Multi-energy management with hierarchical distributed multi-scale strategy for pelagic islanded microgrid clusters

Energy 185 (2019) 910e921

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Multi-energy management with hierarchical distributed multi-scale strategy for pelagic islanded microgrid clusters Mian Hu a, b, Yan-Wu Wang a, b, *, Jiang-Wen Xiao a, b, Xiangning Lin c a

School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, 430074, China Key Laboratory of Image Processing and Intelligent Control (Huazhong University of Science and Technology), Ministry of Education, Wuhan, 430074, China c State Key Laboratory of Advanced Electromagnetic Engineering and Technology, School of Electrical and Electronic Engineering, Huazhong University of Science and Technology, Wuhan, 430074, China b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 1 June 2019 Received in revised form 6 July 2019 Accepted 14 July 2019 Available online 17 July 2019

The pelagic islands usually include resource islands and load islands with electricity and natural gas transactions among them. Since they are apart from the main land, the finiteness of energy resources results in crucial need of developing efficient energy management framework for the pelagic islanded microgrid clusters (PIMGCs). In this paper, we introduce a novel multi-energy management framework for a PIMGC, where the operators on resource islands sell energy resources, while the aggregators and users on load islands dispatch and consume energy resources, respectively. The multi-scale energy management strategy is proposed, where the operators determine their daily optimal energy supply in a distributed collaborative way by adopting the primal-dual subgradient method, each aggregator determines its daily optimal energy demand and hourly optimal energy usage, and each user determines its hourly optimal energy consumption. A hierarchical day-ahead distributed algorithm is proposed to obtain the Nash equilibrium strategy, where the operators minimize their aggregate operational cost, each aggregator maximizes its revenue and each user maximizes its payoff. Simulation results are provided to show the effectiveness and benefits of the proposed multi-energy management framework for the PIMGCs. © 2019 Elsevier Ltd. All rights reserved.

Keywords: Microgrid cluster Energy management Multi-energy Multi-scale Hierarchical optimization Nash equilibrium

1. Introduction The pelagic island serves as a seabase for connecting the land and marine territories, with abundant land and ocean resources. It is not only the defense outpost of a nation, but also a symbol of national sovereignty [1]. Maintaining a stable power supply to the pelagic islands is crucial for the work and life on the islands. The energy resources on the pelagic islands may be abundant or insufficient, and are unlikely to resemble the main grid to provide reliable energy supply, due to their randomness, volatility, and intermittent characteristics. Due to the long-distance from the islands to the main land, it is extremely difficult for the pelagic islands to access the main grid on the main land. Therefore, it is crucial to develop effective energy management strategy for the

* Corresponding author. School of Artificial Intelligence and Automation, Huazhong University of Science and Technology, Wuhan, 430074, China. E-mail address: [email protected] (Y.-W. Wang). https://doi.org/10.1016/j.energy.2019.07.087 0360-5442/© 2019 Elsevier Ltd. All rights reserved.

reliability of the pelagic islands considering the limited energy resources (LERs) [2e6]. More recently, many researchers have focused on the energy management in islanded microgrids. In Refs. [7,8], two different methods are proposed for both the grid-connected and islanded microgrid energy management systems. In Ref. [9], a scenariobased investment planning model is proposed for the isolated multi-energy microgrids. A coordinated control strategy for the reserve management of the isolated microgrids is presented in Ref. [10]. In Ref. [11], a two-stage decision process is proposed for the isolated microgrid energy management system. A stochasticpredictive energy management system for isolated microgrids is presented in Ref. [12]. An integrated energy management system for isolated microgrid is presented in Ref. [13]. These studies are concerned with the efficient operation of a single islanded microgrid, where the electricity producers and consumers are both exist and the energy resources are enough. However, due to the nonuniformity of the scale and resource distribution of the islands, there exist two types of pelagic islands rather than a single islanded

M. Hu et al. / Energy 185 (2019) 910e921

Nomenclature

Edem ; Gdem Total electricity and total natural gas demand of i i aggregator i

Indices and sets U; Y; F Strategy sets of the operators, aggregators and users ε Index of iteration error kk; m; r Indices of iterations p Set of payoff functions of users at time slot t R Set of revenue functions of aggregators L Set of users Li Set of users on load island i M ;N Sets of resource islands and load islands T Set of time slots j; i Indices of resource island and load island k Index of user Number of users on load island i Li M; N Number of resource islands and load islands T Number of time slots Parameters and variables of upper level r1j ; r2j Dual variables in robust problem g

g

g

aej ; aj ; bej ; bj ; cej ; cj Production cost parameters Cost Ej

Aggregate operational cost function of operators Aggregate REGs during one time period

max Emin Minimum and maximum capacity of electricity ij ; Eij

Esup ; Gsup ij ij Gmax j max Gmin ij ; Gij

u uj fij 6

Gj mej ; mgj g g pej ; pj ; dej ; dj

transported by ships from resource island j to load island i Electric power and natural gas supplied from operator j to aggregator i Maximum cubic meters of natural gas Minimum and maximum capacity of natural gas transported by ships from resource island j to load island i Total decision of operators Total decision of operator j Energy supply and price decision of operator j to aggregator i Dual variables using primal-dual subgradient method Conservative or robustness parameter Electricity and natural gas price parameters Maximum predicted error of REGs

