discussion and these observations, it is observed that, by adding the terms, the response can be controlled. Hence, in this paper, a cubic realization of a grey model is proposed. A detailed analysis and development of the model is explained. The evaluation of the model was conducted on the MCP data of Indian Energy Exchange Limited data. On the basis of this review, the following objectives are framed for this work.

• To develop a mathematical model based on a cubic realization of a grey differential equation and present the solution of the proposed CGM.
• To present the data analysis of the MCP signals on the basis of different attributes.
• To develop a forecasting model for predicting the MCPs for different data samples from different grey models. In addition, to present a comparative analysis of the performance of these models based on mean absolute percentage error (MAPE) analysis and other relevant error analyses.

The remaining part of the paper is organized as follows. In Section 2, the development process of the cubic model is explained. In Section 3, the data analysis of the MCP signal is presented and, in Section 4, simulation results and an analysis of the proposed model is presented. In the last section, the major contributions of the paper are summarized.

2. Development of New Proposed Grey Model Comprising Cubic Polynomial (CGM)

A time series prediction analysis through a grey model is executed in many approaches. The main idea of grey models lies in accumulating the information of the past and passing it on to the next steps. Due to this, the approaches where these models are applied are intrinsically monotonic increasing functions. When signals are nonlinear and contain noise, the prediction becomes difficult. MCP signals are volatile and highly dependent on the market sentiments.

Definition 1.

Let us assume an initial sequence that indicates that MCP is considered as

$$Y^{\left(0\right)}=y^{\left(0\right)}\left(1\right),y^{\left(0\right)}\left(2\right),\cdots y^{\left(0\right)}\left(r\right),$$

(1)

It is essential that all terms of the time series considered for the grey prediction should be positive; hence, we have ignored the concept of negative cost here.

The one-time accumulated generating term is given by

$$Y^{\left(1\right)}=y^{\left(1\right)}\left(1\right),y^{\left(1\right)}\left(2\right),\cdots y^{\left(1\right)}\left(r\right),$$

(2)

$$y^1\left(r\right)=\sum _{i=1}^r{y}^{\left(0\right)}\left(i\right),r=1,\cdots n,$$

(3)

Definition 2.

The inverse process of finding the accumulated generation sequence can be given as the sequence mean of $Y^{\left(0\right)}$, which can be given as

$$Z^{\left(1\right)}=z^{\left(1\right)}\left(1\right),z^{\left(1\right)}\left(2\right),\cdots z^{\left(1\right)}\left(r\right),$$

(4)

where

$$Z^{\left(1\right)}\left(k\right)=\frac{y^{\left(1\right)}\left(k\right)+y^{\left(1\right)}(k-1)}{2}\mathit{for}k=2,\cdots r,$$

(5)

The Grey Model with Cubic Polynomial The first-order linear differential equation known as the whitening equation of the proposed model CGM is given by

$$\frac{d{y}^{\left(1\right)}\left(t\right)}{dt}+a{y}^{\left(1\right)}\left(t\right)=\alpha t^3+\beta t^2+\gamma t+\delta$$

(6)

where a is the development coefficients and the right hand side term is known as the grey action quantity of the grey model.

It can be easily observed that, when $\alpha =0$, the CGM model reduced to the QGM model. When $\alpha =0,\beta =0$, it reduced to the NGM(1,1,k,c) model. Upon putting $\alpha =0,\beta =0,\gamma =0$, one can find the classical GM(1,1) model.

Theorem 1.

If $y^{\left(0\right)}\left(r\right)$ is a term of the non-negative sequence and $z^{\left(1\right)}\left(r\right)$ is the $r^{th}$ term of mean sequence $Z^{\left(1\right)}\left(r\right)$ defined by (5), then

$$\begin{array}{c}\hfill y^{\left(0\right)}\left(r\right)+a{z}^{\left(1\right)}\left(r\right)=\alpha \left(r^3-\frac{3}{2}{r}^2+r-\frac{1}{4}\right)+\\ \hfill \beta \left(r^2+r-\frac{1}{3}\right)+\gamma \left(r-\frac{1}{2}\right)+\delta \end{array}$$

(7)

Proof. 

