Coordination of bioenergy supply chains under government incentive policies: a game-theoretic analysis

Abstract

Biomass as an abundant renewable energy source can play a vital role in controlling the greenhouse gas emissions. The distributed nature of biomass and its low energy density have complicated the utilization of this cheap and available source of energy. Governments can stimulate the bioenergy industry and remove barriers for adoption of bioenergy by implementing supporting regulations and incentives. In this paper, two types of government incentives, representing direct and indirect incentives, are analyzed and their efficiencies in fostering bioenergy generation are compared. A Stackelberg (leader–follower) game is proposed to formulate the integration of incentives as a bi-level problem in coordination of biomass supply chains. We further illustrate the applicability of the proposed approach through an empirical case study of three Canadian remote communities. The case study demonstrates the effects of incentives on coordination of biomass suppliers and end-user communities and promoting bioenergy share in electricity generation mix of communities. The findings of this study highlight the importance of government’s support, in form of indirect incentives, for provisioning of infrastructure needed for biomass supply and conversion, with a significant impact on increasing the share of bioenergy generation.

Graphical abstract

Appendices

Appendix 1

The supply chain model incorporating a direct incentive scheme for communities is presented by Eqs. (9)–(24).

$${\text{Min}}\;F_{1} = \mathop \sum \limits_{j} \left\{ {\mathop \sum \limits_{t} \left[ {\left( {\left( {\mathop \sum \limits_{j} y_{kj}^{t} B_{kj}^{t} } \right) + (I_{j}^{t} \;a_{j} ) + \left( {LB_{j} \;z_{j}^{t} } \right) + \left( {LD_{j} (D_{j}^{t} - z_{j}^{t} } \right)} \right)} \right]} \right\}$$

(9)

Subject to:

$$I_{j}^{t} = I_{j}^{t - 1} + \mathop \sum \limits_{k} y_{kj}^{t - rp} - \frac{{z_{j}^{t} }}{{fc_{j} }},\;I_{j}^{0} = 0$$

(10)

$$z_{j}^{t} \le \min \;\left( {D_{j}^{t} ,\;Lf_{j} *Z_{j} *720} \right)$$

(11)

$$I_{j}^{t} \le Ib_{j}$$

(12)

$$y_{kj}^{t} ,I_{j}^{t} ,z_{j}^{t} \ge 0$$

(13)

$${\text{Max}}\;F_{2} = \mathop \sum \limits_{i} \left\{ {\mathop \sum \limits_{t} \left[ {\left( {\mathop \sum \limits_{k} X_{ik}^{t} (P_{ik}^{t} + ) - hs_{i} S_{i}^{ } - H_{i} IS_{i}^{t} - \mathop \sum \limits_{k} X_{ik}^{t} T_{ik} } \right)} \right]} \right\}$$

(14)

$$P_{ik}^{t} = P_{i}^{u} - \left( {P_{i}^{u} - P_{i}^{l} } \right)\frac{{X_{ik}^{t} }}{{S_{i}^{ } }}$$

(15)

$$IS_{i}^{t} = IS_{i}^{t - 1} + S_{i}^{ } - \mathop \sum \limits_{k} X_{ik}^{t} IS_{i}^{0} = 0$$

(16)

$$\mathop \sum \limits_{k} X_{ik}^{t} \le S_{i}^{ }$$

(17)

$$IS_{i}^{t} \le S_{i}^{ }$$

(18)

$$P_{ik}^{t} ,\;IS_{i}^{t} \ge 0$$

(19)

$${\text{Max}}\;F_{3} = \mathop \sum \limits_{k} \left\{ {\mathop \sum \limits_{t} \left[ {\left( {\mathop \sum \limits_{j} y_{kj}^{t} B_{kj}^{t} - \mathop \sum \limits_{i} X_{ik}^{t} P_{ik}^{t} - Hc_{k} h_{k}^{t} } \right)} \right] } \right\}$$

