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Regional Collaborative Electricity Consumption Management: an Urban Operations Research Model

Abstract

Electricity generation can be a major source of pollution. In a compact region where pollutants can easily transfer from one city to another, a unilateral response—on the part of one city to improve its environmental conditions—is often ineffective. This paper develops an urban operations research model for collaborative management of a reduction in electricity consumption. This model internalizes the external costs of electricity consumption in a region; derives the optimal level of electricity consumption; and sets up a scheme to compensate for the externalities of electricity consumption. This analysis is the first urban operations research model for collaborative electricity consumption management, which internalizes the external costs of electricity consumption. This study is also the first attempt to derive the optimal level of electricity consumption within regional collaboration. This is the first time that a scheme to compensate for the externalities is proposed to ensure that agreed-upon optimality principle could be maintained throughout the entire duration of cooperation.

Introduction

Industries consume vast amounts of electricity, and the world’s energy demand for electricity increases year by year. Electricity generation is one of the major emission sources of air pollutants [11, 35]. Moreover, in a compact region where cities cluster together, air pollutants can easily transfer from one city to another. Pollutants emitted in one city will not just affect the city’s own air quality but will also diffuse to its surrounding cities rapidly, therefore, causing the regional environmental problems. A unilateral response on the part of one city to improve its environmental conditions is often ineffective. Cooperation in environmental control holds out the best promise of effective action. The application of collaborative environmental management schemes can be found in Yeung and Petrosyan [53] and Yeung [52].

However, there are various hurdles in regional collaboration and environmental externality is one of them. The environmental costs of electricity generation are “external” because the real costs are not taken into account when making decisions. Some models have been developed to examine these environmental externalities. For example, Cian and Tavoni [17] examined international emissions trading by studying how a cap on the trade of carbon offsets influences innovation, technological change, and welfare. Antoci et al. [3] proposed a taxonomy of different structural changes based on distributive, environmental, and economic outcomes, and they studied a two-sector model with environmental externalities to identify under which conditions each structural change could occur. Banzhaf and Chupp [7] provided a simple model and demonstrated that the choice of pricing policy also depended crucially on the shape of the marginal costs of providing the public good. Do et al. [20] introduced a special class of games with externalities and issue linkages to promote cooperation on transboundary water resources. Other studies can be found in Chander and Tulkens [13, 14] and Lera-López et al. [37].

Urban operations research has been proven to be one of the most effective instruments in designing an optimal scheme for problematic urban issues. Operations research (OR) is well known for the plethora of tools and techniques it offers to address complex and sometimes non-intuitive problems [25]. Much of the work in urban operations research conducted was summarized and extended in Larson and Odoni [36] and its updated edition in 2007. During the last few years, urban operations research has extended its scope to include notable environmental applications. Several operations research methods and tools—such as optimization techniques, control theory, and interactive strategic analysis—are employed either to identify the type and intensity of environmental problems or to assist environmental planners to effectively cope with these problems [18]. For example, an application of OR in Israel’s water sector can be found in Shamir [46]. In Daniel et al. [18], they specified the contribution of OR to environmental planning. They illustrated that OR’s relationship with environmental planning was dynamic and interactive.

Urban operations research has also been applied to solve conflicts where different agents use a common resource. This kind of problem includes transboundary air/water pollution and so on; for instance, a cooperative differential game model of transboundary industrial pollution was presented by Yeung [51]. Li [38] extended Yeung’s model to a more general model, in which emission permits trading is taken into account. More examples can be found in Golden and Wasil [26], ReVelle and Hugh [45], Tulkens [49], Chander and Tulkens [12], Yeung et al. [54], and a special volume of Annals of Operations Research on collaborative environmental management and modeling edited by Haurie et al. [31].

Urban operations research techniques solve problems with analytical OR methods and help decision-makers decompose problems into basic components and solve them via analytical and numerical techniques. In designing an optimal scheme for urban problems, urban operations research has been proven to be one of the most effective instruments.

In this study, we propose to develop an urban operations research model for collaborative electricity consumption management in a region with clustered cities. The main objective of this study is to conduct a cooperative scheme that internalizes the external cost of electricity consumption through optimal electricity consumption reduction. The study attempts to (i) explore the relation between electricity consumption and environmental costs, (ii) construct an urban operations research model of collaborative electricity consumption management for practicable applications, and (iii) set up a scheme to compensate the externalities of electricity consumption.

This analysis is the first urban operations research model for collaborative electricity consumption management, which internalizes the external costs of electricity consumption. This study is also the first attempt to derive the optimal level of electricity consumption within regional collaboration. This is also the first time that a scheme to compensate for the externalities is proposed to ensure that the agreed-upon optimality principle could be maintained throughout the entire duration of cooperation.

The organization of the paper is as follows: Section 2 presents an urban operations research model for collaborative electricity consumption management. A numerical application of this model of the PRD region is conducted in Section 3. Concluding remarks are given in Section 4.

An Operations Research Modeling Framework for Electricity Consumption Reduction

In this section, we will investigate the market-implied relationship between GDP and electricity consumption’s variation, and each individual city’s external cost of electricity consumption. The optimal electricity consumption cut for each individual city will be computed and a compensation plan will be constructed.

Market-Implied Implicit Valuation of Electricity Consumption

Let us consider the case of n asymmetric cities in a region. We first investigate the market-implied relationship between GDP and electricity consumption’s variation in these cities. We denote a change in the electricity consumption of city i at time t by \( {\omega}_t^i\equiv \varDelta {C}_t^i \). Applying the concept of the elasticity of electricity consumption [1, 2, 10], which is the ratio of the percentage change in electricity consumption to the percentage change in GDP, we can define the elasticity of electricity consumption as:

$$ \xi =\frac{\frac{\omega_t^i}{\left[{C}_t^i+\left({C}_t^i+{\omega}_t^i\right)\right]}}{\frac{Y_t^i- GD{P}_t^i}{Y_t^i+ GD{P}_t^i}} $$
(1)

where \( {C}_t^i \) is the electricity consumption of city i in year t; \( GD{P}_t^i \) is the GDP of city i in year t; and \( {Y}_t^i \) is the GDP of city i if its electricity consumption changes from \( {C}_t^i \) to \( {C}_t^i+{\omega}_t^i \). In particular, \( \frac{\omega_t^i}{\left[{C}_t^i+\left({C}_t^i+{\omega}_t^i\right)\right]} \) is the percentage change in electricity consumption and \( \frac{Y_t^i- GD{P}_t^i}{Y_t^i+ GD{P}_t^i} \) is the percentage change in GDP.