Integrating the whitening equation defined in (6) both sides with regard to between the interval $[r-1,r]$,

$$\begin{array}{c}\hfill {\int }_{r-1}^r\left[\frac{d{y}^{\left(1\right)}\left(t\right)}{dt}+a{y}^{\left(1\right)}\left(t\right)\right]dt=\alpha {\int }_{r-1}^r{t}^{3}dt+\\ \hfill \beta {\int }_{r-1}^r{t}^{2}dt+\gamma {\int }_{r-1}^{r}tdt+\delta {\int }_{r-1}^{r}dt\end{array}$$

(8)

which gives us

$$\begin{array}{c}\hfill y^{\left(0\right)}\left(r\right)+a{\int }_{r-1}^r{y}^{\left(1\right)}\left(t\right)dt=\alpha \frac{\left(r^4-{(r-1)}^4\right)}{4}\\ \hfill +\beta \frac{\left(r^3-{(r-1)}^3\right)}{3}+\gamma \frac{\left(r^2-{(r-1)}^2\right)}{3}+\delta \end{array}$$

(9)

Using trapezoidal formula ${\int }_{r-1}^r{y}^{\left(1\right)}\left(t\right)dt=\frac{y^{\left(1\right)}\left(r\right)+y^{\left(1\right)}(r-1)}{2}=z^{\left(1\right)}\left(r\right)$ and after some simplification, we have the RHS of Theorem 1. □

Theorem 2.

If the initial sequence and its inverse accumulated sequence are given by Definition 1 and Definition 2 and the mean sequence is represented by (5), then the values of parameters $a,\alpha ,\beta ,\gamma$ and δ in terms of matrix A and X are given by

$${\left(a,\alpha ,\beta ,\gamma ,\delta \right)}^T={\left(A^{T}A\right)}^{-1}{A}^{T}X$$

(10)

where

$$A=\left[\begin{array}{ccccc}-z^1\left(2\right)& \frac{15}{4}& \frac{7}{3}& \frac{3}{2}& 1\\ -z^{1}3& \frac{77}{4}& \frac{19}{3}& \frac{5}{2}& 1\\ ⋮& ⋮& ⋮& ⋮\\ -z^1\left(r\right)& \left(r^3-\frac{3}{2}{r}^2+r-\frac{1}{4}\right)& \left(r^2-r+\frac{1}{3}\right)& \left(r-\frac{1}{2}\right)& 1\end{array}\right]$$

(11)

and

$$X=\left[\begin{array}{c}{y}^0\left(2\right)\\ y^0\left(3\right)\\ ⋮\\ y^0\left(r\right)\end{array}\right]$$

(12)

Proof. 

Using the concept of mathematical induction on taking $r=2,3,\cdots n$ in Theorem 1, we find

$$\left\{\begin{array}{c}{y}^0\left(2\right)=-a{z}^{\left(2\right)}+\frac{15}{4}\alpha +\frac{7}{3}\beta +\frac{3}{2}\gamma t+\delta \hfill \\ y^0\left(3\right)=-a{z}^{\left(3\right)}+\frac{77}{4}\alpha +\frac{19}{3}\beta +\frac{5}{2}\gamma t+\delta \hfill \\ ⋮\hfill \\ y^0\left(n\right)=-a{z}^{\left(n\right)}+\left(n^3-\frac{3}{2}{n}^2+n-\frac{1}{4}\right)\alpha +\hfill \\ \left(n^2-n+\frac{1}{3}\right)\beta +\left(n-\frac{1}{2}\right)\gamma t+\delta \hfill \end{array}\right.$$

(13)