C ej ; C gj

Production cost functions of electricity and natural

pej ; pj

gas Electricity and natural gas prices

sij ssj

Distance between resource island j and load island i Slack variable

Ej

Predicted aggregate REGs during one time period

Parameters and variables of middle level

ddi;st Deflating efficiency of GSS hci;st ; hdi;st Charging and discharging efficiencies of ESS lei ðtÞ; lgi ðtÞ; lhi ðtÞ Electricity, natural gas and heat prices determined by aggregator i

j1i ðtÞ; j2i ðtÞ Dual variables in robust problem

Yi

in Ein i ðtÞ; Gi ðtÞ Total electric power and total natural gas usage of energy hub out out Eout i ðtÞ; Gi ðtÞ; H i ðtÞ Output electric energy, natural gas and thermal energy of energy hub Aggregate REGs Eri ðtÞ dem Edem Electricity and natural gas demand of aggregator i ij ; Gij

from operator j Residual charge level of ESS at the end of last scheduling period

q0i

; qmax Lower and upper bounds of energy level of ESS qmin i i Ri Revenue function of aggregator i Vi ðtÞ Gas reserve of GSS at the end of time slot t V 0i

Residual gas reserve of GSS at the end of last scheduling period

; V max Lower and upper bounds of gas reserve of GSS V min i i r

Ei ðtÞ Predicted aggregate REGs bi ðtÞ; ai ðtÞ Dispatch factors qi ; yi Price decision and total decision of aggregator i y Total decision of aggregators 9i Energy demand decision of aggregator i hi;F Electrical efficiency of gas furnace hi;G Efficiency of compressor

hei;MT ; hgi;MT Electrical and thermal efficiencies of micro turbine Gri Conservative or robustness parameters

εri qi ð0Þ qi ðtÞ Vi ð0Þ

Maximum predicted error of REGs Initial charge level of ESS Charge level of ESS at the end of time slot t Initial gas reserve of GSS

Parameters and variables of lower level

kei;k ðtÞ; kgi;k ðtÞ; khi;k ðtÞ Nonnegative parameters in satisfaction functions Payoff function of user k at time slot t

pi;k ðtÞ

cmin ðtÞ; cmax ðtÞ Minimum and maximum electricity usage i;k i;k Esell i ðtÞ

Total renewable energy sold by users to aggregator i

U ei;k ðtÞ; U gi;k ðtÞ; U hi;k ðtÞ Satisfaction functions attained from using

Transportation cost parameters

εj

g

911

Feasible decision space for aggregator i

electric, gas-based and thermal appliances Electricity traded between user k and aggregator i

yei;k ðtÞ f

Total decision of users Conservative or robustness parameters Decision of user k at time slot t Maximum predicted error of PV energy generation

Gui;k gi;k ðtÞ εui;k

z1i;k ðtÞ; z2i;k ðtÞ Dual variables in robust problem g

ci;k ðtÞ; yi;k ðtÞ; yhi;k ðtÞ Aggregate electricity, gas and heat gi;k ðtÞ

consumption PV energy generation

ðtÞ xbuy i;k

Electricity demand from aggregator i

xsell ðtÞ i;k

Renewable energy sold to aggregator i

g;min

yi;k

g;max

ðtÞ; yi;k

ðtÞ Minimum and maximum natural gas usage

yh;min ðtÞ; yh;max ðtÞ i;k i;k g i;k ðtÞ

Minimum and maximum heat power usage

Predicted PV energy generation

912

M. Hu et al. / Energy 185 (2019) 910e921

microgrid, that is, resource islands and load islands. Vast renewable energy generators and natural gas exploration projects are deployed on or near the resource islands. The load islands have large land area and are suitable for human habitation. Due to the weather and other factors, the energy resources produced on the resource islands may be sufficient or insufficient, which will be abandoned if cannot be used locally. Meanwhile, the local energy resources on the load islands may satisfy part of the energy demand of end users. The pelagic islanded microgrid cluster (PIMGC) consists of multiple pelagic islanded microgrids, with each island being regarded as an islanded microgrid. The energy flows (power flows and gas flows) from resource islands to load islands are realized by ships that transport storage batteries and natural gas from resource islands to load islands. Therefore, it is obvious that the energy management frameworks for a single islanded microgrid in the above works are not suitable for the PIMGCs. As a further study, the microgrid clusters consisting of multiple microgrids have also received considerable attention recently [14]. In Ref. [15], a two-stage adaptive robust optimization based collaborative operation approach is presented for a residential multi-microgrid (MMG). In Ref. [16], a two-stage energy management strategy is developed for networked microgrids with high renewable penetration. A multistep hierarchical optimization algorithm for optimal MMGs operation is proposed based on a multiagent system in Ref. [17]. A novel energy management method for a MMG system is developed based on a bi-level system of systems architecture in Ref. [18]. In Ref. [19], a new prioritybased energy trading mechanism is proposed in a distribution network with multiple microgrids. The above works assume that all the microgrids are connected to the main grid and have sufficient energy supply. However, this situation may not be true in the PIMGCs, which is physically disconnected from the main grid. With the aforementioned observations, this paper proposes a novel multi-energy management framework for a PIMGC, which includes a number of resource islands and load islands. Each resource island contains an operator. Each load island contains a set of users, an aggregator and an energy hub. Considering the aggregators’ potential of increasing the flexibility of the energy market [20e23], the aggregators on load islands act as intermediaries between users on load islands and operators on resource islands. The users may have their individually owned photovolatic (PV) systems. The energy hub is owned by the aggregator. The LERs (i.e., storage batteries and natural gas) are partly transported by ships from the resource islands and partly from the local intermittent renewable energy generations (REGs), such as PV, wind, wave and ocean current power generations, and they are all stored in the energy hubs on the load islands. Note that there exist three main obstacles for solving the multi-energy management problem in a PIMGC. Firstly, compared with the energy management systems for islanded microgrids studied in Refs. [10,11,24] and MMGs studied in Refs. [25e28], where the energy resources are sufficient and the energy demand of users can be fully satisfied at each time slot, the energy supply in PIMGCs is not always sufficient to meet the energy demand of all users. Secondly, in contrast with the multi-energy microgrids studied in Refs. [29e31], where the energy supplied to the microgrid come from the single source, the energy of the load islands is supplied by multiple resource islands. Thirdly, unlike the existing works [15,16,32], where the energy transmission is in real time, the energy dispatch between the resource islands and the load islands is usually in a larger time scale, for example, daily rather than hourly. To tackle the first two challenges, the transportation expenses caused by the distances between resource islands and load islands are considered in the aggregate operational cost of operators, and the satisfaction of users attained from using multi-energy resources