(20)

$$B_{kj}^{t} = B_{kj}^{u} - \left( {B_{kj}^{u} - B_{kj}^{l} } \right)\frac{{y_{kj}^{t - rp} }}{{h_{k}^{ } }}$$

(21)

$$h_{k}^{t} = h_{k}^{t - 1} + \mathop \sum \limits_{i} X_{ik}^{t - rs} - \mathop \sum \limits_{j} y_{kj}^{t} ,h_{k}^{0} = 0$$

(22)

$$h_{k}^{t} \le hk\left( k \right)$$

(23)

$$X_{ik}^{t} ,y_{kj}^{t} ,z_{j}^{t} \ge 0\;\left( {{\text{Decision}}\;{\text{Variables}}} \right)$$

(24)

Equations (9)–(13) are representing the optimization problem of communities, which are the leader of the game. To formulate the problem as a bi-level problem, the optimization problem of the suppliers (Eqs. 14–19) and hubs (Eqs. 20–24) is considered as the constraints of the leaders' problem. To formulate hubs’ direct incentive scenario, Eq. (20) is replaced by Eq. (5), and in case of suppliers’ direct incentive scenario, Eq. (14) is replaced by Eq. (3).

Appendix 2

Table of symbols and nomenclatures.

Symbols	Definitions	Units
Sets
i	Set of suppliers
k	Set of hubs
j	Set of energy convertor facilities
t	Time periods
Parameters
${\varvec{T}}_{{{\varvec{ik}}}}$	Transportation cost from supplier ‘i’ to hub ‘k’	$/kg
Si	Capacity of supplier ‘i’	kg
${\varvec{P}}_{{\varvec{i}}}^{{\varvec{u}}}$	Biomass price of supplier ‘i’ without discount	$/kg
${\varvec{P}}_{{\varvec{i}}}^{{\varvec{l}}}$	Biomass price of supplier ‘i’ with discount	$/kg
${\mathbf{hs}}_{{\varvec{i}}}$	Biomass harvesting cost at supplier ‘i’	$/kg
${\mathbf{H}}_{{\varvec{i}}}$	Holding cost for supplier ‘i’	$/kg
${\varvec{h}}_{{\varvec{k}}}^{\user2{ }}$ k	Capacity of hub ‘k’	kg
${\varvec{Hc}}_{{\varvec{k}}}$	Holding cost at hub ‘k’	$/kg
${\varvec{B}}_{{{\varvec{kj}}}}^{{\varvec{u}}}$	Biomass ordering cost from hub ‘k’ to energy convertor ‘j’ without discount	$/kg
${\varvec{B}}_{{{\varvec{kj}}}}^{{\varvec{l}}}$	Biomass ordering cost from hub ‘k’ to energy convertor ‘j’ with discount	$/kg
${\varvec{Ib}}_{{\varvec{j}}}$	Capacity of biomass inventory at energy convertor ‘j’	kg
${\varvec{a}}_{{\varvec{j}}}$	Holding cost at energy convertor ‘j’	$/kg
${\varvec{fc}}_{{\varvec{j}}}$	Conversion rate of biomass to electricity at energy convertor ‘j’	kWh/kg
${\varvec{Lf}}_{{\varvec{j}}}$	Loading factor of energy convertor ‘j’	%
${\varvec{LB}}_{{\varvec{j}}}$	Electricity generation cost from biomass	$/kWh
${\varvec{LD}}_{{\varvec{j}}}$	Electricity generation cost from diesel	$/kWh
${\varvec{D}}_{{\varvec{j}}}^{{\varvec{t}}}$	Demand in energy convertor ‘j’ at time t	kWh
${\varvec{Z}}_{{\varvec{j}}}$	Capacity of electricity generation	kW
${\varvec{rs}}$	Delivery time between supplier ‘i’ and hub ‘k’	Month
${\varvec{rp}}$	Delivery time between hub ‘k’ and energy convertor ‘j’	Month
γ	Government subsidy's rate to suppliers	$/kg
${\varvec{\gamma}}_{{\varvec{i}}}^{{\varvec{t}}}$	Government subsidy’s rate to supplier ‘i’ at time ‘t’	$/kg
$\overline{\gamma }_{{\varvec{i}}}$	Average government subsidy’s rate to supplier ‘i’	$/kWh
β	Government subsidy’s rate to hubs	$/kg
${\varvec{\beta}}_{{\varvec{k}}}^{{\varvec{t}}}$	Government subsidy’s rate to hub ‘k’ at time ‘t’	$/kg
$\overline{{\varvec{\beta}}}_{{\varvec{k}}}$	Average government subsidy’s rate to hub ‘k’	$/kWh
λ	Government subsidy’s rate to communities	$/kWh
${\varvec{\lambda}}_{{\varvec{j}}}^{{\varvec{t}}}$	Government subsidy’s rate to community ‘j’ at time ‘t’	$/kWh
$\overline{{\varvec{\lambda}}}_{{\varvec{j}}}$	Average government subsidy’s rate to community ‘j’	$/kWh
Decision variables
${\varvec{X}}_{{{\varvec{ik}}}}^{{\varvec{t}}}$	Quantity of biomass delivered from supplier ‘i’ to hub ‘k’ at time ‘t’	kg
${\varvec{y}}_{{{\varvec{kj}}}}^{{\varvec{t}}}$	Quantity of biomass delivered from hub ‘k’ to energy convertor ‘j’ at time ‘t’	kg
${\varvec{IS}}_{{\varvec{i}}}^{{\varvec{t}}}$	Biomass inventory level at supplier ‘i’ at time ‘t’	kg
${\varvec{h}}_{{\varvec{k}}}^{{\varvec{t}}}$	Biomass inventory level at hub ‘k’ at time ‘t’	kg
${\varvec{P}}_{{{\varvec{ik}}}}^{{\varvec{t}}}$	Biomass price offered by supplier ‘i’ to hub ‘k’ at time ‘t’	$/kg
${\varvec{B}}_{{{\varvec{kj}}}}^{{\varvec{t}}}$	Biomass ordering price offered by hub ‘k’ to energy convertor ‘j’ at time ‘t’	$/kg
${\varvec{I}}_{{\varvec{j}}}^{{\varvec{t}}}$	Biomass inventory level at energy convertor ‘j’ at time ‘t’	kg
${\varvec{z}}_{{\varvec{j}}}^{{\varvec{t}}}$	Electricity generation from biomass in community ‘j’ at time ‘t’	kWh
ς	Percentage of suppliers’ capacity increase	%
τ	Percentage of communities’ bioenergy generation capacity increase	%