On expressing the above system of linear equations in matrix form, we obtain

$$\begin{array}{c}\hfill \left[\begin{array}{ccccc}-z^1\left(2\right)& \frac{15}{4}& \frac{7}{3}& \frac{3}{2}& 1\\ -z^{1}3& \frac{77}{4}& \frac{19}{3}& \frac{5}{2}& 1\\ ⋮& ⋮& ⋮& ⋮\\ -z^1\left(r\right)& \left(r^3-\frac{3}{2}{r}^2+r-\frac{1}{4}\right)& \left(r^2-r+\frac{1}{3}\right)& \left(r-\frac{1}{2}\right)& 1\end{array}\right]\\ \\ \hfill \times \left[\begin{array}{c}a\\ \alpha \\ \beta \\ \gamma \\ \delta \end{array}\right]=\left[\begin{array}{c}{y}^0\left(2\right)\\ y^0\left(3\right)\\ ⋮\\ y^0\left(n\right)\end{array}\right]\end{array}$$

(14)

□

Theorem 3.

The time response sequence of the proposed model CGM is given by

$$\begin{array}{c}\hfill {\hat{y}}^{\left(1\right)}\left(r\right)=e^{-a\left(r-1\right)}\left(y^{\left(1\right)}\left(0\right)+P\right)+\frac{\alpha }{a}{r}^3-\left(3\frac{\alpha }{a^2}-\frac{\beta }{a}\right)r^2\\ \hfill +\left(6\frac{\alpha }{a^3}-2\frac{\beta }{a^2}+\frac{\gamma }{a}\right)r-Q\end{array}$$

(15)

and its restored value is

$$\begin{array}{c}\hfill {\hat{y}}^{\left(0\right)}\left(r\right)=e^{-a\left(r-2\right)}\left(y^{\left(1\right)}\left(0\right)+P\right)+\frac{\alpha }{a}\left(3r^2-3r-1\right)\\ \hfill -\left(\frac{3\alpha }{a}-\frac{\beta }{a}\right)\left(2r-1\right)+Q\end{array}$$

(16)

where

$$P=-\frac{\alpha }{a}+\frac{3\alpha }{a^2}-\frac{6\alpha }{a^3}+\frac{6\alpha }{a^4}+\frac{\beta }{a}-\frac{2\beta }{a^2}+\frac{2\beta }{a^3}+\frac{\gamma }{a}-\frac{\gamma }{a^2}+\frac{\delta }{a}$$

(17)

$$Q=\frac{6\alpha }{a^3}-\frac{2\beta }{a^2}+\frac{\gamma }{a}-\frac{\delta }{a}$$

(18)

Proof. 

The general solution of the whitening equation can be obtained easily from the theory of first-order ordinary differential equation and is given by

$$y^1\left(t\right)=y^1\left(0\right)e^{-{\int }_1^{t}adu}+{\int }_1^t\left(\alpha s^3+\beta s^2+\gamma s+\delta \right)e^{{\int }_t^{s}adu}ds$$

(19)

Now, using the formula of simple integration, we can easily obtain

$$\begin{array}{c}\hfill {\hat{y}}^{\left(1\right)}\left(t\right)=e^{-a\left(r-1\right)}\left(y^{\left(1\right)}\left(0\right)+P\right)+\frac{\alpha }{a}{t}^3-\left(3\frac{\alpha }{a^2}-\frac{\beta }{a}\right)t^2\\ \hfill +\left(6\frac{\alpha }{a^3}-2\frac{\beta }{a^2}+\frac{\gamma }{a}\right)t-Q\end{array}$$

(20)

We can easily obtain its restore value by using the relation ${\hat{y}}^{\left(0\right)}\left(r\right)=y^{\left(0\right)}\left(r\right)-y^{\left(1\right)}(r-1)$. □

Discussion

In conventional grey models, the action quantity is a constant number that is a homogeneous exponent model, so when the data are non-homogeneous in nature, these models are not very useful. On the other hand, adding time-varying terms in the grey action quantity modifies the model and makes it compatible for non-homogeneous data. Further, it is observed from Equation (20) that the term with negative signs is a correction term, which is making the time response as per the change in the variable. From the solution, it is observed that NGM and GM models do not have such correction terms in the solution; hence, in some cases, these models fail to predict nonlinear and non-stationary time series models.