is considered in the payoff of users. The operators determine the optimal energy supply to each load island in a distributed collaborative way by adopting the primal-dual subgradient method. To tackle the third challenge, the multi-scale energy management strategy is proposed, where each operator determines its daily optimal energy supply, each aggregator determines its daily optimal energy demand and hourly optimal energy usage, and each user determines its hourly optimal energy consumption. A hierarchical day-ahead distributed algorithm is proposed to obtain the Nash equilibrium strategy, where the operators minimize their aggregate operational cost, each aggregator maximizes its revenue and each user maximizes its payoff. Hence, the main contributions of this paper are as follows: 1. A novel multi-energy management framework for the PIMGCs is proposed, taking the LERs and the multiple sources of multienergy dispatch into consideration. 2. A hierarchical multi-scale bidirectional interaction mechanism is designed, and a tri-level day-ahead distributed algorithm is proposed to obtain the Nash equilibrium strategy for operators, aggregators and users. 3. Using the realistic demand and REG traces, the effectiveness and benefits of the proposed multi-energy management framework for the PIMGCs are illustrated. The rest of this paper is organized as follows. Section 2 describes the system model. Section 3 solves the optimal responses of users, aggregators and operators, and designs a tri-level day-ahead distributed algorithm. Section 4 presents the simulation results to illustrate the effectiveness and benefits of the proposed multienergy management framework, and conclusions are drawn in Section 5.

2. System model The structure of the multi-energy management framework in a PIMGC is shown in Fig. 1. As has been presented in the Introduction part, we consider a PIMGC consisting of a set M bf1; 2; /; Mg of resource islands and a set N bf1; 2; /; Ng of load islands. The electricity and natural gas on the resource island j2M are owned and dispatched by its operator j. The load island i2N contains a set L i bf1i ; 2i ; /; Li g of users, an aggregator i and an energy hub i. The aggregators buy electricity and natural gas from the operators, with these energy resources (i.e., storage batteries and natural gas)

Fig. 1. Multi-energy management framework in a PIMGC.

M. Hu et al. / Energy 185 (2019) 910e921

periodically transported by ships. Each ship is equipped with the high-capacity energy storage system and gas storage tank for transporting storage batteries and natural gas from resource islands to load islands. The directions of the energy flows are expressed by the shipping routes shown in Fig. 1. The transported energy resources as well as the local REGs (PV, wind, wave, and ocean current power generations) are stored in the energy hubs. The electricity and gas-based/thermal appliances of users are energized by the electricity, gas and heat stored and converted in the energy hubs. Each aggregator controls the energy scheduling and conversion in the energy hub and determines the electricity, natural gas and heat prices traded with users, as well as informs these prices to users. All users own the decision-making controllers to participate in the energy management optimization by determining the electricity, gas and heat traded with the aggregators. The hierarchical multi-scale structure of bidirectional interaction in a PIMGC is shown in Fig. 2, where i) at the lower level each user modifies its electricity, gas and thermal demand pattern to maximize its payoff according to the electricity, natural gas and heat prices advertised by the aggregator, ii) at the middle level each aggregator determines its electricity and natural gas demand from every operator, the electricity and natural gas usage in the energy hub, as well as the electricity, natural gas and heat prices traded with users to maximize its revenue, given the energy supply and pricing schemes of every operator, as well as the electricity, gas and thermal energy demand of every user, iii) at the upper level each operator computes the electricity and natural gas supply to every aggreagtor, as well as the electricity and natural gas pricing schemes to minimize the aggregate operational cost of all operators according to the total electricity and natural gas demand of every aggregator. We assume that the round trip time of each ship is less than a day, which includes the loading time and unloading time of the transported energy resources from resource island to ship and from ship to load island, respectively. We focus on the day-ahead dispatch, with one dispatch period being equally divided into a series T bf1; 2; /; Tg of time slots. For example, if we choose T ¼ 24 (as in our simulation setup), then the dispatch is executed every hour. To tackle the fitness of energy resources, the interactions between the upper level and middle level are implemented by the day, while the interactions between the middle level and lower level are implemented by the time slot, which is the difference from the other hierarchical frameworks, such as [23,33].