Appendix 3

The equilibrium values of dynamic direct incentives with various levels of communities' incentivization ($\lambda_{j}^{t}$).

Time period (t)	λ = 0			λ = 0.01			λ = 0.02			λ = 0.03			λ = 0.04			λ = 0.05
	j = 1	j = 2	j = 3	j = 1	j = 2	j = 3	j = 1	j = 2	j = 3	j = 1	j = 2	j = 3	j = 1	j = 2	j = 3	j = 1	j = 2	j = 3
1	0	0	0	0.01	0.01	0.01	0.02	0.02	0.02	0.03	0.03	0.03	0.04	0.04	0.04	0.05	0.05	0.05
2	0	0	0	0	0.0041	0	0	0.0083	0	0	0.0124	0	0	0.0166	0	0	0.0207	0
3	0	0	0	0	0.0005	0	0	0.0010	0	0	0.0015	0	0	0.0021	0	0	0.0026	0
4	0	0	0	0	0.0010	0	0	0.0021	0	0	0.0031	0	0	0.0041	0	0	0.0052	0
5	0	0	0	0	0.0010	0	0	0.0020	0	0	0.0029	0	0	0.0039	0	0	0.0049	0
6	0	0	0	0	0.0004	0	0	0.0008	0	0	0.0012	0	0	0.0015	0	0	0.0019	0
7	0	0	0	0	0.0009	0	0	0.0018	0	0	0.0028	0	0	0.0037	0	0	0.0046	0
8	0	0	0	0	0.0017	0	0	0.0033	0	0	0.0050	0	0	0.0067	0	0	0.0084	0
9	0	0	0	0	0.0025	0	0	0.0050	0	0	0.0076	0	0	0.0101	0	0	0.0126	0
10	0	0	0	0	0.0034	0	0	0.0069	0	0	0.0103	0	0	0.0137	0	0	0.0172	0
11	0	0	0	0	0.0028	0	0	0.0057	0	0	0.0085	0	0	0.0114	0	0	0.0142	0
12	0	0	0	0	0.0026	0	0	0.0052	0	0	0.0079	0	0	0.0105	0	0	0.0131	0