3. Data Analysis

For evaluating the performance of the proposed cubic model, the data of MCP of Indian Energy Exchange Limited (IEX) were deployed [22]. IEX is a platform for automated energy trading for the physical delivery of the electricity. The whole IEX is subdivided into five sub regions, including, east, west, north–south and north–east regions. The main statistical attributes, including the mean, standard deviation, kurtosis and skewness of the obtained data, along with the sample size, are exhibited in the

Table 1. Data sample details and statistical attributes.

Cases	Data Indicator	Time Duration	Specifcation	Skewness	Kurtosis	Mean	SD
Case-1	D-1	1-06-2021 to 30-06-2021	30 samples	0.6069	3.0648	3061.8813	604.4347
	D-2	1-07-2021 to 30-07-2021	30 samples	0.7097	3.1813	2970.2250	780.1299
	D-3	1-01-2019 to 01-02-2019	32 samples	0.3974	3.2540	3316.9563	317.0100
	D-4	1-03-2019 to 01-04-2020	32 samples	−0.3324	1.9490	2448.4938	278.9202
	D-5	1-05-2021 to 01-06-2021	32 samples	−0.0267	2.1373	2838.6059	270.8984
	D-6	1-03-2021 to 01-04-2021	32 samples	0.6606	2.8131	4051.9459	615.3318
Case-2	D-7	1-01-2020	24 samples	−0.0716	1.9723	2978.2167	735.0516
	D-8	1-03-2020	24 samples	0.7272	3.1536	2636.5188	307.6723
	D-9	24-10-2020	24 samples	2.5992	9.8850	2552.3067	480.6052
	D-10	24-10-2019	24 samples	2.2243	7.2330	2685.7975	349.8738
	D-11	14-08-2020	24samples	1.3077	3.4041	3671.4146	1354.6820
	D-12	15-07-2020	24 samples	1.1664	4.6392	2587.3438	362.8971
Case-3	D-13	7-08-2021	96 samples	1.4135	3.5530	3887.9233	1596.6424
	D-14	9-09-2021	96 samples	1.5307	4.1393	3778.4663	1627.1399
	D-15	14-08-2019	96 samples	1.3947	3.8070	3671.4132	1359.4739
	D-16	20-08-2020	96 samples	1.1191	4.9793	2271.0991	410.5623
	D-17	10-09-2019	96 samples	0.7981	2.8497	2546.1361	759.2563
	D-18	15-05-2019	96 samples	0.5152	2.8808	3423.6879	589.5000
Case-4	D-19	Monthly average of 2015	12 samples	2.0346	6.6163	2820.7017	302.7745
	D-20	Monthly average of 2016	12 samples	1.1875	4.0746	2400.7325	206.3804
	D-21	Monthly average of 2017	12 samples	0.9295	2.4451	3018.3925	585.8632
	D-22	Monthly average of 2018	12 samples	1.3335	3.9808	3930.0533	815.6535
	D-23	Monthly average of 2019	12 samples	−0.4774	1.7014	3114.3142	241.0883
	D-24	Monthly average of 2020	12 samples	0.0988	1.5406	2619.4675	196.1199

. Kurtosis of the data sample is a measure of the data regarding how the tail of distribution is different from the normally distributed data. Skewness determines the symmetry of the distribution. Negative values of the kurtosis indicate that the distribution is flat and has a thin tail. In addition, the negative values of skewness indicate that the data are skewed left. These distinct values are identified in the boldface. For the evaluation and the data obtained and sampled in different cases, the details of the cases are as follows:

• Case 1. Thirty samples of a time duration of a month, considering the average MCP of whole day (monthly values of MCP). The data distribution of the case is showcased in Figure 1.
• Case 2. Twenty-four samples of a trading day (hourly values of MCP). The data distribution of the case is showcased in Figure 2.
• Case 3. Ninety-six samples of a day with the values of MCP measured for every 15 min (per 15 min values of MCP for a day). The data distribution of the case is showcased in Figure 3.
• Case 4. Twelve samples of the mean of monthly MCP for a particular year. The data distribution of the case is showcased in Figure 4.

