Consumer's behavior determinants after the electricity market liberalization: the Portuguese case

Débora Maravilha; Susana Silva; Erika Laranjeira; Débora Maravilha; Susana Silva; Erika Laranjeira

doi:10.3934/GF.2022021

Green Finance

2022, Volume 4, Issue 4: 436-449. doi: 10.3934/GF.2022021

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Research article Special Issues

Consumer's behavior determinants after the electricity market liberalization: the Portuguese case

1.
GKN Automotive
2.
CEF.up and Faculdade de Economia da Universidade do Porto (FEP), Porto, Portugal
3.
COMEGI, Centro Universitário Lusíada – Norte, Porto, Portugal and CEF.up

Received: 25 August 2022 Revised: 26 October 2022 Accepted: 01 November 2022 Published: 10 November 2022
JEL Codes: D10, D40, L94, Q48

Electricity markets have been liberalized worldwide, but the success of country specific experiences varied widely. Consumers' behavior is among the key factors for successful liberalization experiences namely regarding the decision to switch operator. This decision has been shown to be influenced by a multiplicity of factors. The goal of this article is to explore the analysis of the drivers for switching operator in a liberalized electricity market. With that purpose, we focused on the residential Portuguese case using a questionnaire. The logit estimation showed that men are more likely to switch supplier than women and that larger families are less likely to do so probably, due to the perception of high information search costs. Other sociodemographic variables were not found to be statistically significant. Regarding specific determinants, our results showed that past experiences with a supplier, dissatisfaction with the current operator, and family and friends' experiences were the most important determining factor for the decision to switch operator. Hence, the price was not the most important determinant. We also explored if different income groups had differentiated responses regarding the main drivers but concluded that there was no evidence that the income group affected the importance given to the price or to the other determinants for the decision.

Keywords:

Citation: Débora Maravilha, Susana Silva, Erika Laranjeira. Consumer's behavior determinants after the electricity market liberalization: the Portuguese case[J]. Green Finance, 2022, 4(4): 436-449. doi: 10.3934/GF.2022021

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Abstract

1. Introduction

Lots of physical phenomena can be expressed by non-linear partial differential equations (PDE), including, inter alia, dissipative and dispersive PDE. In this paper, we consider the Kuramoto-Sivashinsky (KS) equation

$\begin{eqnarray} \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi}{\partial s} = \varphi(s, t)\; 0 \le s\le 1, \; 0 \le t \le T, \; \gamma > 0, \end{eqnarray}$

(1.1)

$\begin{eqnarray} &&\phi(0, t) = 0, \; \phi(1, t) = 0, \; \phi_{ss}(0, t) = 0, \phi_{ss}(1, t) = 0, \; 0 < t < T, \end{eqnarray}$

(1.2)

$\begin{eqnarray} &&\phi(s, 0) = \varphi(s), \; 0\le s \le 1, \end{eqnarray}$

(1.3)

where $\gamma \in R$ is the constant.

The KS equation plays an important role in physics such as in diffusion, convection and so on. Lots of attention has been paid by researchers in recent years. An $H_1$ -Galerkin mixed finite element method for the KS equation was proposed in ^[1], lattice Boltzmann models for the Kuramoto-Sivashinsky equation were studied in ^[2], Backward difference formulae (BDF) methods for the KS equation were investigate in ^[3]. Stability regions and results for the Korteweg-de Vries-Burgers and Kuramoto-Sivashinsky equations were given in ^[4,5], respectively. In ^[6], an improvised quintic B-spline extrapolated collocation technique was used to solve the KS equation, and the stability of the technique was analyzed using the von Neumann scheme, which was found to be unconditionally stable. In ^[7], a septic Hermite collocation method (SHCM) was proposed to simulate the KS equation, and the nonlinear terms of the KS equation were linearized using the quasi-linearization process. In ^[8], a semidiscrete approach was presented to solve the variable-order (VO) time fractional 2D KS equation, and the differentiation operational matrices and the collocation technique were used to get a linear system of algebraic equations. In ^[9] the discrete Legendre polynomials (LPs) and the collocation scheme for nonlinear space-time fractional KdV-Burgers-Kuramoto equation were presented.

In order to avoid the Runge's phenomenon, barycentric interpolation ^[10,11,12] was developed. In recent years, linear rational interpolation (LRI) was proposed by Floater ^[13,14,15], and error of linear rational interpolation was also proved. The barycentric interpolation collocation method (BICM) has been developed by Wang et al.^{[22,23,24,25]}, and the algorithm of BICM has been used for linear/non-linear problems ^[21]. Volterra integro-differential equation (VIDE)^[16,20], heat equation (HE) ^[17], biharmonic equation (BE) ^[18], the Kolmogorov-Petrovskii-Piskunov (KPP) equation ^[19], fractional differential equations ^[20], fractional reaction-diffusion equation ^[28], semi-infinite domain problems ^[27] and biharmonic equation ^[26], plane elastic problems ^[29] have been studied by the linear barycentric interpolation collocation method (LBICM), and their convergence rates also have been proved.

In order to solve the KS equation efficiently, the LBRIM is presented. Because the nonlinear part of the KS equation cannot be solved directly, three kinds of linearization methods, including direct linearization, partial linearization and Newton linearization, are presented. Then, the nonlinear part of the KS equation is translated into the linear part, three kinds of iterative schemes are presented, and matrix equation of the linearization schemes are constructed. The convergence rate of the LBRCM for the KS equation is also given. At last, two numerical examples are presented to validate the theoretical analysis.

2. Linearization for KS equation

In the following, the KS equation is changed into the linear equation by the linearization scheme, including direct linearization, partial linearization and Newton linearization.