Using (1) the GDP of city i can be expressed as:

$$ {Y}_t^i=\frac{2\xi {C}_t^i+\left(\xi +1\right){\omega}_t^i}{2\xi {C}_t^i+\left(\xi -1\right){\omega}_t^i} GD{P}_t^i\equiv {Y}_t^i\left({\omega}_t^i\right) $$
(2)

Implicit valuation is one form of an implicit cost, which is represented by the lost opportunity in the use of a resource. The implicit valuation can be thought of as the opportunity cost related to undertaking a certain project or decision (see also [11, 39]). To reflect that the observed GDP and electricity consumption in a city are consistent with a market equilibrium outcome, we use \( {\phi}_t^i\left({C}_t^i+{\omega i}_t\right) \) to denote the market-implied implicit valuation of electricity consumption. The derivative of \( {\phi}_t^i\left({C}_t^i+{\omega i}_t\right) \) with respect to \( {\omega}_t^i \), \( \frac{d{\phi}_t^i\left({C}_t^i+{\omega i}_t\right)}{d\left({\omega i}_t\right)} \), yields the marginal market-implied implicit valuation of electricity consumption to city i.

For a market equilibrium, the marginal value product of electricity consumption has to be equal to its marginal market-implied implicit valuation, that is:

$$ \frac{d{Y}_t^i}{d{\omega}_t^i}=\frac{d{\phi}_t^i\left({C}_t^i+{\omega i}_t\right)}{d{\omega}_t^i} $$
(3)

The derivative of \( {Y}_t^i\left({\omega}_t^i\right) \) with respect to \( {\omega}_t^i \), \( \frac{d{Y}_t^i\left({\omega}_t^i\right)}{d{\omega}_t^i} \), yields the marginal value product of electricity consumption to city i, reflecting the addition to its GDP by the additional electricity consumed. By definition, the marginal value product of an input (factor of production) is the extra output that can be produced by using one more unit of the input (for instance, in our case, the difference in GDP when electricity consumption is increased by \( {\omega}_t^i \)), assuming that the quantities of no other inputs to production change [24].

Invoking (2), we perform the differentiation operator of the left side of Eq. (3):

$$ \frac{d{Y}_t^i\left({\omega}_t^i\right)}{d{\omega}_t^i}=\frac{4\xi {C}_t^i}{{\left[2\xi {C}_t^i+\left(\xi -1\right){\omega}_t^i\right]}^2} GD{P}_t^i $$
(4)

Using a linear approximation of the market-implied value, \( {\phi}_t^i\left({C}_t^i+{\omega i}_t\right) \) \( ={\hat{\phi}}_t^i\left({C}_t^i+{\omega i}_t\right) \), where \( {\hat{\phi}}_t^i \) is a positive constant, we perform the differentiation operator of the right side of Eq. (3):

$$ \frac{d{\phi}_t^i\left({C}_t^i+{\omega i}_t\right)}{d{\omega}_t^i}={\hat{\phi}}_t^i $$
(5)

Substituting (4) and (5) into Eq. (3), we have:

$$ \frac{4\xi {C}_t^i}{{\left[2\xi {C}_t^i+\left(\xi -1\right){\omega}_t^i\right]}^2} GD{P}_t^i={\hat{\phi}}_t^i $$
(6)

In a market equilibrium, the observed GDP and electricity consumption are consistent optimal choices of the market, under the implicit valuation of electricity consumption. Hence, there is no need to make any change, which leads to \( {\omega}_t^i=0 \). Solving (6), the market-implied marginal implicit valuation of electricity consumption can be obtained as:

$$ {\hat{\phi}}_t^i=\frac{GD{P}_t^i}{\xi {C}_t^i} $$
(7)

External Cost of Electricity Consumption and Electricity Levy

In economics, an externality is a cost or benefit that is not included in the current pricing system and is incurred by a party who was not involved as either a buyer or seller of the goods or services causing the cost or benefit [22, 29, 41, 42, 50]. Externalities of electricity are environmental and social costs that are not accounted for in the market price of electricity (European Commission, 1999; [5, 19]).

To internalize the costs of environmental degradation, pollution tax is one of the economic approaches that is applied in environmental management. Pollution taxes provide effective incentives to reduce emissions and improve the environmental conditions, minimize the total abatement costs, internalize the environmental costs, and provide a source of revenue (see [4, 8, 21, 28, 34, 48]). In this study, we adopt a pollution tax levied on electricity consumption rather than on the total amount of pollution emission.

Individual City’s External Cost of Electricity Consumption

In this study, we focus on the external cost of electricity consumption instead of the external cost of production. There are three compelling reasons. First, the external cost of electricity consumption covers a wider spectrum than the cost of electricity production. Second, as stated in Weinzettel et al. [50], the external cost of production may not reflect the actual damage if there are exports and imports of electricity. Finally, consumer responsibility for electricity emissions contributes to enhanced electricity use efficiency by consumers, which will lead to actual reductions in electricity production [6].

None of the current estimates of the external costs of electricity include all the effects of environmental damage. In this study, we follow the indicator suggested in European Environment Agency [23], which included three components (climate change damage costs associated with emissions of CO2; damage costs (such as impacts on health, crops) associated with other air pollutants; and other non-environmental social costs for non-fossil electricity–generating technologies), and define c as external costs per unit of electricity consumption. Let \( {C}_t^i \) be the electricity consumption of city i in year t and then city i’s own external costs in year t is \( c{\eta}_t^i{C}_t^i \), where \( {\eta}_t^i \)is the percentage of electricity that comes from non-renewable sources.

In a region where the cities are closely clustered together, the spillover of pollution among the cities is intensive. Pollution stocks in a city are contributed by its own emission, and by its surrounded cities as well (see [38, 51, 52]). The total damage from electricity consumption for a city would not just include the external costs of electricity incurred by its own consumption but also the spillover damages it gets from other cities. Let \( {C}_t^{i(j)} \) be city i’s external costs received from city j in year t and then city i’s external costs received from other cities in year t is \( {\sum}_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}{\eta}_t^j{C}_t^{i(j)} \). Combining Weinzettel et al. [50] and Chen and Ye [15], we formulate the total damage from electricity consumption borne by city i:

$$ {TD}_t^{i(j)}=c{\eta}_t^i{C}_t^i+\sum \limits_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}{\eta}_t^j{C}_t^{i(j)} $$
(8)

for i, j ∈ N and j ≠ i,

where \( c{\eta}_t^i{C}_t^i \) is city i’s own external costs in year t and \( {\sum}_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}{\eta}_t^j{C}_t^{i(j)} \) is city i’s external costs received from other cities in year t; \( {\eta}_t^j \) is the percentage of electricity that comes from non-renewable sources at year t.

Equilibrium Under an Electricity Levy

Pigouvian tax, first proposed by Pigou [44], is a tax placed on any good which creates negative externalities and aims to make the price of the good equal to the social marginal cost and create a more socially efficient allocation of resources. In the context of Pigouvian tax, an electricity levy is a tax whose rate is set to equal the total damage caused by the electricity consumption.

In this study, we consider an inter-city collaborative environmental scheme in which an electricity consumption levy equaling the unit external cost c is imposed across all the cities to internalize the externalities of electricity generated from non-renewable resources. In an equilibrium under the electricity consumption levy scheme, the marginal value product has to be equal to the marginal market-implied implicit valuation with the levy included, that is:

$$ \frac{d{Y}_t^i}{d{\omega}_t^i}=\frac{d\left({\phi}_t^i+{\eta}_t^ic\right)\left({C}_t^i+{\omega i}_t\right)}{d{\omega}_t^i} $$
(9)

The market equilibrium under a levy implied by (2.9) can be obtained by solving the problem which maximizes:

$$ {B}_t^i= GD{P}_t^i-\left({\hat{\phi}}_t^i+{\eta}_t^ic\right){C}_t^i={Y}_t^i-\left({\hat{\phi}}_t^i+{\eta}_t^ic\right)\left({C}_t^i+{\omega i}_t\right) $$
(10)

because the first-order condition for a maximum of \( {B}_t^i \) is condition (9).