By observing the data analysis, it is concluded that the data employed for the testing possess diverse properties and vary in a wide range. Hence, for the validation, these 24 data sets are reliable. For each case, we took six different data samples. MCP analysis is also represented through pictorial representation in Figure 1. By observing the plot, it can be seen that the data are nonlinear in nature; hence, they pose a challenge to the conventional grey models.

4. Results and Discussion

In this section, we present a comparative analysis of different conventional grey models along with the proposed cubic-polynomial-based grey model. For validating the effectiveness of the approach, we took the data of the Indian Electricity market (MCP) for different time periods as described in Table 1. Some recently published grey models, along with the conventional models, wre chosen to validate the efficacy of the proposed approach. These models are the quadratic grey model (QGM) [12], discrete grey model (DGM) [23,24] and conventional GM(1,1) (CGM) [5].

The evaluation of the performance of these forecasters was carried out on the basis of the evaluation through two major error indices. These indices are the average percentage error (APE) and mean absolute percentage error (MAPE).

Both of these indices are defined as:

$$APE\left(m\right)=\frac{\left|{\widehat{y}}^{\left(0\right)}\left(m\right)-y^{\left(0\right)}\left(m\right)\right|}{y^{\left(0\right)}\left(m\right)}\times 100$$

(21)

$$MAPE=\frac{1}{n}\sum _n^{k=1}\left|\frac{{\widehat{y}}^{\left(0\right)}\left(m\right)-y^{\left(0\right)}\left(m\right)}{y^{\left(0\right)}\left(m\right)}\right|\times 100.$$

(22)

The results of data sample D-1 are depicted in

Table 2. Forecasted results of D-1.

Original	GM	QGM	DGM	CGM
3198.1	3198.1	3198.1	3198.1	3198.1
3236.8	2696.623	3631.085	2708.123	3650.643
3331.72	2720.255	3440.045	2731.1	3447.441
3589.98	2744.094	3269.987	2754.272	3266.313
3781.45	2768.143	3120.119	2777.641	3106.671
2834.2	2792.401	2989.677	2801.208	2967.909
2899.43	2816.873	2877.929	2824.975	2849.411
2709.61	2841.559	2784.169	2848.943	2750.544
2694.05	2866.461	2707.716	2873.115	2670.659
2652.28	2891.581	2647.917	2897.492	2609.095
3151.76	2916.922	2604.143	2922.076	2565.173
2249.54	2942.484	2575.788	2946.868	2538.199
2039.33	2968.271	2562.271	2971.871	2527.463
2171.33	2994.284	2563.03	2997.086	2532.237
2473.47	3020.524	2577.527	3022.515	2551.777
2601.42	3046.995	2605.243	3048.159	2585.321
2661.71	3073.697	2645.678	3074.022	2632.09
2906.26	3100.634	2698.352	3100.103	2691.285
2772.75	3127.806	2762.802	3126.406	2762.089
2482.96	3155.217	2838.585	3152.932	2843.668
2729.16	3182.868	2925.27	3179.683	2935.164
3299.16	3210.761	3022.448	3206.661	3035.702
4009.4	3238.899	3129.722	3233.868	3144.386
4119.55	3267.283	3246.709	3261.306	3260.299
3186.65	3295.916	3373.044	3288.977	3382.501
3465.28	3324.8	3508.373	3316.882	3510.031
2799.51	3353.938	3652.357	3345.024	3641.905
3475.56	3383.33	3804.668	3373.405	3777.118
3688.21	3412.98	3964.992	3402.027	3914.638
4645.81	3442.89	4133.027	3430.891	4053.41

. It is observed that the values of MCP for the month of June 2021 predicted by the proposed model are aligned with the original values. In addition, the calculated MAPE is optimal for the proposed CGM. We also observed that higher values of APE for these data have been observed than for the NGM model. The possible reasons for the failure of the NGM have been discussed in reference [13] at length.