2.1. Direct linearization

For the Kuramoto-Sivashinskyr equation with the initial value of nonlinear term $\phi\frac{\partial \phi}{\partial s}$ is changed to $\phi_{0}\frac{\partial \phi_{0}}{\partial s}$ ,

$\begin{eqnarray} \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi_{0}\frac{\partial \phi_{0}}{\partial s} = \varphi(s, t), \end{eqnarray}$

(2.1)

and then we get the linear scheme as

$\begin{eqnarray} \frac{\partial \phi_{n}}{\partial t}+\gamma \frac{\partial^4 \phi_{n}}{\partial s^4}+\frac{\partial^2 \phi_{n}}{\partial s^2} = -\phi_{n-1}\frac{\partial \phi_{n-1}}{\partial s}+\varphi(s, t), a \le s\le b, 0 \le t \le T. \end{eqnarray}$

(2.2)

2.2. Partial linearization

By the partial linearization, nonlinear term $\phi\frac{\partial \phi}{\partial s}$ is changed to $\phi_{0}\frac{\partial \phi}{\partial s}$ ,

(2.3)

and then we have

$\begin{eqnarray} \frac{\partial \phi_{n}}{\partial t}+\gamma \frac{\partial^4 \phi_{n}}{\partial s^4}+\frac{\partial^2 \phi_{n}}{\partial s^2}+\phi_{n-1}\frac{\partial \phi_{n}}{\partial s} = \varphi(s, t), a \le s\le b, 0 \le t \le T. \end{eqnarray}$

(2.4)

2.3. Newton linearization

For the initial value $\phi\frac{\partial \phi}{\partial s} = \phi_{0}\frac{\partial \phi_{0}}{\partial s}+(\frac{\partial \phi_{0}}{\partial s}+\phi_{0}\frac{\partial \phi_{0}}{\partial s})(\phi-\phi_{0})$ , we have

$\begin{eqnarray} \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi_{0}}{\partial s}+ \phi_{0}\frac{\partial \phi_{0}}{\partial s}\phi = \varphi(s, t)+\phi_{0}\frac{\partial \phi_{0}}{\partial s}\phi_{0}, \end{eqnarray}$

(2.5)

and then we have

$\begin{eqnarray} \frac{\partial \phi_{n}}{\partial t}+\gamma \frac{\partial^4 \phi_{n}}{\partial s^4}+\frac{\partial^2 \phi_{n}}{\partial s^2}+\phi_{n}\frac{\partial \phi_{n-1}}{\partial s}+\phi_{n-1}\frac{\partial \phi_{n-1}}{\partial s}\phi_{n} = \varphi(s, t)+\phi_{n-1}\frac{\partial \phi_{n-1}}{\partial s}\phi_{n-1}, \end{eqnarray}$

(2.6)

where $n = 1, 2, \cdots.$

3. Differentiation matrices of KS equation

Interval $[a, b]$ is divided into $a = s_{0} < s_{1} < s_{2} < \cdots < s_{m-1} < s_{m} = b$ , for uniform partition with $h_{s} = \frac{b-a}{m}$ and nonuniform partition to be the second kind of Chebychev point. Time $[0, T]$ is divided into $0 = t_{0} < t_{1} < t_{2} < \cdots < t_{n-1} < t_{n} = T$ and $h_{t} = \frac{T}{n}$ for uniform partition. Then, we take $\phi_{nm}(s, t)$ to approximate $\phi(s, t)$ as

$\begin{eqnarray} \phi_{nm}(s, t) = \sum\limits_{i = 0}^{m} \sum\limits_{j = 0}^{n} r_{i}(s) r_{j}(t) \phi_{ij} \end{eqnarray}$

(3.1)

where $\phi_{ij} = \phi(s_{i}, t_{j})$ ,

$\begin{eqnarray} r_{i}(s) = \frac{ \frac{w_{i}}{s-s_{i}} }{ \sum\limits_{j = 0}^{m} \frac{w_{j}}{s-s_{j}}}, \quad r_{j}(t) = \frac{ \frac{w_{j}}{t-t_{j}} }{ \sum\limits_{i = 0}^{n} \frac{w_{i}}{t-t_{i}}} \end{eqnarray}$

(3.2)

is the barycentric interpolation basis ^[26], and

$\begin{eqnarray} w_{i} = \sum\limits_{k \in J_{i}}(-1)^{k} \prod\limits_{j = k, j \neq i}^{k+d_{s}} \frac{1}{s_{i}-s_{j}}, \quad w_{j} = \sum\limits_{k \in J_{j}}(-1)^{k} \prod\limits_{i = k, k \neq j}^{k+d_{t}} \frac{1}{t_{j}-t_{i}} \end{eqnarray}$

(3.3)

where $J_{i} = \{k \in I, i-d_{s} \le k \le i\}, I = \{0, 1, \cdots, m-d_{s} \}$ . See ^[26]. We get the barycentric rational interpolation.

For the case

$\begin{eqnarray} w_{i} = \frac{1}{\prod\nolimits_{i \neq k}\left(s_{i}-s_{k}\right)}, w_{j} = \frac{1}{\prod\nolimits_{j \neq k}\left(t_{j}-t_{k}\right)}, \end{eqnarray}$

(3.4)

we get the barycentric Lagrange interpolation.