In Eq. (10), \( {B}_t^i \) is the market equilibrium of city i, which is its GDP net of the market-implied electricity cost and levy c; \( GD{P}_t^i \) is the GDP of city i with an electricity consumption equal to \( {C}_t^i \); and \( {Y}_t^i \) is the GDP of city i with an electricity consumption equal to \( \left({C}_t^i+{\omega}_t^i\right) \).

Now consider the case when a city would like to improve its environmental conditions by reducing its electricity consumption. A notation \( {\varpi}_t^i \) is used to denote the reduction in the amount of electricity consumed by city i at period t, and \( {\hat{Y}}_t^i \) is used to denote the GDP of city i where the electricity consumption was reduced by \( {\varpi}_t^i \). Applying the concept of the elasticity of electricity consumption, we have:

$$ \xi =\frac{\frac{\left({C}_t^i-{\varpi}_t^i\right)-{C}_t^{\mathrm{i}}}{\left({C}_t^i-{\varpi}_t^i\right)+{C}_t^i}}{\frac{{\hat{Y}}_t^i- GD{P}_t^i}{{\hat{Y}}_t^i+ GD{P}_t^i}}=\frac{\frac{-{\varpi}_t^i}{2{C}_t^i-{\varpi}_t^i}}{\frac{{\hat{Y}}_t^i- GD{P}_t^i}{{\hat{Y}}_t^i+ GD{P}_t^i}} $$
(11)

where ξ is the elasticity of electricity consumption; \( {C}_t^i \) is the electricity consumption of city i in year t; \( GD{P}_t^i \) is the GDP of city i when the city’s electricity consumption is \( {C}_t^i \) in year t; and \( {\hat{Y}}_t^i \) is the GDP of city i when the electricity consumption is \( {C}_t^i-{\varpi}_t^i \). In particular, \( \frac{\left[\left({C}_t^i-{\varpi}_t^i\right)-{C}_t^i\right]}{\left[\left({C}_t^i-{\varpi}_t^i\right)+{C}_t^i\right]} \) is the percentage change in electricity consumption reduction, and \( \frac{{\hat{Y}}_t^i- GD{P}_t^i}{{\hat{Y}}_t^i+ GD{P}_t^i} \) is the percentage change in GDP.

Using (11) the GDP of city i can be expressed as:

$$ {\hat{Y}}_t^i=\frac{2\xi {C}_t^i-\left(\xi +1\right){\omega}_t^i}{2\xi {C}_t^i-\left(\xi -1\right){\omega}_t^i} GD{P}_t^i\equiv {\hat{Y}}_t^i\left({\varpi}_t^i\right) $$
(12)

The condition in (12) shows the relation between the GDP of a city and its reduction in electricity consumption: city i’s GDP changes from \( GD{P}_t^i \) to \( {\hat{Y}}_t^i \) if its electricity consumption reduces from \( {C}_t^i \) to \( \left({C}_t^i-{\varpi}_t^i\right) \).

The market equilibrium of city i under the levy implied can be obtained by solving the problem which maximizes:

$$ {\hat{Y}}_t^i\left({\varpi}_t^i\right)-\left({\hat{\phi}}_t^i+{\eta}_t^ic\right)\left({C}_t^i-{\varpi}_t^i\right) $$
(13)

where \( {\hat{Y}}_t^i\left({\varpi}_t^i\right) \) is given in (12).

Optimal Electricity Consumption Cut Under Cooperation Scheme

Now consider the case when all the cities in a region want to collaborate and tackle the pollution problem together, by reducing their electricity consumption. We derive the optimal electricity consumption under the levy c. Substitute (12) into (13); the market equilibrium of city i becomes:

$$ \frac{2\xi {C}_t^i-\left(\xi +1\right){\omega}_t^i}{2\xi {C}_t^i-\left(\xi -1\right){\omega}_t^i} GD{P}_t^i-\left({\hat{\phi}}_t^i+{\eta}_t^ic\right)\left({C}_t^i-{\varpi}_t^i\right) $$
(14)

The optimal electricity consumption reduction can be obtained by solving the problem:

$$ \underset{\varpi_t^i}{\max}\left[\frac{2\xi {C}_t^i-\left(\xi +1\right){\omega}_t^i}{2\xi {C}_t^i-\left(\xi -1\right){\omega}_t^i} GD{P}_t^i-\left({\hat{\phi}}_t^i+{\eta}_t^ic\right)\left({C}_t^i-{\varpi}_t^i\right)\right] $$
(15)

First-order condition for a maximum in the problem (15) yields:

$$ \frac{-4\xi {C}_t^i}{{\left[2\xi {C}_t^i-\left(\xi -1\right){\varpi}_t^i\right]}^2} GD{P}_t^i+{\hat{\phi}}_t^i+{\eta}_t^ic=0 $$
(16)

We denote \( {\varpi}_t^{\ast}\equiv \left[{\varpi}_t^{1\ast },{\varpi}_t^{2\ast },\cdots, {\varpi}_t^{n\ast}\right] \) as the solution to (16); then, \( {\varpi}_t^{\ast } \) is the set of the optimal electricity consumption cuts. The optimal electricity consumption cut for each city in a region can be obtained in Proposition 1.

Proposition 1

The optimal electricity consumption cut for city i in a region is:

$$ {\varpi}_t^{i\ast }=\left[2\xi {\mathrm{C}}_t^i-\sqrt{\frac{4\xi {C}_t^i GD{P}_t^i}{{\hat{\phi}}_t^i+{\eta}_t^ic}}\right]/\left(\xi -1\right) $$
(17)

for i ∈ N and ξ > 1.

Proof. See Appendix 1.

Moreover, under the cooperation agreement, the total damage (own plus spillover from cities) borne by each city with an optimal electricity consumption cut for each city can be expressed as:

$$ {TD}_t^{i(j)\ast }=c{\eta}_t^i{C}_t^{i\ast }+\sum \limits_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}{\eta}_t^j{C}_t^{i(j)\ast } $$
(18)

for i, j ∈ N and j ≠ i,

where\( c{\eta}_t^i{C}_t^{i\ast } \) is city i’s own external costs with its optimal electricity consumption cut and \( {\sum}_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}{\eta}_t^j{C}_t^{i(j)\ast } \) is city i’s external costs received from other cities when the other cities are part of the cooperation scheme with optimal electricity consumption.

Inter-city Compensation

From Section 2.2.1, we know that external costs are not just incurred from a city’s own electricity consumption but also from other cities’ spillover. In order to offer a long-term solution and guarantee that all participants will always be better off within the entire duration of the agreement, we propose a compensation scheme in this subsection. Under the compensation scheme, spillover external cost of electricity consumption between cities in the planning years under cooperation will be calibrated. Net damage of individual cities caused by spillover external cost of electricity consumption in the planning years under cooperation will be computed, and compensation for all the cities in the region will be designed.