2.1. Model of upper level

913

N is subject to

Emin  Esup  Emax ij ij : ij

(1)

The electricity supply Esup is coupled by ij

X

sup

Eij

 Ej :

(2)

i2N

Considering the seasonal and periodic characteristics of the REGs, we build a robust model to construct an uncertain set of Ej of resource island j2M :

8 > < E  E þ G ε : r1 ; j j j j j > Ej  Ej  Gj εj : r2j : :

(3a,3b)

By duality theory, for any feasible r1j ; r2j , it holds that Ej  r1j ðEj þ Gj εj Þ  r2j ðEj  Gj εj Þ and at the optimum the terms on both sides of the inequality are equal. Therefore, we can replace (2) with the following constraints:

8 X sup     > Eij þ r1j Ej þ Gj εj  r2j Ej  Gj εj  0; > > < i2N

(4a,4b,4c)

r1j  r2j  1; > > > : r1 ; r2  0: j

j

By introducing ssj  0 that satisfies r1j  r2j ¼  1 þ ssj . Using this transformation, the constraints (4a)-(4c) are equivalent to the following constraints:

8 X sup    > Eij þ 2r1j Gj εj þ ssj  1 Ej  Gj εj  0; > > < i2N

r1j þ ssj  1  0; > > > : r1 ; ss  0:

(5a,5b,5c)

j

j

Similarly, The natural gas supply Gsup from operator j2M to ij aggregator i2N is subject to sup

Gmin  Gij ij

 Gmax ij :

(6)

The natural gas supply Gsup is coupled by: ij

X

sup

Gij

 Gmax : j

(7)

i2N

The electricity supply Esup from operator j2M to aggregator i2 ij

The operators adopt the appropriate electricity and natural gas pricing schemes to motivate the aggregators toward buying electricity and natural gas to minimize their aggregate operational cost. We consider the following linearized electricity and natural gas pricing schemes

X

pej ¼ pej þ dej

sup

(8)

Gsup : ij

(9)

Eij ;

i2N

pgj ¼ pgj þ dgj

X i2N

Fig. 2. Hierarchical multi-scale structure of bidirectional interaction in a PIMGC.

The operators aim to minimize their aggregate operational cost, which includes the production costs, selling profits and transportation expenses of electricity and natural gas. The transportation expenses are introduced by the multiple sources of multienergy dispatch. The production cost functions of electricity and natural gas are assumed to be increasing and strictly convex [34] and modeled as:

914

M. Hu et al. / Energy 185 (2019) 910e921

C ej ¼ aej

X

 sup 2

Eij

þ bej

i2N

C gj

¼ agj

X

X

ESS is bounded as given below: sup

Eij

þ cej ;

(10)

i2N

2 Gsup ij

þ

X

bgj

i2N

Gsup ij

þ

Xh

(11)

i2N

þ

X

 X  sup Eij pej þ Gsup pgj ij

C ej þ C gj 

j2M

i2N

mej sij Esup þ mgj sij Gsup ij ij

i

(12)

i2N

Given Edem ; Gdem of aggregator i2N , the electricity and natural i i gas supplied from operator j to aggregator i are subject to

X

Esup ij

(13)

Gsup ¼ Gdem : i ij

(14)

j2M

X j2M

2.2. Model of middle level The energy hub i on load island i can be equipped with an energy management system (EMS) controlled by aggregator i to effectively schedule and usage the energy carriers [29,35], as well as to determine the electricity, gas and heat prices traded with users, as shown in Fig. 3. The total amount of electricity purchased by aggregator i from all operators and the local intermittent REGs are stored in the energy storage system (ESS). The total amount of natural gas purchased by the aggregator i from all operators is stored in the gas storage system (GSS).

2.2.1. Energy storage constraints Since the optimal strategy of each aggregator is determined in one day cycle, the leakage rate of the ESS is ignorable, then the energy level dynamics of the ESS in the energy hub i is obtained as

qi ðtÞ ¼ qi ðt  1Þ þ

8 <

r

Eri ðtÞ  Ei ðtÞ þ Gri εri ; : Eri ðtÞ  Eri ðtÞ  Gri εr : i

hci;st

where qi ð0Þ ¼ q0i þ



Esell i ðtÞ

Edem , i

þ

Esell i ðtÞ

Eri ðtÞ ¼





P k2L

hdi;st Ein i ðtÞ;

(17a,17b)

Considering the temporally coupled characteristic of charge level in (15), we can transform (17a) and (17b) into the following constraints: t t X 8X r Eri ðtÞ  Ei ðtÞ þ t*Gri εri : j1i ðtÞ; > <

t¼1

t > :X

t¼1

Eri ðtÞ



t¼1

Edem ; i

¼

(16)

We build a robust model to construct an uncertain set of Eri ðtÞ:

cgj :

Consequently, the aggregate operational cost function of operators in M is formulated as

Cost ¼

qmin  qi ðtÞ  qmax : i i

t X

(18a,18b) r Ei ðtÞ

t¼1



t*Gri εri

:

j2i ðtÞ:

By duality theory and robust optimization method, we can replace (15) and (16) with the following constraints: t t   X X r r 1 Ei ðtÞ þ t Gri εri þ j2i ðtÞ Ei ðtÞ  t Gri εri 8 ji ðtÞ t¼1 t¼1 > > > t t > X X > upp sell > c d >  h E ð t Þ þ h Ein  q > i i ðtÞ; i;st i;st i > > > t¼1 t¼1 > > < t t   X X r r j1i ðtÞ Ei ðtÞ þ t Gri εri þ j2i ðtÞ Ei ðtÞ  t Gri εri > > t¼1 t¼1 > > > t t > X X > > sell low c d > þ h E ð t Þ  h Ein  q > i i ðtÞ; i i;st i;st > > > t¼1 t¼1 : j1i ðtÞ  j2i ðtÞ  hci;st ;

j1i ðtÞ; j2i ðtÞ  0; (19a,19b,19c,19d) where qupp ¼ qmax  qi ð0Þ and qlow ¼ qmin  qi ð0Þ. i i i i 2.2.2. Gas storage constraints Since the optimal strategy of each aggregator is determined in one day cycle, the leakage rate of the GSS is ignorable, then the gas reserve dynamics of the GSS in the energy hub i is obtained as