So,

$\begin{eqnarray} r_{j}^{\prime }\left(s_{i}\right) = \frac{w_{j} / w_{i}}{s_{i}-s_{j}}, \quad j \neq i, \quad r_{i}^{\prime }\left(s_{i}\right) = -\sum\limits_{j \neq i} r_{j}^{\prime }\left(s_{i}\right), \end{eqnarray}$

(3.5)

$\begin{eqnarray} r_{j}^{(k)}\left(s_{i}\right) = k \left(r_{i}^{(k-1)}\left(s_{i}\right) r_{i}^{\prime}\left(s_{j}\right) - \frac{r_{i}^{(k-1)}\left(s_{j}\right) }{s_{i}-s_{j}}\right), \quad j \neq i, \end{eqnarray}$

(3.6)

$\begin{eqnarray} r_{i}^{(k)}\left(s_{i}\right) = -\sum\limits_{j \neq i} r_{j}^{(k)}\left(s_{i}\right). \end{eqnarray}$

(3.7)

Then, we have

$\begin{eqnarray} \mathrm{{D}}^{(0, 1)}_{ij}& = & r_{i}^{\prime }\left(t_{j}\right), \end{eqnarray}$

(3.8)

$\begin{eqnarray} \mathrm{{D}}^{(1, 0)}_{ij}& = & r_{i}^{\prime }\left(s_{j}\right), \end{eqnarray}$

(3.9)

$\begin{eqnarray} \mathrm{{D}}^{(k, 0)}_{ij}& = & r_{i}^{(k) }\left(s_{j}\right), k = 2, 3, \cdots. \end{eqnarray}$

(3.10)

3.1. Matrix equation of direct linearization

Combining (3.1) and (2.2), we have

$\begin{eqnarray} \left[\mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n} \right] \mathcal{\phi}_{n} = {\Psi}-diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}, \end{eqnarray}$

(3.11)

and then we have

$\begin{eqnarray} \mathcal{L} \mathcal{\phi}_{n} = {\Psi}_{n-1} \end{eqnarray}$

(3.12)

where

$\mathcal{L} = \mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n},$

${\Psi}_{n-1} = {\Psi}-diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}$

and $\otimes$ is the Kronecher product ^[17].

3.2. Matrix equation of partial linearization

Combining (3.1) and (2.4), we have

$\begin{eqnarray} \left[\mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n} \right] \mathcal{\phi}_{n} = {\Psi}, \end{eqnarray}$

(3.13)

$n = 1, 2, \cdots$ , and then we have

$\begin{eqnarray} \mathcal{L} \mathcal{\phi} = {\Psi} \end{eqnarray}$

(3.14)

where $\mathcal{L} = \mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}$ .

3.3. Matrix equation of Newton linearization

Combining (3.1) and (2.6), we have

$\begin{eqnarray} &&\left[\mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n} \right] \mathcal{\phi}_{n} \\&& = {\Psi} + [diag(\mathcal{\phi}_{n})-diag(\mathcal{\phi}_{n-1})]\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}, \end{eqnarray}$

(3.15)

and then we get

$\begin{eqnarray} \mathcal{L} \mathcal{\phi} = {\Psi}_{n-1} \end{eqnarray}$

(3.16)

where

$\mathcal{L} = \mathcal{{I}}_{m} \otimes \mathcal{{D}}^{(0, 1)} + \mathcal{{D}}^{(2, 0)}\otimes\mathcal{{I}}_{n}+\gamma \mathcal{{D}}^{(4, 0)}\otimes\mathcal{{I}}_{n}+diag(\mathcal{\phi}_{n-1})\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n},$

and

${\Psi}_{n-1} = {\Psi} + [diag(\mathcal{\phi}_{n})-diag(\mathcal{\phi}_{n-1})]\mathcal{{D}}^{(1, 0)}\otimes\mathcal{{I}}_{n}\cdot\mathcal{\phi}_{n-1}.$

4. Convergence rate of KS equation

In this part, an error estimate of the KS equation is given with $r_n(s) = \sum\limits_{i = 0}^{n} r_{i}(s) \phi_{i}$ to replace $\phi(s)$ , where $r_{i}(s)$ is defined as (3.2), and $\phi_{i} = \phi(s_{i})$ . We also define

$\begin{eqnarray} e(s): = \phi(s)-r_n(s) = (s-s_{i}) \cdots(s-s_{i+d})\phi\left[s_{i}, s_{i+1}, \ldots, s_{i+d}, s\right]. \end{eqnarray}$

(4.1)

Then, we have the following.

Lemma 1. For $e(s)$ defined by (4.1) and $\phi(s)\in C^{d+2}[a, b]$ , there is

$\begin{equation} \left|e^{(k)}(s)\right| \leq C h^{d-k+1}, k = 0, 1, \cdots. \end{equation}$

(4.2)

For KS equation, rational interpolation function of $\phi(s, t)$ is defined as $r_{mn}(s, t)$

$\begin{equation} r_{mn}(s, t) = \frac{ \sum\limits_{i = 0}^{m+d_{s}} \sum\limits_{j = 0}^{n+d_{t}} \frac{w_{i, j}}{\left(s-s_{i}\right)\left(t-t_{j}\right)} {\phi}_{i, j}}{ \sum\limits_{i = 0}^{m+d_{s}} \sum\limits_{j = 0}^{n+d_{t}} \frac{w_{i, j}}{\left(s-s_{i}\right)\left(t-t_{j}\right)}} \end{equation}$

(4.3)

where

$\begin{equation} w_{i, j} = (-1)^{i-d_{s}+j-d_{t}} \sum\limits_{k_{1} \in J_{i}}\prod\limits_{h_{1} = k_{1}, h_{1} \neq j}^{k_{1}+d_{s}}\frac{1}{\left|s_{i}-s_{h_{1}}\right|} \sum\limits_{k_{2} \in J_{i}} \prod\limits_{h_{2} = k_{2}, h_{2} \neq j}^{k_{2}+d_{t}} \frac{1}{\left|t_{j}-t_{h_{2}}\right|}. \end{equation}$

(4.4)