General Form of Cost Allocation Scheme

One of the most commonly used cost allocation schemes in practical policy making is proportional sharing and its extension in cost allocation (see Pennisi [43], Si et al. [47] and Harigovindan et al. [30]). Following Si et al. [47], supposing there are n units (cities) jointly using a common platform, whose corresponding fixed cost to be shared is a positive value R, and the allocated cost to unit j is denoted as Rj, we have:

$$ \sum \limits_{j=1}^n{R}_j=R,{R}_j\ge 0,\forall j $$
(19)

It ensures that the allocated costs precisely sum to R, and the amount of the allocated cost Rj to each unit (city) is from zero to R.

Based on the proportional sharing method, each unit is allocated a share of the fixed cost in direct proportion to the benefit it makes to the common platform in the case where there is no external spillover. Suppose the benefit of unit j from the use of the common platform is yj, j = 1, 2, …n, then, the proportional sharing criteria suggests the allocated cost for unit j should be:

$$ {R}_j=\frac{y_j}{\sum_{j=1}^n{y}_j}R $$
(20)

for j = 1, 2, …n.

Besides its simple computation of the allocation, formula (20) has at least two other advantages:

  1. (a)

    It seems fair for the use of exactly the same weight to yj’s use, j = 1, 2, …n, across all the units. To further illustrate this point, we change the formula (20) to the following equation:

$$ {\displaystyle \begin{array}{c}{R}_j={u}_j{y}_j,j=1,2,\dots n.\\ {}{\sum}_{j=1}^n{R}_j=R,{u}_j,{y}_j\ge 0,\forall j\end{array}} $$
(21)

In (21), all yj, j = 1, 2, …n, are attached with a common weight u, and actually, uis uniquely determined as \( {u}_j=R/{\sum}_{j=1}^n{y}_j \) in this case;

  1. b

    There is an implicit assumption that the more earnings a unit gains from the common platform, the more allocated cost it should afford. The last constraint uj ≥ 0 in (21) makes this implicit assumption effective.

Compensation Formula

In this subsection, we aim to construct a compensation formula and consider allocating city i’s levy revenue between city i and the cities which received a net spillover. We first consider the difference between the spillover of external cost generated by city i to the neighboring city j and the spillover of external cost city i received from city j, that is:

$$ {\vartheta}_t^{i(j)}= IS{O}_{i,j}\times c{\eta}_t^i{C}_t^{i\ast }- IS{O}_{j,i}\times c{\eta}_t^j{C}_t^{j\ast } $$
(22)

where ISOi, j is the intensity of spillover from city i to city j and ISOj, i is the intensity of spillover from city j to city i.

Furthermore, we define the net spillover generated by city i to city j as the value of \( {\vartheta}_t^{i(j)} \) which is positive, that is:

$$ {\vartheta}_t^{i(j)+}= IS{O}_{i,j}\times c{\eta}_t^i{C}_t^{i\ast }- IS{O}_{j,i}\times c{\eta}_t^j{C}_t^{j\ast }>0 $$
(23)

The total own damage and net spillover costs generated by city i, \( {T}_t^i, \) becomes the sum of its own external cost and the net spillover generated to neighboring cities:

$$ {T}_t^i=c{\eta}_t^i{C}_t^{i\ast }+\sum \limits_{j\in {N}^i}{\vartheta}_t^{i(j)+} $$
(24)

The levy on electricity consumption in city i yields the levy revenue:

$$ RE{V}_t^i=c{\eta}_t^i{C^{\ast}}_t^{\mathrm{i}}. $$
(25)

To construct a compensation formula, we consider allocating city i’s levy revenue between city i and the cities which received a net spillover from it. It involves a transfer payment from city i to city j, if \( {\vartheta}_t^{i(j)}>0 \), as follows:

$$ T{P}_t^{i(j)}=\frac{\vartheta_t^{i(j)}}{T_t^i} RE{V}_t^i=\frac{\vartheta_t^{i(j)}}{c{\eta}_t^i{C}_t^{i\ast }+{\sum}_{j\in {N}^i}{\vartheta}_t^{i(j)+}} RE{V}_t^i $$
(26)

The main spirit of the proportional sharing method is that each unit is allocated a share of the fixed cost, in direct proportion to the benefit it makes of the common platform in the case. In particular, the transfer payment \( T{P}_t^{i(j)} \) shows that transfer payments received by city j are proportional to the damage from city i’s electricity consumption.

Numerical Application and Implementation of the Operations Research Model in the PRD Region, China

The Pearl River Delta (PRD) region in China has been developed into a large industrial center since the onset of China’s economic reforms in the late 1970s. PRD region has experienced more than three decades of unprecedented economic growth. Consumption of electricity in this region increased sharply due to the booming economic growth. The environmental impact from power development is a major source of air problems and the emission share of the power sector increased along with power demand growth. The PRD region is an example of the fact that booming economic growth is inevitably associated with environmental issues, among which the deterioration in air quality is the most common one [40]. A major contributor to air pollution and acid rain in the PRD region is the use of electricity [32]. Moreover, in a compact region like the PRD, air pollutants can easily transfer from one city to another.

In this section, we provide a numerical application of the implementation of the operations research model. We consider the case where there are 11 cities in the PRD region: Guangzhou, Shenzhen, Zhuhai, Foshan, Jiangmen, Zhaoqing, Huizhou, Dongguan, Zhongshan, Hong Kong, and Macao (GZ, SZ, ZH, FS, JM, ZQ, HZ, DG, ZS, HK, and MC). These 11 cities are illustrated in Fig. 1 (see Appendix Fig 1). The economic structures of the PRD cities have remained stable since 2000, and the postulate that the “macro-structure” of the cities remains unchanged is adopted.

Market-Implied Implicit Valuation of Electricity Consumption in the PRD Region

As we know, the elasticity of electricity consumption is an index which measures the ratio of the percentage change in electricity consumption to the percentage change in GDP. Equation (1) can be expressed as:

$$ \xi =\frac{\frac{C_t-{C}_{t-1}}{\frac{1}{2}\left({C}_t+{C}_{t-1}\right)}}{\frac{GD{P}_t- GD{P}_{t-1}}{\frac{1}{2}\left( GD{P}_t+ GD{P}_{t-1}\right)}}=\frac{\frac{C_t-{C}_{t-1}}{\left({C}_t+{C}_{t-1}\right)}}{\frac{GD{P}_t- GD{P}_{t-1}}{\left( GD{P}_t+ GD{P}_{t-1}\right)}} $$
(27)

where Ct is the electricity consumption in year t and GDPt is the GDP in year t; \( \frac{C_t-{C}_{t-1}}{\frac{1}{2}\left({C}_t+{C}_{t-1}\right)} \) is the growth rate of electricity consumption; and \( \frac{GD{P}_t- GD{P}_{t-1}}{\frac{1}{2}\left( GD{P}_t+ GD{P}_{t-1}\right)} \) is the growth rate of GDP.