The MAPE calculated for all data samples is exhibited in

Table 3. MAPE values for all forecasted data samples.

	GM	QGM [12]	DGM [24]	CGM	NGM [23]
D-1	15.18395	9.270227	15.18601	9.194595	212.4705
D-2	13.07943	12.5719	13.11626	11.38833	96728.94
D-3	4.943737	4.966602	4.940739	4.624694	956.5385
D-4	6.987991	6.782471	6.983572	6.748549	151.9457
D-5	6.992565	9.914095	6.975714	3.945423	2.15 × 1010
D-6	11.84488	9.993427	11.8372	10.15447	13.84575
D-7	19.44901	9.683639	19.52746	9.837551	15.35724
D-8	9.136202	7.988542	9.141686	7.588309	10.26142
D-9	9.703319	9.892095	9.682081	10.5868	15.53634
D-10	6.955285	6.943842	6.943957	8.097218	11.86641
D-11	27.15006	16.96214	27.06731	15.40635	52.48131
D-12	7.809574	8.075531	7.798247	7.056193	10.49084
D-13	24.66103	16.28119	24.62086	15.21046	222.4543
D-14	25.41172	15.68077	25.39927	15.21671	58.44042
D-15	28.20147	17.41897	28.16229	16.37026	217.168
D-16	11.54929	9.324261	11.5432	10.3146	24.1126
D-17	24.29577	14.34776	24.26541	14.5537	380.3055
D-18	12.24162	5.921934	12.24239	5.841004	90.67132
D-19	6.394083	6.462649	6.353714	4.753319	12.13664
D-20	5.760683	5.503576	5.800825	5.074644	9.191136
D-21	8.57278	9.446656	8.544085	31.38439	16.1163
D-22	14.08237	13.16862	13.94257	13.17557	17.83858
D-23	5.041229	3.683989	5.041314	42.6211	10.45466
D-24	4.766011	2.319754	4.793671	2.30511	8.837341

. Inspecting the results shown in Table 3, it is observed thatthe prediction by CGM is most accurate in the cases of 15 data samples. Optimal MAPE results are highlighted in boldface and in colored cells. Lower values of MAPEs compared to other forecasters indicate that the proposed model is able to match the predicted data with forecasted data accurately.

4.1. Average Rank-Based Analysis

For conducting a rank-based analysis, the method that obtained the optimal MAPE was ranked first and the method that obtained the highest MAPE was given last rank (i.e., fifth). On the basis of this, all ranks were stacked in an array and the average of MAPEs obtained by the models was obtained. It was concluded that the average rank obtained by the proposed CGM is (1.71) compared with GM (3.25), QGM (2.17), DGM (2.92) and NGM (5). A rank-based analysis of these 24 time series data revealed that the optimal rank is obtained by CGM. From this analysis, it can be concluded that the proposed cubic model is able to deal with the diversified data. For establishing the efficacy of the proposed model, the results in terms of APE are plotted and depicted through Figure 5. From the figure, it is observed that the NGM model requires much improvement, as the APE observed for this model is highest.

4.2. Box Plot Analysis

A box plot analysis of the APE of signal D-20 is presented in Figure 6. It is observed that the interquartile range (IQR) is optimal for CGM compared to other grey methods. In order to have greater insight on the statistics related to this plot, various parameters, such as Q1 (the values of APE that are between the smallest number and the median), Q3 (the values of APE that are between the maximum and median), median and box numerical characters tics, along with whisker statistics, are depicted in

Table 4. Box plot statistics of APE (signal D-20).