We define $e(s, t)$ to be the error of $\phi(s, t)$ as

$\begin{eqnarray} e(s, t) :& = & \phi(s, t)-r_{mn}(s, t) \\ & = & \left(s-s_{i}\right) \cdots\left(s-s_{i+d_{s}}\right)\phi\left[s_{i}, s_{i+1}, \ldots, s_{i+d_1}, s, t\right] \\ &+&\left(t-t_{j}\right) \cdots\left(t-t_{j+d_{t}}\right)\phi\left[s, t_{j}, t_{j+1}, \ldots, t_{j+d_2}, t\right]. \end{eqnarray}$

(4.5)

With similar analysis of Lemma 1, we have the following

Theorem 1. For $e(s, t)$ defined as (4.5) and $\phi(s, t) \in C^{d_{s}+2}[a, b]\times C^{d_{t}+2}[0, T]$ , we have

$\begin{equation} \left|e^{(k_{1}, k_{2})}\left(s, t\right)\right| \leq C (h_{s}^{d_{s}-k_{1}+1}+h_{t}^{d_{t}-k_{2}+1}), k_{1}, k_{2} = 0, 1, \cdots. \end{equation}$

(4.6)

We take the direct linearization of the KS equation as an example prove the convergence rate. Let $\phi(s_{m}, t_{n})$ be the approximate function of $\phi(s, t)$ and $L$ be a bounded operator. There holds

$\begin{eqnarray} L \phi(s_{m}, t_{n}) = \varphi(s_{m}, t_{n}), \end{eqnarray}$

(4.7)

and

$\begin{eqnarray} \lim\limits_{m, n\rightarrow \infty} \phi(s_{m}, t_{n}) = \phi(s, t). \end{eqnarray}$

(4.8)

Then, we get the following

Theorem 2. For $\phi(s_{m}, t_{n}):L\phi(s_{m}, t_{n}) = \varphi(s, t)$ and $L$ defined as (4.7), there

$|\phi(s, t)-\phi(s_{m}, t_{n})|\le C(h^{d_{s}-3}+\tau^{d_{t}}).$

Proof. As

$\begin{eqnarray} \begin{array}{ll} &L \phi(s, t)-L \phi(s_{m}, t_{n}) \\[0.5cm]& = \frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}- \phi_{0}\frac{\partial \phi_{0}}{\partial s}-\varphi(s, t) \\[0.5cm]&- \left[\frac{\partial \phi(s_{m}, t_{n})}{\partial t}+\gamma \frac{\partial^4 \phi(s_{m}, t_{n})}{\partial s^4}+\frac{\partial^2 \phi(s_{m}, t_{n})}{\partial s^2}+ \phi_{0}(s_{m}, t_{n})\frac{\partial \phi_{0}(s_{m}, t_{n})}{\partial s}-\varphi(s, t)\right] \\[0.5cm]& = \frac{\partial \phi}{\partial t}-\frac{\partial \phi}{\partial t}(s_{m}, t_{n}) +\gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right] \\[0.5cm]&+ \frac{\partial^2 \phi}{\partial s^2}-\frac{\partial^2 \phi}{\partial s^2}(s_{m}, t_{n})+\left[\phi_{0}\frac{\partial \phi_{0}}{\partial s}- \phi_{0}(s_{m}, t_{n})\frac{\partial \phi_{0}}{\partial s}(s_{m}, t_{n}) \right] \\[0.5cm]&: = E_{1}(s, t)+E_{2}(s, t)+E_{3}(s, t)+E_{4}(s, t). \end{array} \end{eqnarray}$

(4.9)

Here,

$E_{1}(s, t) = \frac{\partial \phi}{\partial t}-\frac{\partial \phi}{\partial t}(s_{m}, t_{n}),$

$E_{2}(s, t) = \gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right],$

$E_{3}(s, t) = \frac{\partial^2 \phi}{\partial s^2}-\frac{\partial^2 \phi}{\partial s^2}(s_{m}, t_{n}),$

$E_{4}(s, t) = \phi_{0}\frac{\partial \phi_{0}}{\partial s}- \phi_{0}(s_{m}, t_{n})\frac{\partial \phi_{0}}{\partial s}(s_{m}, t_{n}) .$

With $E_{2}(s, t)$ , we have

$\begin{equation} \begin{array} {ll} E_{2}(s, t)& = \gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right] \\& = \gamma \left[\frac{\partial^4 \phi}{\partial s^4}- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t)+ \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t)- \frac{\partial^4 \phi}{\partial s^4}(s_{m}, t_{n})\right] \\& = \frac{ \sum\limits_{i = 0}^{m-d_{s}} (-1)^{i}\frac{\partial^4 \phi}{\partial s^4}[s_{i}, s_{i+1}, \ldots, s_{i+d_1}, s_{m}, t]} { \sum\limits_{i = 0}^{m-d_{s}} \lambda_{i}(s)} \\ \nonumber&+ \frac{ \sum\limits_{j = 0}^{n-d_{t}}(-1)^{j}\frac{\partial^4 \phi}{\partial s^4}[ t_{j}, t_{j+1}, \ldots, t_{j+d_2}, s_{m}, t_{n}]} { \sum\limits_{j = 0}^{n-d_{t}}\lambda_{j}(t)} \\& = \frac{\partial^4 e}{\partial s^4}(s_{m}, t)+\frac{\partial^4 e}{\partial s^4}(s_{m}, t_{n}). \end{array} \end{equation}$

For $E_{2}(s, t)$ we get

$\begin{equation} |E_{2}(s, t)|\le \left|\frac{\partial^4 e}{\partial s^4}(s_{m}, x)+\frac{\partial^4 e}{\partial s^4}(s_{m}, t_{n})\right|\le C (h^{d_{s}-3}+\tau^{d_{t}+1}). \end{equation}$