Using the GDP and electricity consumption of China (price base year = 2016) from 2000 to 2015, the growth rate in GDP and electricity consumption can be computed. Applying Eq. (29), the elasticity of electricity consumption in China is calculated. The average elasticity of electricity consumption in China is 1.03.

Using ξ = 1.03 as a proxy of the elasticity of electricity consumption in the PRD region, the forecast of the electricity consumption by the 11 PRD cities in the planning years is:

$$ {C}_t^i={C}_{t-1}^i\left(1+1.03\frac{GD{P}_t^i- GD{P}_{t-1}^i}{GD{P}_t^i+ GD{P}_{t-1}^i}\right)\div \left(1-1.03\frac{GD{P}_t^i- GD{P}_{t-1}^i}{GD{P}_t^i+ GD{P}_{t-1}^i}\right) $$
(28)

for i ∈ {DG, FS, GZ, HK, HZ, JM, MC, SZ, ZQ, ZS, ZH}; \( GD{P}_t^i \) is GDP of city i in year t; and \( {C}_t^i \) is the electricity consumption of city i in year t. Using Eq. (30), the projected GDP of the eleven cities of the PRD, and the electricity consumption of each city, and the electricity consumption of the PRD cities for 2020 and 2021 can be forecast. The projected data are presented in Table 1.

Table 1 Electricity consumption forecast (unit: billion RMB)

Moreover, the market-implied marginal implicit valuation of electricity consumption in Eq. (7) becomes:

$$ {\hat{\phi}}_t^i=0.970\frac{GD{P}_t^i}{C_t^i} $$
(29)

Invoking (31) and the forecasted GDP and electricity consumption \( {C}_t^i \) of each city, the marginal market-implied implicit valuation of electricity consumption \( {\hat{\phi}}_t^i \) for the eleven PRD cities can be obtained in Table 2.

Table 2 Marginal market-implied implicit valuation of electricity consumption (unit: billion RMB) \( {\hat{\phi}}_t^{i.} \)

Individual City’s External Cost of Electricity Consumption in the PRD Region

Since the PRD cities are closely clustered together, the spillover of pollution among the cities is intensive. The distribution of air pollution and attendant impacts vary across Chinese provinces due to differences in physical geography, meteorology, population density, level of economic development, production structure, and available technologies [27, 33]. According to Chen and Ye [15], a city’s average air pollution index is expected to increase by 0.40–0.51 if the average air pollution index in its surrounding cities increases by one unit. City i’s air pollution index is positively correlated with its own and all of its surrounding cities’ air pollution level. Base on the model in Chen and Ye [15], the spillover effect from city is surrounding city j is:

$$ IS{O}_{i,j}=\frac{1}{D_{i,j}} $$

where Di, j is the geographical distance between the centroid of city i and the centroid of city j and Di, j < 500.

Using the distance between the PRD cities, we compute the ISO of cities (see Table 3).

Table 3 Spillover effect between each pair of cities

A quantification of the environmental impacts from electricity in monetary terms has become the core of many research projects, and their results provide relevant indicators for policy [50]. Based on the spillover effect between each pair of cities, we calculate the spillover social cost of electricity borne by city i due to city j with the following equation:

$$ {\hat{C}}_t^{i(j)}= IS{O}_{i,j}\times c{\eta}_t^j{C}_t^j $$
(30)

for i, j ∈ {DG, FS, GZ, HK, HZ, JM, MC, SZ, ZQ, ZS, ZH} and j ≠ i, where \( {\hat{C}}_t^{i(j)} \) is the spillover social cost of electricity consumption that city i received from city j; \( {C}_t^j \) is electricity consumption of city j in year t; \( c{\eta}_t^j{C}_t^j \) is the social cost of electricity consumption of city j in year t; and ISOi, j is the spillover effect between cities i and j. In PRD region, the spillover external cost of electricity borne by city i due to all of the other cities is:

$$ \sum \limits_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}{\hat{C}}_t^{i(j)}=\sum \limits_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}} IS{O}_{i,j}\times c{\eta}_t^j{C}_t^j=\sum \limits_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}\frac{1}{D_{i,j}}\times c{\eta}_t^j{C}_t^j $$
(31)

The total damage from electricity consumption borne by each city can be obtained from the following equation:

$$ {\hat{TD}}_t^{i(j)}=c{\eta}_t^i{C}_t^i+\sum \limits_{\begin{array}{c}j\in N\\ {}j\ne i\end{array}}{\hat{C}}_t^{i(j)} $$
(32)

where \( c{\eta}_t^i{C}_t^i \) is city i’s own social costs in year t and \( {\sum}_{\begin{array}{c}j\in {N}^i\\ {}j\ne i\end{array}}{\hat{C}}_t^{i(j)} \) is city i’s external costs received from other cities in year t.

To estimate the external costs of electricity consumption, this study adopts a unit external cost of electricity consumption c. Reviews in Zhang [55] show that external costs of electricity consumption in different countries differ widely, ranging from 13 to 700% of electricity price. This study adopts the lower quartile of this range, which is 13% of electricity price, as the unit external cost of electricity consumption. According to China Energy Portal [16], China’s renewable energy sector is growing very fast. The proportion of electricity generated from non-renewable sources dropped from 80.7% in 2000 to 68.9% in 2019. By using regression analysis to predict the reference η for the year 2020 and 2021, we have η2020 = 71.1% and η2021 = 70.4%.

Using (32) and (33), we obtain estimates for the social cost of each city’s own electricity consumption, the spillover external costs of electricity consumption borne by each city due to all other cities, and the total damage (own plus spillover from other cities) of electricity consumption borne by each PRD city (Table 4).

Table 4 Total Ddamage of Eelectricity Cconsumption Bborne by Eeach Ccity (Uunit: billion RMB)

Optimal Electricity Consumption Cut Under Cooperation Scheme in the PRD Region

Involving ξ =1.03, the optimal electricity consumption cut in Proposition 1 can be expressed as:

$$ {\varpi}_t^{i\ast }=\left[2.06{C}_t^i-\sqrt{\frac{4.12{C}_t^i GD{P}_t^i}{{\hat{\phi}}_t^i+{\eta}_t^ic}}\right]\div 0.03 $$
(33)

Applying (34), the optimal electricity consumption cut for each city is computed and listed in Table 5. The reductions in electricity consumption for each city in percentage terms are also presented in this table.

Table 5 Optimal electricity consumption cut under cooperation

Moreover, under the cooperation agreement, the total damage (own plus spillover from other cities) borne by each city in the PRD region, with the optimal electricity consumption cut for each city, can be expressed as:

$$ {\hat{TD}}_t^{i(j)\ast }=c{\eta}_t^i{C}_t^{i\ast }+\sum \limits_{\begin{array}{c}j\in {N}^i\\ {}j\ne i\end{array}}{\hat{C}}_t^{i(j)\ast }=c{\eta}_t^i{C}_t^{i\ast }+\sum \limits_{\begin{array}{c}j\in {N}^i\\ {}j\ne i\end{array}} IS{O}_{i,j}\times c{\eta}_t^j{C}_t^{j\ast } $$
(34)

where \( c{\eta}_t^i{C}_t^{i\ast } \) is city i’s own external costs with its optimal electricity consumption cut and \( {\sum}_{\begin{array}{c}j\in {N}^i\\ {}j\ne i\end{array}}{\hat{C}}_t^{i(j)\ast } \)is city i’s external costs received from its neighboring cities when they are part of the cooperation scheme with optimal electricity consumption.