	GM	QGM	DGM	CGM	NGM
Min	0.00	0.00	0.00	0.00	0.00
Q1	2.87	2.47	2.96	2.72	3.31
Median	4.90	4.23	4.87	4.89	3.96
Q3	8.90	8.05	9.01	7.24	11.4
Max	15.6	15.6	15.5	12.2	40.1
Box Characteristics
Box1 (hidden)	2.87	2.47	2.96	2.72	3.31
Box2 (lower)	2.03	1.76	1.91	2.17	0.654
Box3 (upper)	4.00	3.82	4.14	2.35	7.47
Whisker Statistics
Whisker Top	6.69	7.60	6.49	4.96	28.7
Whisker Bottom	2.87	2.47	2.96	2.72	3.31

. It is observed that the values of Q3, i.e., APE values higher than the median, are optimal for CGM. These values are highlighted in boldface. This indicates that IQR is optimal and distributed in a narrow range. In addition, the maximum values of APE are also optimal. The box plot analysis on signal D-12 is shown in Figure 7. Due to space limitation, the details pertaining to quartile-1 for all signals and corresponding grey methods are exhibited in

Table 5. Quartile-1 details of signals (D-1 to D-24).

Data-Sample	GM	QGM	DGM	CGM	NGM
D-1	5.252159	1.620585	5.518314	1.652447	125.0758
D-2	4.363537	3.454967	4.486975	3.396315	503.5049
D-3	2.151343	2.172548	2.137846	1.845989	188.3042
D-4	1.646958	0.996663	1.651975	1.433694	110.3121
D-5	2.227184	2.726051	2.196458	1.858524	15885.64
D-6	3.894088	3.345588	3.805262	3.633281	4.630946
D-7	5.572624	3.100285	5.80758	2.099111	5.221259
D-8	3.298597	5.199594	3.148692	3.254303	2.939336
D-9	3.36086	2.832068	3.322605	4.683704	4.689639
D-10	2.315975	2.230016	2.171853	2.402969	4.131091
D-11	19.32161	8.371239	19.1147	6.050676	30.90563
D-12	2.436188	4.239803	2.503431	2.178686	1.511468
D-13	11.91881	10.21054	11.90255	7.965385	129.2233
D-14	15.03638	6.916835	14.66576	6.946082	36.37131
D-15	15.18945	8.042814	15.26803	6.831108	126.8702
D-16	4.763766	2.902926	4.744045	3.691789	13.27542
D-17	10.57584	5.694458	10.5343	5.965359	153.128
D-18	6.906703	1.701408	6.897625	1.672846	88.30793
D-19	2.647887	3.149512	2.498998	0.494169	4.285859
D-20	2.866761	2.466218	2.958643	2.716202	3.307033
D-21	1.034131	1.955302	0.937485	8.818959	5.760807
D-22	5.594835	8.168625	5.580734	5.603662	10.87225
D-23	2.392003	1.268331	2.355543	1.879085	3.99963
D-24	2.023046	0.621496	2.09803	0.657267	3.093761

. The optimal results are shown in boldface. We observed that these statistics also favor the applicability of CGM in MCP prediction. The analysis based on the average rank method, along with box plot characteristics, indicates that, for almost all of the signals, the CGM can be a better choice for MCP prediction compared to other grey methods.

5. Conclusions

Unknown rival behavior, imperfect competition and dynamic marketing conditions make the energy market a grey architecture. MCP prediction at the early stage can be a profit-winning tool for any generating company. This paper has presented a cubic-polynomial-based grey model for predicting MCP in the energy market. The following are the major conclusions from this work:

• A case study of the Indian stock exchange was taken and 24 different data samples of a diverse nature were applied to test the efficacy of the proposed CGM.
• It is observed by error analysis that the proposed method yields accurate results, as prediction MAPEs are optimal. For developing a deep understanding of the performance of CGM, rank-based analysis and box-plot-analysis-based observations were exhibited. All analyses indicate that CGM possesses a better quality regarding the anticipation of market conditions for MCP prediction.
• With the help of the proposed CGM, a generating company can emulate their rivals and use the information for profit accumulation. Further, the development of a neighborhood-algorithm-based cubic model lies in the future scope.

From the observations of case 2 and 4, it is pragmatic to state that the whitening equation realized by the cubic polynomial may sometimes give pessimistic results. Hence, it is advisable to tune the parameter of the whitening equation with the help of any modern optimizer. Hence, we keep this work in the future scope, where an optimization routine can be established to optimize the coefficients of whitening equations.