(4.10)

Then, we have

$\begin{equation} |E_{1}(s, t)|\le \left|\frac{\partial e}{\partial t}(s_{m}, t)+\frac{\partial e}{\partial t}(s_{m}, t_{n})\right|\le C (h^{d_{s}+1}+\tau^{d_{t}}). \end{equation}$

(4.11)

Similarly, for $E_{3}(s, t)$ we have

$\begin{equation} \begin{array}{ll} E_{3}(s, t) = \frac{\partial^2 \phi}{\partial s^2}(s, t)- \frac{\partial^2 \phi}{\partial s^2}(s_{m}, t_{n}) = \frac{\partial^2 e}{\partial s^2}(s, t_{n})+ \frac{\partial^2 e}{\partial s^2}(s_{m}, t_{n}), \end{array} \end{equation}$

(4.12)

and

$\begin{equation} |E_{3}(s, t)|\le \left| \frac{\partial^2 e}{\partial s^2}(s, t_{n})+ \frac{\partial^2 e}{\partial s^2}(s_{m}, t_{n}) \right|\le C (h^{d_{s}-1}+\tau^{d_{t}+1}). \end{equation}$

(4.13)

For $E_{4}(s, t)$ we get

$\begin{equation} \begin{array}{ll} |E_{4}(s, t)|& = \left|\phi_{0}\frac{\partial \phi}{\partial s}- \phi_{0}(s_{m}, t_{n})\frac{\partial \phi}{\partial s}(s_{m}, t_{n})\right| \\ &\le \left|\frac{\partial e}{\partial t}(s_{m}, t)+\frac{\partial e}{\partial t}(s_{m}, t_{n})\right|\le C (h^{d_{s}+1}+\tau^{d_{t}}). \end{array} \end{equation}$

(4.14)

Combining (4.9) and (4.11)–(4.14) together, the proof of Theorem 2 is completed. □

5. Numerical examples

All the examples are carried on a computer with Intel(R) Core(TM) i5-8265U CPU @ 1.60 GHz 1.80 GHz operating system, 16 G radon access running memory and a 512 G solid state disk memory. All simulation experiments were realized by the software Matlab (Version: R2016a). In this part, two examples are presented to test the theorem.

Example 1. Consider the KS equation

$\frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi}{\partial s} = \varphi(s, t)$

with the condition is

$\phi(0, t) = 0, \phi(1, t) = 0,$

and

$\phi(s, 0) = \sin(2\pi s).$

$\phi_{ss}(0, t) = 0, \phi_{ss}(1, t) = 0,$

and

$\varphi(s, t) = e^{-t}\sin(2\pi s)[2\pi e^{-t}\cos(2\pi s)-1+16\pi^4-4\pi^2].$

The solution of the KS equation is

$\phi(s, t) = e^{-t}\sin(2\pi s).$

In Figures 1–3, errors of unform partition with direct linearization, partial linearization, Newton linearization for the KS equation are presented. In Figures 4–6, errors of non-uniform partition with direct linearization, partial linearization, Newton linearization for the KS equation are presented.

Figure 1. Errors of nonuniform partition by direct linearization with

$m = n = 19$ .

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Figure 2. Errors of nonuniform partition by partial linearization with

$m = n = 19$ .

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Figure 3. Errors of nonuniform partition by Newton linearization with

$m = n = 19$ .

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Figure 4. Errors of uniform partition by direct linearization with

$m = n = 19$ .

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Figure 5. Errors of uniform partition by partial linearization with

$m = n = 19$ .

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Figure 6. Errors of uniform partition by Newton linearization with

$m = n = 19$ .

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In Tables 1 and 2, errors of LBCM and LBRCM for the KS equation with boundary condition dealt with by the method of substitution and method of addition are given. From Table 1, we know that the accuracy of LBCM is higher than LBRCM, and from Table 2 the accuracy of the method of additional is higher than the method of substitution.

Table 1. Errors of LBCM for KS equation with

$m = n = 17$ .

	Method of substitution		Method of additional
Linearization	Uniform partition	Nonuniform partition	Uniform partition	Nonuniform partition
direct	1.3278e-07	5.6616e-10	1.7050e-08	4.6293e-10
partial	5.5563e-07	2.6381e-09	1.1492e-07	5.0974e-10
Newton	6.6705e-07	4.8875e-10	8.8609e-08	2.5867e-11

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Table 2. Errors of LBRCM for KS equation with

$m = n = 17, d_{s} = d_{t} = 12$ .

	Method of substitution		Method of additional
Linearization	Uniform partition	Nonuniform partition	Uniform partition	Nonuniform partition
direct	4.4575e-06	3.2280e-08	4.1010e-08	2.2749e-09
partial	4.4573e-06	3.2245e-08	5.4191e-07	1.5951e-07
Newton	4.4560e-06	3.2215e-08	1.2972e-06	3.5137e-07

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In , we choose the Newton linearization to solve the KS equation, and the error of LBRCM for uniform and nonuniform partitions are presented with $t = 0.3, 0.9, 2, 4, 8, 16$ .

Table 3. Errors of Newton linearization for

$t$ .