Using Eq. (35), we obtain estimates of the total damage (own plus spillover from other cities) of electricity under the cooperation agreement accrued to each city in Table 6.

Table 6 Total damage (own plus spillover) of electricity consumption borne by each city under cooperation (unit: billion RMB)

Reductions in the external cost from a city’s own electricity consumption from non-cooperation to cooperative scheme are presented in Table 7.

Table 7 Reduction in social cost from own electricity consumption

Reductions in the spillover external cost borne by city i due to other cities from non-cooperation to cooperative scheme are presented in Table 8.

Table 8 Reduction in spillover external cost of electricity consumption borne by each city

Reductions in the total damage (own plus spillover from other cities) borne by each city from non-cooperation to cooperative scheme are presented in Table 9.

Table 9 Reduction in total (own plus spillover) damage/cost of electricity consumption borne by each city

Inter-city Compensation in the PRD Region

Inter-city Spillover Pattern

Under the cooperation agreement, we measure the spillover social costs of electricity consumption borne by city i due to city j by:

$$ {\hat{C}}_t^{i(j)\ast }= IS{O}_{i,j}\times c{\eta}_t^j{\hat{C}}_t^{j\ast } $$
(35)

where \( {\hat{C}}_t^{i(j)\ast } \) is the spillover external cost of electricity consumption that city i received from city j under the cooperation scheme.

Using Eq. (36), under the cooperative scheme, the spillover external cost of electricity consumption generated to other city and received from other city are shown in Table 10. Each pair of cities would generate a spillover external cost of electricity consumption to each other.

Table 10 Inter-city spillover of external cost of electricity consumption under cooperation (unit: billion RMB)

Compensation Plan

Let us consider the difference between the spillover of external cost generated by city i to city j and the spillover of external cost city i received from city j by Eq. (22), for i, j ∈ {DG, FS, GZ, HK, HZ, JM, MC, SZ, ZQ, ZS, ZH}, and j ≠ i.

For example, in 2020, under the cooperative scheme, Guangzhou would generate 0.369 billion RMB of external cost of electricity to Foshan, while at the same time, it would receive 0.253 billion RMB of external cost of electricity from Foshan. The difference in the spillover for this pair of cities, i.e., Guangzhou(Foshan), is:

$$ {\displaystyle \begin{array}{c}{\vartheta}_{2020}^{GZ(FS)}= IS{O}_{GZ, FS}\times c{\eta}_{2020}^{GZ}{C}_{2020}^{GZ\ast }- IS{O}_{FS, GZ}\times c{\eta}_{2020}^{FS}{\eta}^{FS}{C}_{2020}^{FS\ast}\\ {}=0.369-0.253=0.116\ \mathrm{billion}\ \mathrm{RMB}.\end{array}} $$

In 2018, under the cooperative scheme, Foshan would generate 2 billion RMB of external cost of electricity to Guangzhou, and it would receive 3.576 billion of external cost from Guangzhou. The difference in the spillover for this pair of cities, i.e., Foshan(Guangzhou), is:

$$ {\displaystyle \begin{array}{c}{\vartheta}_{2020}^{FS(GZ)}= IS{O}_{FS, GZ}\times c{\eta}_{2020}^{FS}{C}_{2020}^{FS\ast }- IS{O}_{GZ, FS}\times c{\eta}_{2020}^{GZ}{C}_{2020}^{GZ\ast}\\ {}=0.253-0.369=-0.116\ \mathrm{billion}\ \mathrm{RMB}.\end{array}} $$

Involving Eq. (22), the difference in the spillover of external cost for the pairs of cities under the cooperation scheme is computed and presented in Table 11.

Table 11 Difference in the spillover for pairs of cities under cooperation (unit: billion RMB)

Furthermore, by using Eq. (23), the net spillover generated by city i to city j in the PRD region can be obtained. The total self damage and net spillover costs generated by the PRD cities can also be calculated. Table 12 presents the breakdowns for each city’s total self-damage and net spillover (\( {T}_t^i \)) generated by city i to city j.

Table 12 Breakdowns of total self-damage and net spillover generated (unit: billion RMB)

In this numerical application, the levy of 13% on electricity consumption in the PRD cities (of those generated by thermal power) yields the levy revenue. Table 13 presents each city’s electricity levy under the cooperation agreement.

Table 13 Electricity levy of each city under cooperation (unit: billion RMB)

Transfer payments from city i to city j among the PRD cities can be obtained by Eq. (26). Transfer payments paid and received by each of the 11 PRD cities are summarized in Table 14 (the number in brackets represents the transfer payment received).

Table 14 Transfer payments paid and received between cities (Unit: billion RMB)

In theory, city i would pay compensation to city j for its net spillover. The total amounts of the transfer payments for each city can be found in Table 14. Projected data from 2020 to 2021 show that, under the cooperation scheme, the five cities, Guangzhou, Shenzhen, Foshan, Dongguan, and Hong Kong, would have to pay for the net spillover external costs of electricity consumption they caused to the other PRD cities. The other six cities of the region, Zhuhai, Jiangmen, Huizhou, Zhaoqing, Zhongshan, and Macao, would receive compensation.

Conclusion

In this study, we construct an optimal and mutual agreeable scheme to reduce electricity consumption in a compact region. This model internalizes the external costs of electricity consumption in the region; derives the optimal level for electricity consumption; and sets up a scheme to compensate the externalities of electricity consumption.

The urban operations research model for regional cooperation in electricity consumption reduction, developed in this study, provides an instrument to deal with the pollution problems in the PRD region. The numerical application involves the collaboration of all the cities in the PRD region and internalizes the external social costs of electricity consumption through implementation of collaborative electricity consumption management and an appropriate reduction in electricity consumption by the PRD region. The numerical application estimates the non-cooperative market outcome and evaluates the external cost of electricity consumption. Then the market-implied condition with the marginal market-implied implicit valuation of electricity consumption has been deliberated. The optimal electricity consumption reduction and levy are thus derived. Under the cooperative scheme which includes individual cities’ external costs of electricity and the level of spillover compensation for each city, transfer payments have been proposed among the PRD cities.

This research represents the first attempt at the application of an urban operations research model for a collaborative management scheme for a reduction in the electricity consumption of a region. It facilitates the exploration of hitherto intractable problems in regional environmental cooperation and establishes solution plans. In particular, the urban operations model is expected to provide practical policy choices.

References

  1. 1.

    Adams FG, Miovic P (1968) On relative fuel efficiency and the output elasticity of energy consumption in Western Europe. J Ind Econ 17(1):41–56

    Article  Google Scholar 

  2. 2.