Vi ðtÞ ¼ Vi ðt  1Þ  ddi;st Gin i ðtÞ;

(20)

(15)

xsell ðtÞ. The energy level of i;k

where Vi ð0Þ ¼ V 0i þ Gdem . The gas reserve of GSS is bounded as i given below:

i

V min  Vi ðtÞ  V max : i i

(21)

2.2.3. Power balance constraints The energy hub i dispatches Gin i ðtÞ to the combined heat and power (CHP) and the users by bi ðtÞ2½0; 1 at time slot t. Then at the CHP the natural gas is dispatched to the micro turbine and the gas furnace by ai ðtÞ2½0; 1. Given xbuy ðtÞ; ygi;k ðtÞ; yhi;k ðtÞ of user k at time i;k P P buy g out slot t, we have Eout i ðtÞ ¼ k2L i xi;k ðtÞ; Gi ðtÞ ¼ k2L i yi;k ðtÞ; P h Hout i ðtÞ ¼ k2L i yi;k ðtÞ. The

relationship

in Ein i ðtÞ; Gi ðtÞ of

Fig. 3. Energy hub of load islanded microgrid with EMS and communication network.

between

out out Eout i ðtÞ; Gi ðtÞ; H i ðtÞ

and

energy hub i at time slot t2T satisfies the following matrix equation:

M. Hu et al. / Energy 185 (2019) 910e921

2 6 6 6 6 4 2

Eout i ðtÞ Gout i ðtÞ H out i ðtÞ

3

slot t is subject to

7 7 7 7 5

max cmin i;k ðtÞ  ci;k ðtÞ  ci;k ðtÞ:

3

1 6 6 ¼ 60 4 0

bi ðtÞai ðtÞhei;MT 7 7 ð1  bi ðtÞÞhi;G 7  5 g bi ðtÞ ai ðtÞhi;MT þ ð1  ai ðtÞÞhi;F 2

(22)

natural gas demand of aggregator i from operator j are subject to



Esup ; ij

(23)

Gdem  ij

Gsup ; ij

(24) P j2M

Edem and Gdem ¼ i ij

P j2M

Gdem ij . The aggregator

i2N aims to maximize its revenue, which includes the selling profits of electricity, natural gas and thermal energy, as well as the purchase cost of electricity and natural gas. The revenue function of aggregator i is formulated as

Ri ¼

Xh

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ci;k ðtÞ  cmin ðtÞ: i;k

(29)

3

Given Esup ; Gsup of operator j to aggregator i, the electricity and ij ij

where Edem ¼ i

(28)

The satisfaction of each user attained from using electric appliances depends on the aggregate electric power usage. The satisfaction function U ei;k ðtÞ of user k at time slot t is

U ei;k ðtÞ ¼ kei;k ðtÞ

6 Ein ðtÞ 7 4 iin 5: Gi ðtÞ

Edem ij

915



2.3.2. Gas-based appliances constraints There are some gas-based appliances at the users’ premise, such g as the stove and furnace. The aggregate gas consumption yi;k ðtÞ of user k at time slot t is bounded as g;min

yi;k

g

g;max

ðtÞ  yi;k ðtÞ  yi;k

ðtÞ:

(30)

The satisfaction of each user attained from using the gas-based appliances depends on the aggregate natural gas usage. The satisfaction function U gi;k ðtÞ of user k at time slot t is

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U gi;k ðtÞ ¼ kgi;k ðtÞ

ygi;k ðtÞ  yg;min ðtÞ: i;k

(31)



g out sell lei ðtÞ Eout i ðtÞ  Ei ðtÞ þ li ðtÞGi ðtÞ

t2T  i X dem g e Edem þlhi ðtÞHout i ðtÞ  ij pj þ Gij pj :

(25)

j2M

2.3.3. Thermal appliances constraints Thermal demand of users often consists of hot water or lowpressure steam demand in the winter and a cooling demand in the summer. The aggregate heat consumption yhi;k ðtÞ of user k at time slot t is bounded as

2.3. Model of lower level

yh;min ðtÞ  yhi;k ðtÞ  yh;max ðtÞ: i;k i;k

Each user k2L i on load island i participates in the energy management optimization by trading electric energy, natural gas and thermal energy with the aggregator i. Each user aims to maximize its payoff, which includes the satisfaction from energy consumption and the purchase cost. The satisfaction function determines how much each user is satisfied (in monetary units) from using the electric, gas-based and thermal appliances, and is generally the increasing and concave function [36].

The satisfaction of each user attained from using the thermal appliances depends on the aggregate heat power usage. The

2.3.1. Electric appliances constraints There are some electric appliances at the users’ premise, such as lights, refrigerators, dishwashers, washers and dryers. The energy balance for user k at time slot t is given below:

yei;k ðtÞ ¼  gi;k ðtÞ þ ci;k ðtÞ;

satisfaction function U hi;k ðtÞ of user k at time slot t is

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi yhi;k ðtÞ  yh;min ðtÞ: i;k

U hi;k ðtÞ ¼ khi;k ðtÞ

(33)

The payoff function of user k2L i at time slot t from using electric appliances, gas-based appliances and thermal appliances is given below:

pi;k ðtÞ ¼ U ei;k ðtÞ þ U gi;k ðtÞ þ U hi;k ðtÞ  lei ðtÞyei;k ðtÞ g

g

(34)

h

li ðtÞyi;k ðtÞ  li ðtÞyhi;k ðtÞ:

(26)

where yei;k ðtÞ ¼ xbuy ðtÞ  0 denotes the user k is buying the electric i;k energy from aggregator i, or else yei;k ðtÞ ¼ xsell ðtÞ  0 denotes the i;k user k is selling the surplus electric energy to aggregator i. We build a robust model to construct an uncertain set of gi;k ðtÞ:

8 > < g ðtÞ  g ðtÞ þ Gu εu : z1 ðtÞ; i;k i;k i;k i;k i;k 2 u u > g ðtÞ  g : i;k i;k ðtÞ  Gi;k εi;k : zi;k ðtÞ;

(32)

3. Hierarchical distributed day-ahead energy management strategy 3.1. Optimal responses of users e

(27a,27b)

The aggregate electricity consumption ci;k ðtÞ of user k at time

g

h

e

Given qi ¼ ½li ; li ; li  of aggregator i, where li ¼ ½lei ð1Þ; lei ð2Þ; /;

lei ðTÞ,

lgi

g g g ½li ð1Þ; li ð2Þ; /; li ðTÞ

lhi

½lhi ð1Þ; lhi ð2Þ; /; lhi ðTÞ.

¼ and ¼ Substituting (26) into (34), the payoff function of user k2L i at time slot t can be rewritten as

916

M. Hu et al. / Energy 185 (2019) 910e921

pi;k ðtÞ ¼ U ei;k ðtÞ þ U gi;k ðtÞ þ U hi;k ðtÞ  lei ðtÞci;k ðtÞ lgi ðtÞygi;k ðtÞ  lhi ðtÞyhi;k ðtÞ þ lei ðtÞgi;k ðtÞ:

(35)

g

Let 6i;k ðtÞ ¼ ½ci;k ðtÞ;gi;k ðtÞ;yi;k ðtÞ;yhi;k ðtÞ, the payoff maximization problem of user k at time slot t is formulated as

maxpi;k ðtÞ

6i;k ðtÞ

s:t: ð27aÞ; ð27bÞ; ð28Þ; ð30Þ; ð32Þ:

(36)

Considering the separability of the optimization problem (36), the decision variables ci;k ðtÞ; ygi;k ðtÞ; yhi;k ðtÞ and gi;k ðtÞ can be separately solved. Note that problem (36) is a convex optimization ðtÞ and yh* ðtÞ can be found problem, the best response ci;k ðtÞ, yg* i;k i;k

½gi;k ð1Þ; gi;k ð2Þ; /; gi;k ðTÞ of user k2L i, where gi;k ðtÞ is defined in Section 3.1. To guarantee (38a), (38b), (38c) of user k satisfies (28), g

(30), (32), the energy prices lei ðtÞ, li ðtÞ and lhi ðtÞ are subject to

8 9 8 > > e < = > k ðtÞ > i;k e > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi q ; l ðtÞ  max > i > k2L i > > :2 cmax ðtÞ  cmin ðtÞ> ; > > i;k i;k > > > 8 9 > > > > > > g < = < k ðtÞ i;k lgi ðtÞ  max qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; > k2L i > :2 yg;max ðtÞ  yg;min ðtÞ> ; > > > i;k i;k > > > 8 9 > > > > > > h < = > k ðtÞ > i;k h > > ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi q l ðtÞ  max : > i : k2L i > :2 yh;max ðtÞ  yh;min ðtÞ> ; i;k i;k

using

in dem Let yi ¼ ½qi ; E in ; Gdem , where qi is defined in i i ; Gi ; ai ; bi ; ji ; E i

8   > > > vpi;k ðtÞ > ¼ 0;  > >  > ðtÞ vc > > i;k ci;k ðtÞ¼ci;k ðtÞ > > >  > > > < vpi;k ðtÞ ¼ 0;  g > vyi;k ðtÞ yg ðtÞ¼yg* ðtÞ > > i;k i;k > > >  > > > vpi;k ðtÞ > > > ¼ 0:  > > vyh ðtÞ  > : i;k yh ðtÞ¼yh* ðtÞ i;k

in in Section 3.1, similarly to qi , we can also define Ein i from Ei ðtÞ, Gi 1 2 dem from Gin i ðtÞ, ai from ai ðtÞ, bi from bi ðtÞ, ji from ji ðtÞ and ji ðtÞ, E i dem from Edem from Gdem ij . The revenue maximization problem of ij , Gi

aggregator i is formulated as

(37a,37b,37c) max yi

Ri

s:t: ð19Þ  ð19bÞ; ð20Þ  ð24Þ; ð41aÞ  ð41cÞ:

i;k

Then we can obtain the optimal response of user k at time slot t, which can be written as the functions of decision values lei ðtÞ, lgi ðtÞ h

and li ðtÞ of aggregator i and given below:

8 > > > kei;k ðtÞ2 > > > ci;k ðtÞ ¼ e 2 þ cmin > > i;k ðtÞ; > > 4li ðtÞ > > > > > < kgi;k ðtÞ2 g* g;min yi;k ðtÞ ¼ g 2 þ yi;k ðtÞ; > > 4 l ðtÞ > i > > > > h > > ki;k ðtÞ2 > > > ðtÞ ¼ h þ yh;min ðtÞ: yh* > i;k i;k > > 4li ðtÞ2 :

(38a,38b,38c)

In Appendix, we show that the revenue maximization problem (42) is a convex optimization problem defined over Yi. Thus, the revenue maximization problem (42) has unique global optimal strategy yi [38], and can be solved by using the nonlinear programming methods [39].