	Uniform partition		Nonuniform partition
$t$	$(8, 8) d_{s}=d_{t}=7$	$(16, 16) d_{s}=d_{t}=15$	$(8, 8) d_{s}=d_{t}=7$	$(16, 16) d_{s}=d_{t}=15$
0.3	1.5449e-01	1.3163e-06	6.2692e-02	2.4769e-08
0.9	1.4211e-01	1.1737e-06	6.1721e-02	2.3846e-08
2	1.2162e-01	1.0785e-06	5.8680e-02	2.3685e-08
4	9.1544e-02	9.4383e-07	5.3241e-02	2.3353e-08
8	5.1798e-02	7.2283e-07	4.3721e-02	2.2440e-08
16	1.6540e-02	4.1712e-07	2.9435e-02	1.9220e-08

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The errors of LBRCM of uniform and Chebyshev partitions are presented with $(m, n, d_{s}, d_{t}) = (8, 8, 7, 7), (16, 16, 15, 15)$ . From the table, comparing $(m, n) = (8, 8)$ with $(m, n) = (16, 16)$ , the accuracy was higher when the number became bigger.

In the following table, we take Newton linearization to present numerical results. From and , with errors of Newton linearization for uniform partition $d_{t} = 6;t = 1$ are given and convergence rate is $O(h^{d_{s}})$ . From , with space variable $s, d_{s} = 6$ , and there is superconvergence rate $O(h^{d_{s}-1})$ at $t = 1$ .

Table 4. Errors of Newton linearization for uniform partition

$d_{t} = 6$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	4.1317e-01		3.2652e-03		3.3180e-01
16, 16	1.8608e-01	1.1508	3.1257e-02	-	3.3919e-02	3.2902
32, 32	9.5437e-02	0.9633	1.0198e-02	1.6159	3.3873e-03	3.3239
64, 64	4.7221e-02	1.0151	2.6490e-03	1.9448	3.5472e-04	3.2554

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Table 5. Errors of Newton linearization for uniform partition

$d_{s} = 6$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	1.3997e-01		1.4004e-01		1.4008e-01
16, 16	5.4923e-03	4.6716	5.4957e-03	4.6714	5.4973e-03	4.6714
32, 32	1.2850e-04	5.4176	1.2883e-04	5.4148	1.2891e-04	5.4143
64, 64	2.9976e-06	5.4218	3.0728e-06	5.3898	3.0798e-06	5.3874

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For and , the errors of Chebyshev partition for Newton linearization with $s$ and $t$ are presented. For $d_{t} = 6$ , the convergence rate is $O(h^{d_{s}})$ in Table 6, while in Table 7, there are also superconvergence phenomena.

Table 6. Errors of Newton linearization for Chebyshev partition

$d_{t} = 6$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	5.4754e-01		2.9399e-02		8.5922e-02
16, 16	1.0318e-01	2.4078	4.6815e-03	2.6507	1.2658e-03	6.0849
32, 32	9.6912e-02	0.0904	8.0675e-04	2.5368	1.9577e-05	6.0148
64, 64	4.8014e-01	-	1.7672e-03	-	2.2716e-05	-

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Table 7. Errors of Newton linearization for Chebyshev partition

$d_{s} = 6$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	6.1344e-02		6.1386e-02		6.1415e-02
16, 16	8.1492e-05	9.5561	8.1163e-05	9.5629	8.0977e-05	9.5669
32, 32	1.4204e-07	9.1642	1.4183e-07	9.1606	1.5487e-07	9.0303
64, 64	6.3190e-06	-	3.8960e-06	-	1.4861e-06	-

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Example 2. Consider the KS equation

$\frac{\partial \phi}{\partial t}+\gamma \frac{\partial^4 \phi}{\partial s^4}+\frac{\partial^2 \phi}{\partial s^2}+ \phi\frac{\partial \phi}{\partial s} = 0,$

with the analytic solution

$\phi(s, t) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(s-ct+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(s-ct+s_{0})\right],$

and boundary condition

$\phi(-10, t) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(-10-ct+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(-10-ct+s_{0})\right],$

$\phi(10, t) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(10-ct+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(10-ct+s_{0})\right],$

and initial condition

$\phi(s, 0) = c+\frac{15\sqrt{11}}{19\sqrt{19}}\left[-3\tanh \frac{\sqrt{11}}{2\sqrt{19}}(s+s_{0}) +\tanh^3 \frac{\sqrt{11}}{2\sqrt{19}}(s+s_{0})\right],$

with $c = 2, x_{0} = 10$ .

In –, errors of direct linearization, partial linearization, Newton linearization with $m = n = 19$ KS equation are presented, respectively.

Figure 7. Errors of direct linearization with

$m = n = 19$ .

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Figure 8. Errors of partial linearization with

$m = n = 19$ .

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Figure 9. Errors of Newton linearization with

$m = n = 19$ .

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In the following table, direct linearization is chosen to present numerical results. From and , errors of direct linearization for uniform partition $d_{t} = 7$ with different $d_{s}$ are given and the convergence rate is $O(h^{d_{s}}-1)$ . From , with space variable $s, d_{s} = 7$ , and there are also superconvergence phenomena.

Table 8. Errors of direct linearization for uniform partition for

$d_{t} = 7$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	1.3587e+00		8.9361e-01		6.3703e-01
16, 16	2.1617e-01	2.6520	2.7467e-01	1.7019	2.5682e-01	1.3106
32, 32	6.7743e-02	1.6740	6.8822e-02	1.9967	4.7078e-02	2.4476
64, 64	2.5175e-02	1.4281	1.3216e-02	2.3806	4.3739e-03	3.4281

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Table 9. Errors of direct linearization for uniform partition for

$d_{s} = 7$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	3.6253e-01		3.6380e-01		3.6446e-01
16, 16	1.8147e-01	0.9984	1.8124e-01	1.0052	1.8121e-01	1.0081
32, 32	6.4076e-02	1.5019	6.4158e-02	1.4982	6.4141e-02	1.4983
64, 64	8.9037e-04	6.1692	8.9840e-04	6.1581	8.9863e-04	6.1574

| Show Table

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For and , the errors of Chebyshev partition for direct linearization with $s$ and $t$ are presented. For $d_{t} = 7$ , the convergence rate is $O(h^{d_{s}})$ in Table 10, while in Table 11, there are also superconvergence phenomena.