    Ang BW (1991) A statistical analysis of energy coefficients. Energy Econ 13(2):93–110

    Article  Google Scholar 

  3. 3.

    Antoci A, Russu P, Ticci E (2012) Environmental externalities and immiserizing structural changes in an economy with heterogeneous agents. Ecol Econ 81:80–91

    Article  Google Scholar 

  4. 4.

    Arnold FS (1995) Economic analysis of environmental policy and regulation. Wiley, New York

    Google Scholar 

  5. 5.

    ATSE (2009) The hidden costs of electricity: externalities of power generation in Australia. A report by the Australian Academy Of Technological Sciences And Engineering

  6. 6.

    Bai HT, Zhang YX, Wang H, Huang Y, Xu H (2014) A hybrid method for provincial scale energy-related carbon emission allocation in China. Environ Sci Technol 48(5):2541–2550

  7. 7.

    Banzhaf HS, Chupp BA (2012) Fiscal federalism and Interjurisdictional externalities: new results and an application to US air pollution. J Public Econ 96:449–464

    Article  Google Scholar 

  8. 8.

    Barde J-P (2000) Environmental policy and policy instruments. In: Folmer H, Gabel HL, Aldershot HO (eds) Principles of environmental and resource economics: a guide for students and decision-makers. Hants, E. Elgar

    Google Scholar 

  9. 9.

    Bernow SS, Marron DB (1990) Valuation of environmental externalities for energy planning and operations. Tellus Institute, Boston

    Google Scholar 

  10. 10.

    Brookes LG (1972) More on the output elasticity of energy consumption. J Ind Econ 21(1):83–92

    Article  Google Scholar 

  11. 11.

    Carbaugh RJ (2006) Contemporary economics: an applications approach, 4th edn. Thomson/South-Western, Mason, Ohio

    Google Scholar 

  12. 12.

    Chander P, Tulkens H (1992) Theoretical foundations of negotiations and cost sharing in transfrontier pollution problems. Eur Econ Rev 36(2/3):388–399

    Article  Google Scholar 

  13. 13.

    Chander P, Tulkens H (1995) A core-theoretic solution for the design of cooperative agreements on transfrontier pollution. Int Tax Public Financ 2(2):279–294

    Article  Google Scholar 

  14. 14.

    Chander P, Tulkens H (1997) The Core of an economy with multilateral environmental externalities. Int J Game Theory 26:379–401

    Article  Google Scholar 

  15. 15.

    Chen X and Ye J (2015) When the wind blows: spatial spillover effects of urban air pollution. Environment for Development, EFD DP 15

  16. 16.

    China Energy Portal (2020) 2019 electricity & other energy statistics. Retrieved from. https://chinaenergyportal.org/2019-electricity-other-energy-statistics-preliminary/. Accessed 18 Sept 2020

  17. 17.

    Cian ED, Tavoni M (2012) Do technology externalities justify restrictions on emission permit trading? Resour Energy Econ 34:624–646

    Article  Google Scholar 

  18. 18.

    Daniel SE, Diakoulaki DC, Pappis CP (1997) Operations research and environmental planning. Eur J Oper Res 102:248–263

    Article  Google Scholar 

  19. 19.

    Ding S, Zhang Q, Xu W, Guo H, Xiong S, Jiang H (2006) External costs of electricity generation in China. Environ Inform Arch 4:475–482

    Google Scholar 

  20. 20.

    Do KHP, Dinar A, Mckinney D (2012) Transboundary water management: can issue linkage help mitigate externalities? Int Game Theory Rev 14(1):1250002. https://doi.org/10.1142/S0219198912500028

    Article  Google Scholar 

  21. 21.

    Eskeland G, Kong C (1998) Protecting the environment and the poor: a public goods framework and an application to Indonesia (development research group working paper). World Bank, Washington

    Google Scholar 

  22. 22.

    European Commission (2003) External costs-research results on socio-environmental damages due to electricity and transport. EUR 20198

  23. 23.

    European Environment Agency (2008) EN35 external costs of electricity production. Available online: https://www.eea.europa.eu/data-and-maps/indicators/en35-external-costs-of-electricity-production (accessed on 20 Feb 2013)

  24. 24.

    Frank RH, Bernanke BS (2001) Principles of micro-economics. McGraw-Hill/Irwin, Boston, Mass

    Google Scholar 

  25. 25.

    García-Ortiz A, Amin SM, Wootton JR (1999) Operations research and semantic control: application to advanced traffic management. Int Trans Oper Res 6(1):5–20

    Article  Google Scholar 

  26. 26.

    Golden BL and Wasil, EA. (1994) “Managing fish, forests, wildlife, and water: applications of management science and operations research to natural resource decision problems.” In S.M. Pollock, A.D. Barnett, M.H. Rothkopf (eds.), Handbooks in Operations Research and Management Science: Operations Research and the Public Sector, 6: 289–363

  27. 27.

    Gong ZZ, Zhang XP (2017) Assessment of urban air pollution and spatial spillover effects in China: Cases of 113 key environmental protection cities. J Resour Ecol 8(6):584–594

  28. 28.

    Goulder LH (1995) Environmental taxation and the “double dividend”: a reader’s guide. Int Tax Public Financ 2(2):157–183

    Article  Google Scholar 

  29. 29.

    Griffin JM, Steele HB (1986) Energy economics and policy, 2nd edn. Academic Press; Orlando, Fla

  30. 30.

    Harigovindan VP, Babu AV, Jacob L (2014) Proportional fair resource allocation in vehicle-to-infrastructure networks for drive-thru internet applications. Comput Commun 40:33–50

    Article  Google Scholar 

  31. 31.

    Haurie A, Yeung DWK, Zaccour G (2014) Special volume on collaborative environmental management and Modelling. Ann Oper Res 220(1):1–3

    Article  Google Scholar 

  32. 32.

    Hopkinson L, Stern R (2003) One country, two systems, one smog cross-boundary air pollution policy challenges for Hong Kong and Guangdong. China Environ Ser 6:19–36

  33. 33.

    Jiang ZY, Cheng JP, Liu SL, Pan ZQ (2008) The external cost of coal power chain in China. J China Coal Soc 33(11):1325–1328

    Google Scholar 

  34. 34.

    Kahn JR (1998) The economic approach to environmental and natural resources, 2nd edn. Dryden Press, Fort Worth

    Google Scholar 

  35. 35.

    Kim S-H (2007) Evaluation of negative environmental impacts of electricity generation: neoclassical and institutional approaches. Energy Policy 35:413–423

    Article  Google Scholar 

  36. 36.

    Larson RC, Odoni AR (1981) Urban operations research: logistical and transportation planning methods. Prentice-Hall, Englewood Cliffs

    Google Scholar 

  37. 37.

    Lera-López F, Faulin J, Sánchez M (2012) Determinants of the willingness-to-pay for reducing the environmental impacts of road transportation. Transp Res D 17:215–220

    Article  Google Scholar 

  38. 38.

    Li S (2014) A differential game of transboundary industrial pollution with emission permits trading. J Optim Theory Appl 163:642–659

    Article  Google Scholar 

  39. 39.