3.3. Optimal responses of operators

. Let u ¼ ½u1 ; u2 ; /; uM , where uj ¼ ½E sup ; Gsup ; rj , Esup ¼ Gdem i j j j ; Esup ; /; Esup , Gsup ¼ ½Gsup ; Gsup ; /; Gsup , rj ¼ ½r1j ; ssj . The ½Esup 1j 2j Nj j 1j 2j Nj aggregate operational cost function (12) can be rewritten as

and strong duality property [38]:



Gui;k εui;k :

(39)

The optimal electric energy transaction

ye* ðtÞ i;k

Cost ¼

 X  X  X h e sup 2 sup aej  dj Eij þ bej  pej Eij

j2M

between user k

i2N

ye* i;k ðtÞ ¼

e

4li ðtÞ2

u u þ cmin i;k ðtÞ  g i;k ðtÞ þ Gi;k εi;k : e

g

h

i2N

(43)

i2N

The cost minimization problem of operators is formulated as

(40)

Therefore, under qi ¼ ½li ; li ; li  of aggregator i, we can obtain

i2N

 X 2  X  þ agj  dgj Gsup þ bgj  pgj Gsup ij ij i2N i2N i X X g þ mej sij Esup þ mgj sij Gsup þ cej þ cj ij ij

and aggregator i at time slot t is obtained from (26), (38a) and (39) as

kei;k ðtÞ2

(42)

Given 9 ¼ ½91 ; 92 ; /; 9N  of aggregators N , where 9i ¼ ½Edem ; i

According to the robust optimization method [37] and the duality theory [38], we can find g i;k ðtÞ by using the KKT conditions

g i;k ðtÞ ¼ g i;k ðtÞ

(41a,41b,41c)

minCost u

g*

the optimal strategy gi;k ðtÞ ¼ ½ye* ðtÞ; yi;k ðtÞ; yh* ðtÞ. i;k i;k

3.2. Optimal responses of aggregators sup

sup

g

Given fij ¼ ½Eij ; Gij ; peij ; pij  of operator j2M . Given gi;k ¼

s:t: ð1Þ; ð5aÞ  ð5cÞ; ð6Þ; ð7Þ; ð13Þ; ð14Þ:

(44)

Due to the non-separability of the problem with respect to the decision variables, a primal-dual subgradient method proposed in Ref. [40] is adopted to solve (44) in a distributed way. Let 6 ¼ ½c; s; n. The Lagrangian of (44) is defined as

M. Hu et al. / Energy 185 (2019) 910e921

X

L ðu; c; s; nÞ ¼ Cost þ

cej

hX

Esup þ 2r1j Gj εj ij

ðkkÞ

13: uj

j2M i2N i   i X h þ ssj  1 Ej  Gj εj þ sj  r1j þ ssj  1

þ

X

cgj

hX

j2M

i2N

sup

i

j2M

X

hX

sup nei Eij j2M i2N i X g h X sup ; þ ni Gij  Gdem i j2M i2N

Gij

þ  Gmax j

 Edem i

(45)

h sup;m ¼ Eij  2L

h ¼ Gsup;m  2L Gsup;mþ1 ij ij

ðum ; cm ; sm ; nm Þ Esup ij

iþ

h

iþ

iEmax ij Emin ij

;

iGmax ij Gmin ij

;

to users L i ; broadcasts 19: repeat 20: if r > 1 then ðkk;rÞ

21: aggregator i determines yi

(47)

26: user k2L i broadcasts gi;k to aggregator i; 27: r )r þ 1;         28: until yðkk;r1Þ  yðkk;r2Þ j2 =yðkk;r1Þ j2  ε;

(48)

(49)

(51)

where ½xþ ¼ maxfx; 0g and ½xba ¼ minfmaxfx; ag; bg denote the projection of x [39], L X ðum ; cm ; sm ; nm Þ denotes the subgradient of L at ðum ; cm ; sm ; nm Þ with respect to X. The initial vector u0 is assigned with arbitrary values as the algorithm projects the solution on U at the first step. The initial vectors c0 ; s0 ; n0 should be feasible, e.g., c0 ¼ 0; s0 ¼ 0; n0 ¼ 0. The scalar 2 > 0 is a constant step size. The convergence analysis of the iterates can refer to the convergence result on the primal-dual subgradient method in Ref. [40]. 3.4. Distributed algorithm design

Algorithm 1. Tri-level day-ahead distributed algorithm 1: Upper Level: 2: kk)1; ð1Þ

3: Initialization: The operator j2M selects uj

and broadcasts

to aggregator i;

4: repeat 5: if kk > 1 then 6: m)1; ðkk;1Þ

7: Initialization: Set 6ðkk;1Þ , operator j selects uj

11: update 6ðkk;mÞ based on (49)e(51);

ðkkÞ

ðkk;r1Þ

, gi;k )gi;k

;

;

30: aggregator i broadcasts 9i to operators M ; 31: kk )kk þ 1;         32: until uðkk1Þ  uðkk2Þ j2 =uðkk1Þ j2  ε; 33: u )uðkk1Þ , y )yðkk1Þ , f )fðkk1Þ ; 34: return the Nash equilibrium strategy u, y and f . In this subsection, we formulate a noncooperative game D ≡〈fM ; N ; L g; fU; Y; Fg; fCost; R; pg〉 among operators M , aggregators N and all users L . In this game model, the operator j2 M determines uj to minimize their aggregate operational cost (43) and broadcasts fij to the aggregator i2N . The aggregator i2N determines yi to maximize its revenue (25), then broadcasts qi to the users in L i and 9i to the operators in M . The user k2L i determines gi;k ðtÞ to maximize its payoff (34) and broadcasts gi;k to the aggregator i. Proposition 1. A Nash equilibrium strategy always exists in the noncooperative game D. Proof 1. Nash equilibrium implies no player can gain by unilaterally changing its own strategy while the others play their Nash equilibrium strategy [41]. Once each aggregator i2N selects 9i , the operators determine u to minimize their aggregate operational cost (43), and each user determines gi;k ðtÞ to maximize its payoff (34).