Table 10. Errors of direct linearization for Chebyshev partition for

$d_{t} = 7$ .

$m, n$	$d_{s}=2$	$h^{\alpha}$	$d_{s}=3$	$h^{\alpha}$	$d_{s}=4$	$h^{\alpha}$
8, 8	6.5990e-01		4.0742e-01		3.6175e-01
16, 16	1.1154e-01	2.5646	1.7539e-01	1.2160	2.1752e-01	0.7338
32, 32	4.3052e-02	1.3735	8.6654e-03	4.3391	1.2511e-03	7.4418
64, 64	3.9204e-02	0.1351	2.3776e-03	1.8658	3.5682e-04	1.8099

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Table 11. Errors of direct linearization for Chebyshev partition for

$d_{s} = 7$ .

$m, n$	$d_{t}=2$	$\tau^{\alpha}$	$d_{t}=3$	$\tau^{\alpha}$	$d_{t}=4$	$\tau^{\alpha}$
8, 8	4.3760e-01		4.3745e-01		4.3739e-01
16, 16	1.1801e-01	1.8908	1.1801e-01	1.8902	1.1801e-01	1.8900
32, 32	9.9842e-04	6.8850	9.9854e-04	6.8849	9.9801e-04	6.8857
64, 64	2.5749e-06	8.5990	2.5052e-06	8.6388	4.8401e-06	7.6879

| Show Table

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6. Conclusions

In this paper, LBRCM is used to solve the (1+1) dimensional SK equation. Three kinds of linearization methods are taken to translate the nonlinear part into a linear part. Matrix equations of the discrete SK equation are obtained from corresponding linearization schemes. The convergence rate of LBRCM is also presented. In the future work, LBRCM can be developed for the (2+1) dimensional SK equation and other partial differential equations classes, including Kolmogorov-Petrovskii-Piskunov (KPP) equation and, fractional reaction-diffusion equation and so on.

Acknowledgments

The work of Jin Li was supported by the Natural Science Foundation of Shandong Province (Grant No. ZR2022MA003).

The authors also gratefully acknowledges the helpful comments and suggestions of the reviewers, which have improved the presentation.

Conflict of interest

The author declares no conflict of interest.