    Lipsey RG (1983) An introduction to positive economics, 6th edn. Weidenfeld and Nicolson, London

    Google Scholar 

  40. 40.

    Ma XL, Tao JL (2010) Cross-border environmental governance in the greater Pearl River Delta (GPRD). Int J Environ Stud 67(2):127–136

    Article  Google Scholar 

  41. 41.

    Owen AD (2004) Environmental externalities, market distortions and the economics of renewable energy technologies. Energy J 25:127–156

  42. 42.

    Papandreou AA (1994) Externality and institutions. Clarendon Press, Oxford

  43. 43.

    Pennisi A (1998) Disproportionality indexes and robustness of proportional allocation methods. Elect Stud 17(1):3–19

    Article  Google Scholar 

  44. 44.

    Pigou AC (1924) The economics of welfare, 2nd edn. McMillan, London

    Google Scholar 

  45. 45.

    ReVelle C, Hugh EJ (1994) Models for water and air quality management. In S.M.M. Pollock, Allison D. Barnett, M.H. Rothkopf (eds.), Handbooks in Operations Research and Management Science: Operations Research and the Public Sector, 6:365–401

  46. 46.

    Shamir U (1980) Application of operations research in Israel’s water sector. Eur J Oper Res 5(5):332–345

    Article  Google Scholar 

  47. 47.

    Si X, Liang L, Jia G, Yang L, Wu H, Li Y (2013) Proportional sharing and DEA in allocating the fixed cost. Appl Math Comput 219:6580–6590

    Google Scholar 

  48. 48.

    Toman MA, Withagen C (2000) Accumulative pollution, “clean technology” and policy design. Resour Energy Econ 22:367–384

    Article  Google Scholar 

  49. 49.

    Tulkens H (1979) An economic model of international negotiations relating to transfrontier pollution. In: Krippendorff K (ed) Communication and control in society. Gordon and Breach, New York, pp 199–212

    Google Scholar 

  50. 50.

    Weinzettel J, Havránekb M, Scasny M (2012) A consumption-based Indicator of the external costs of electricity. Ecol Indic 17:68–76

    Article  Google Scholar 

  51. 51.

    Yeung DWK (2007) Dynamically consistent cooperative solution in a differential game of transboundary industrial pollution. J Optim Theory Appl 134:143–160

    Article  Google Scholar 

  52. 52.

    Yeung DWK (2014) Dynamically consistent collaborative environmental management with production technique choices. Ann Oper Res 220(1):181–204. https://doi.org/10.1007/s10479-011-0844-0

    Article  Google Scholar 

  53. 53.

    Yeung DWK, Petrosyan LA (2016) A cooperative dynamic environmental game of subgame consistent clean technology development. Int Game Theory Rev 18(2):164008.01–164008.23

    Article  Google Scholar 

  54. 54.

    Yeung DWK, Petrosyan LA, Zhang YX, Cheung F (2017) BEST SCORES solution to the catastrophe-bound environment. Nova Science Publishers, New York ISBN: 978-1-53610-924-5

    Google Scholar 

  55. 55.

    Zhang YX (2014) Collaborative environmental management in the Pearl River Delta: an urban operation research approach for electricity consumption. University of Hong Kong, Pokfulam, Hong Kong SAR. Retrieved from https://doi.org/10.5353/th_b5328051

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Appendices

Appendix 1: Proof of Proposition 1

The maximization of joint government benefits with electricity consumption controls problem in (15) is a function of \( {\varpi}_t^i \). In order to calculate the maximum, the first-order condition of this function yields:

$$ f{\left({\varpi}_t^i\right)}^{\prime }=\frac{-4\xi {C}_t^i}{{\left[2\xi {C}_t^i-\left(\xi -1\right){\varpi}_t^i\right]}^2} GD{P}_t^i+{\hat{\phi}}_t^i+{\eta}_t^ic=0 $$
(36)

The necessary condition for a relative extremum (maximum or minimum) is that the first-order derivative be zero. This implies:

$$ f{\left({\varpi}_t^i\right)}^{\prime }=\frac{-4\xi {C}_t^i}{{\left[2\xi {C}_t^i-\left(\xi -1\right){\varpi}_t^i\right]}^2} GD{P}_t^i+{\hat{\phi}}_t^i+{\eta}_t^ic=0 $$
(37)

Using \( {\varpi}_t^{i\ast } \) to denote the optimal electricity consumption cut where\( f{\left({\varpi}_t^{h\ast}\right)}^{\prime } \)=0, then \( W\left({\varpi}_t^{i\ast}\right) \) is the extremum.

The second-order condition of the function of \( {\varpi}_t^i \) yields:

$$ f{\left({\varpi}_t^i\right)}^{\prime \prime }=-8\xi {C}_t^i\times GD{P}_t^i\times \left(\xi -1\right)\frac{1}{{\left[2\xi {C}_t^i-\left(\xi -1\right){\varpi}_t^i\right]}^3}. $$

Since \( {C}_t^i \) is the electricity consumption of city i and \( {\varpi}_t^i \) is city i’s electricity consumption cut, \( {C}_t^i \) is always larger than \( {\varpi}_t^i \). If ξ > 1, \( -8\xi {C}_t^i\times GD{P}_t^i\times \left(\xi -1\right)<0 \) and \( \frac{1}{{\left[2\xi {C}_t^i-\left(\xi -1\right){\varpi}_t^i\right]}^3} \)>0. Hence, we have\( f{\left(W\left({\varpi}_t^i\right)\right)}^{\prime \prime }<0 \).

Hence, \( W\left({\varpi}_t^{i\ast}\right) \) is the maximum of joint government benefits when each city has its electricity consumption cut \( {\varpi}_t^{i\ast } \) when ξ > 1.

Working through (37) yields:

$$ {\varpi}_t^i=\left[2\xi {C}_t^i-\sqrt{\frac{4\xi {C}_t^i GD{P}_t^i}{{\hat{\phi}}_t^i+{\eta}_t^ic}}\right]/\left(\xi -1\right) $$
(38)

For notational consistency we denote the set of optimal electricity consumption cuts as:

$$ {\varpi}_t^{i\ast}\equiv \left[{\varpi}_t^{1\ast },{\varpi}_t^{2\ast },\cdots, {\varpi}_t^{n\ast}\right], $$

Hence, the optimal electricity consumption cut for city i is:

$$ {\varpi}_t^{i\ast }=\left[2\xi {C}_t^i-\sqrt{\frac{4\xi {C}_t^i GD{P}_t^i}{{\hat{\phi}}_t^i+{\eta}_t^ic}}\right]/\left(\xi -1\right) $$
(39)

Hence, Proposition 1 follows Q.E.D.

Appendix 2: Fig. 1

Fig. 1
figure1

These 11 cities of PRD Region. Source: Zhang [55]

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Zhang, Y. Regional Collaborative Electricity Consumption Management: an Urban Operations Research Model. SN Oper. Res. Forum 1, 29 (2020). https://doi.org/10.1007/s43069-020-00034-z

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Keywords

  • Urban operations research
  • Electricity consumption
  • Collaborative electricity consumption management
  • Environment