The equilibrium values of dynamic direct incentives with various levels of suppliers’ incentivization ($\gamma_{i}^{t}$).

Time period (t)	γ = 0.05						γ = 0.15						γ = 0.25
	i = 1	i = 2	i = 3	i = 4	i = 5	i = 6	i = 1	i = 2	i = 3	i = 4	i = 5	i = 6	i = 1	i = 2	i = 3	i = 4	i = 5	i = 6
1	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
2	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0	0
3	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.01	0.06	0.06	0.06	0.06	0.06	0.06
4	0.02	0.02	0.02	0.01	0.02	0.02	0.10	0.10	0.10	0.10	0.10	0.11	0.06	0.06	0.06	0.06	0.06	0.06
5	0.01	0.01	0.01	0.01	0.01	0.01	0	0	0	0	0	0	0	0	0	0	0	0
6	0	0	0	0	0	0	0	0	0	0	0	0	0.12	0.13	0.13	0.12	0.13	0.13
7	0.02	0.02	0.02	0.02	0.02	0.02	0.10	0.10	0.10	0.10	0.10	0.10	0.21	0.21	0.21	0.21	0.21	0.21
8	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.03	0.00	0.00	0.00	0.00	0.00	0.01
9	0	0	0	0	0	0	0	0	0	0	0	0	0.03	0.02	0.02	0.02	0.02	0.03
10	0.01	0.01	0.01	0.01	0.01	0.01	0.05	0.05	0.05	0.05	0.05	0.05	0.04	0.05	0.05	0.05	0.05	0.04
11	0.01	0.01	0.01	0.01	0.01	0.01	0.05	0.05	0.05	0.05	0.05	0.05	0.09	0.09	0.08	0.08	0.09	0.09
12	0.02	0.02	0.02	0.02	0.02	0.02	0.07	0.07	0.07	0.07	0.07	0.07	0.13	0.11	0.13	0.14	0.11	0.11

The equilibrium values of dynamic direct incentives with various levels of hubs' incentivization $(\beta_{k}^{t})$.

Time period (t)	β = 0		β = 0.05		β = 0.1		β = 0.15		β = 0.20		β = 0.25
	k = 1	k = 2	k = 1	k = 2	k = 1	k = 2	k = 1	k = 2	k = 1	k = 2	k = 1	k = 2
1	0	0	0.05	0.05	0.1	0.1	0.15	0.15	0.2	0.2	0.25	0.25
2	0	0	0	0	0	0	0	0	0	0	0	0
3	0	0	0	0	0	0	0	0	0	0	0	0
4	0	0	0	0	0	0	0	0	0	0	0	0
5	0	0	0	0	0	0	0	0	0	0	0	0
6	0	0	0	0	0	0	0	0	0	0	0	0
7	0	0	0	0.0290	0	0.0598	0	0	0	0	0	0
8	0	0	0	0.05	0	0.0868	0	0.1086	0	0.1452	0	0
9	0	0	0	0.05	0	0.0912	0	0.15	0	0.2	0	0.0811
10	0	0	0	0.05	0	0.1	0	0.15	0	0.2	0	0.25
11	0	0	0.0130	0.05	0.0393	0.1	0	0.15	0	0.2	0.0017	0.25
12	0	0	0.0387	0.05	0.0721	0.1	0.1198	0.15	0.1598	0.2	0.25	0.25