References

[1]	Aliabadi DE, Chan K (2022) The emerging threat of artificial intelligence on competition in liberalized electricity markets: A deep Q-network approach. Appl Energy 32: 119813. https://doi.org/10.1016/j.apenergy.2022.119813 doi: 10.1016/j.apenergy.2022.119813
[2]	Al-Sunaidy A, Green R (2006) Electricity deregulation in OECD countries. Energy 31: 769–787. https://doi.org/10.1016/j.energy.2005.02.017 doi: 10.1016/j.energy.2005.02.017
[3]	Ariu T, Lewis PE, Goto H, et al. (2012) Impacts and Lessons from the Fully Liberalized European Electricity Market - Residential Customer Price, Switching and Services. Central Research Institute of Electric Power Industry Report, No. Y11018.
[4]	Armstrong M, Sappington D (2006) Regulation, Competition, and Liberalization. J Econ Lit 44: 325–366. https://doi.org/10.1257/jel.44.2.325 doi: 10.1257/jel.44.2.325
[5]	Borenstein S, Bushnell J (2000) Electricity restructuring: deregulation or reregulation. Regulation 23: 46–52.
[6]	Defeuilley C (2009) Retail competition in electricity markets. Energy Policy 37: 377–386. https://doi.org/10.1016/j.enpol.2008.07.025. doi: 10.1016/j.enpol.2008.07.025
[7]	Drosos D, Grigorios LK, Arabatzis G, et al. (2020) Evaluating Customer Satisfaction in Energy Markets Using a Multicriteria Method: The Case of Electricity Market in Greece. Sustainability 12: 3862. https://doi.org/10.3390/su12093862
[8]	ERSE (2013) Extinction of regulated tariffs for electricity and natural gas. Available from: http://www.erse.pt/eng/Paginas/extinctiontariffs.aspx.
[9]	ERSE (2021) Tarifas e preços para a energia elétrica e outros serviços em 2022 e parâmetros para o período de regulação 2022–2025.
[10]	ERSE (2021) Relatório sobre os mercados retalhistas de eletricidade e de gás natural em Portugal.
[11]	Entidade Reguladora dos Serviços Energético (ERSE) (2021) Síntese Mensal, November 2021.
[12]	Ek K, Söderholm P (2008) Households' switching behavior between electricity suppliers in Sweden. Util Policy 16: 254–261. https://doi.org/10.1016/j.jup.2008.04.005. doi: 10.1016/j.jup.2008.04.005
[13]	Ferreira P, Araújo M, O'Kelly MEJ (2006) An overview of the portuguese electricity market. Energy Policy 35: 1967–1977. https://doi.org/10.1016/j.enpol.2006.06.003 doi: 10.1016/j.enpol.2006.06.003
[14]	Flores M, Waddams PC (2013) Consumer behaviour in the British retail electricity market. Centre for Competition Policy Working Paper, University of East Anglia, Norwich (UK). https://ueaeco.github.io/working-papers/papers/ccp/CCP-13-10.pdf
[15]	Gamble A, Juliusson E, Gärling T (2009) Consumer attitudes towards switching supplier in three deregulated markets. J Socio Econ 38: 814–819. https://doi.org/10.1016/j.socec.2009.05.002. doi: 10.1016/j.socec.2009.05.002
[16]	Gärling T, Gamble A, Juliusson E (2008) Consumers' switching inertia in a fictitious electricity market. Int J Consum Stud 32: 613–618. https://doi.org/10.1111/j.1470-6431.2008.00728.x doi: 10.1111/j.1470-6431.2008.00728.x
[17]	Ghazvini M, Ramos S, Soares J, et al. (2019) Liberalization and customer behavior in the Portuguese residential retail electricity market. Util Policy 59: 100919. https://doi.org/10.1016/j.jup.2019.05.005
[18]	Giulietti M, Price C, Waterson M (2005) Consumer choice and competition policy: a study of UK energy markets. Econ J 115: 949–968. https://doi.org/10.1111/j.1468-0297.2005.01026.x doi: 10.1111/j.1468-0297.2005.01026.x
[19]	Hartmann P, Ibáñez VA (2007) Managing customer loyalty in liberalized residential energy markets: The impact of energy branding. Energy Policy 35: 2661–2672. https://doi.org/10.1016/j.enpol.2006.09.016 doi: 10.1016/j.enpol.2006.09.016
[20]	Joskow PL (2011) Deregulation and regulatory reform. In Deregulation of Network Industries: What's Next? Sam Peltzman and Clifford Winston (Eds.). Washington D.C, USA, 113–188.
[21]	Juliusson EA, Gamble A, Gärling T (2007) Loss aversion and price volatility as determinants of attitude towards variable price agreements in the Swedish electricity market. Energy Policy 35: 5953–5957. https://doi.org/10.1016/j.enpol.2007.06.019 doi: 10.1016/j.enpol.2007.06.019
[22]	Kaenzig J, Heinzle SL, Wüstenhagen R (2013) Whatever the customer wants, the customer gets? Exploring the gap between consumer preferences and default electricity products in Germany. Energy Policy 53: 311–322. https://doi.org/10.1016/j.enpol.2012.10.061 doi: 10.1016/j.enpol.2012.10.061
[23]	Littlechild SC (2000) Why do we need electricity retailers: A reply to Joskow on wholesale spot price pass-through. University of Cambridge, Cambridge (UK).
[24]	Macedo DP, Marques AC, Damette O (2020) The impact of the integration of renewable energy sources in the electricity price formation: is the Merit-Order Effect occurring in Portugal? Util Policy 101080. https://doi.org/10.1016/j.jup.2020.101080
[25]	Morey MJ, Kirsch LD (2016) Retail choice in electricity: What have we learned in 20 years? Electric Markets Research Foundation report.
[26]	Pollitt M (2012) The role of policy in energy transitions: Lessons from the energy liberalisation era. Energy Policy 50: 128–137. https://doi.org/10.1016/j.enpol.2012.03.004 doi: 10.1016/j.enpol.2012.03.004
[27]	Roe B, Teisl M, Levy A, et al. (2001) US consumers' willingness to pay for green electricity. Energy policy 29: 917–925. https://doi.org/10.1016/S0301-4215(01)00006-4 doi: 10.1016/S0301-4215(01)00006-4
[28]	Shin KJ, Managi S (2017) Liberalization of a retail electricity market: Consumer satisfaction and household switching behavior in Japan. Energy Policy 110: 675–685. https://doi.org/10.1016/j.enpol.2017.07.048 doi: 10.1016/j.enpol.2017.07.048
[29]	Sioshansi F (2006) Electricity Market Reform: What has the Experience taught us thus far? Util Policy 14: 63–75. https://doi.org/10.1016/j.jup.2005.12.002
[30]	Six M, Wirl F, Wolf J (2017) Information as potential key determinant in switching electricity suppliers. J Bus Econ 87: 263–290. https://doi.org/10.1007/s11573-016-0821-9 doi: 10.1007/s11573-016-0821-9
[31]	Vihalemm T, Keller M (2016) Consumers, citizens or citizen-consumers? Domestic users in the process of Estonian electricity market liberalization. Energy Res Soc Sci 13: 38–48. https://doi.org/10.1016/j.erss.2015.12.004 doi: 10.1016/j.erss.2015.12.004
[32]	Waterson M (2003) The role of consumer in competition and competition policy. Ind J Ind Organ 21: 129–150. https://doi.org/10.1016/S0167-7187(02)00054-1 doi: 10.1016/S0167-7187(02)00054-1
[33]	Woo CK, Lloyd D, Tishler A (2003) Electricity Market Reform Failures: UK, Norway, Alberta and California. Energy Policy 31: 1103–1115. https://doi.org/10.1016/S0301-4215(02)00211-2 doi: 10.1016/S0301-4215(02)00211-2
[34]	Yang Y (2014) Understanding household switching behavior in the retail electricity market. Energy Policy 69: 406–414. https://doi.org/10.1016/j.enpol.2014.03.009 doi: 10.1016/j.enpol.2014.03.009

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Consumer's behavior determinants after the electricity market liberalization: the Portuguese case

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Abstract

1. Introduction

2. Linearization for KS equation

2.1. Direct linearization

2.2. Partial linearization

2.3. Newton linearization

3. Differentiation matrices of KS equation

3.1. Matrix equation of direct linearization

3.2. Matrix equation of partial linearization

3.3. Matrix equation of Newton linearization

4. Convergence rate of KS equation

5. Numerical examples

6. Conclusions

Acknowledgments

Conflict of interest

References

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Abstract

1. Introduction

2. Linearization for KS equation

2.1. Direct linearization

2.2. Partial linearization

2.3. Newton linearization

3. Differentiation matrices of KS equation

3.1. Matrix equation of direct linearization

3.2. Matrix equation of partial linearization

3.3. Matrix equation of Newton linearization

4. Convergence rate of KS equation

5. Numerical examples

6. Conclusions

Acknowledgments

Conflict of interest

References