Once each operator j2M selects fij and each user k2L i selects gi;k , each aggregator determines yi to maximize its revenue (25). The convergence solution to this noncooperative game is regarded as a Nash equilibrium strategy. Therefore, a Nash equilibrium strategy can be found in the game D as soon as the operators, aggregators and users determine their optimal strategy u, y and f , respectively. A Nash equilibrium is achieved when the following conditions are satisfied:

Costðu ; y ; f Þ  Costðu; y ; f Þ; cu2U; usu ;

(52)

  Ri u ; yi ; f  Ri ðu ; yi ; f Þ; ci2N ; yi 2Yi ; yi syi ;

(53)



(46)e(48);

ðkk;r1Þ

)yi

ðkkÞ

m m m m nmþ1 ¼ nm i þ 2L ni ðu ; c ; s ; n Þ; i

based on

by solving (42);

qiðkk;rÞ

(46)

(50)

10: operator j updates

and

qðkk;1Þ i

ðkkÞ

iþ

uðkk;mÞ j

to the aggregator i;

to users L i ; 22: aggregator i broadcasts 23: end if 24: Lower Level: 25: user k2L i determines gi;k ðtÞ by (38b), (38c), (40);

m m m m smþ1 ¼ sm ; j þ 2L sj ðu ; c ; s ; n Þ j

8: repeat 9: m )m þ 1;

;

15: end if 16: Middle Level: 17: r)1;

29: yi

m m m m cmþ1 ¼ cm ; j þ 2L cj ðu ; c ; s ; n Þ j

ð1Þ fij

ðkkÞ fij

ðkk;rÞ

m m m m rmþ1 ¼ rm ; j  2L rj ðu ; c ; s ; n Þ j

h

ðkk;mÞ

)fij

ðkk;1Þ

ðum ; cm ; sm ; nm Þ Gsup ij

h

ðkkÞ

, fij

18: Initialization: The aggregator i2N selects yi

Note that the separability of Lagrangian in (45), the iterates of the primal-dual subgradient method at the mth step, m  0, are generated as follows [40]: sup;mþ1

ðkk;mÞ

)uj

14: operator j broadcasts

i

where c ¼ ½c1 ; c2 ; /; cM ; s ¼ ½s1 ; s2 ; /; sM ; n ¼ ½n1 ; n2 ; /; nN , and cj ¼ ½cej ; cgj ; ni ¼ ½nei ; ngi .

Eij

917

    12: until 6ðkk;mÞ  6ðkk;m1Þ j2  ε;







pi;k u ; yi ; gi;k ðtÞ  pi;k u ; yi ; gi;k ðtÞ ; ck2L i ; t2T :

(54)

The tri-level day-ahead distributed algorithm proposed to determine the Nash equilibrium strategy of the game D is shown in

918

M. Hu et al. / Energy 185 (2019) 910e921

Algorithm 3.4. Given uij of operator j and gi;k of user k2 L i, the aggregator i updates yi through solving the revenue maximization problem (42). Each user updates gi;k ðtÞ based on (38b), (38c), (40) using qi of aggregator i. To achieve the convergence of the middle level  and the lower level, we adopt the termination criterion yðrÞ     yðr1Þ j =yðrÞ j  ε, where yðrÞ denotes the optimal strategy y 2

2

r th

calculated at the iteration. Given 9i of aggregator i, the operator j updates uj through (46)-(48). To achieve the convergence of the upper   level, we adopt the termination criterion 6ðmÞ  6ðm1Þ j  ε, 2

where 6ðmÞ denotes the optimal solution 6 calculated at the mth iteration. To achieve the convergence of those three levels, we adopt         the termination criterion uðkkÞ  uðkk1Þ j2 =uðkkÞ j2  ε, where

uðkkÞ denotes the optimal strategy u calculated at the kkth iteration. Once the strategy of operators converges to the optimal solution u

after a series of iterations, the aggregators and users find their optimal strategy y and f . Therefore, the tri-level day-ahead distributed algorithm converges to the optimal strategy for operators, aggregators and users.

4. Simulation results For numerical simulations, we consider a PIMGC that consists of two resource islands and two load islands. Each load island contains 15 users, with their daily PV power generation and electricity demand profiles randomly chosen from the real data [42] on 1 July 2012 (T ¼ 24). Table 1 presents the values for parameters in the model of lower level. Table 2 presents the values for parameters in the model of middle level. Table 3 presents the values for parameters in the model of upper level. All the following simulation results are obtained by MATLAB R2015b running on a laptop with Inter Core i5-5257U CPU @ 2.7 GHz, 8 GB RAM memory, and 64-bit Windows 10 OS. To demonstrate the performance of the proposed hierarchical distributed multi-energy management framework in a PIMGC, we choose E1 ¼ 1200kW, E2 ¼ 1000kW, Gmax ¼ 800m3 , Gmax ¼ 1 2 1000m3 for the sufficient energy supply of the two resource islands, and choose E1 ¼ 700kW, E2 ¼ 600kW, Gmax ¼ 500m3 , Gmax ¼ 1 2 500m3 for the insufficient energy supply of the two resource islands. Tables 4 and 5 show the energy demand of aggregators and energy supply of operators under the sufficient and insufficient energy supply, respectively. We can observe that the aggregators and operators reach an agreement on the energy transactions, and the LERs with the worst-case scenario of REGs on the resource islands are completely sold to the load islands. Figs. 4 and 5 show the energy demand of users and energy conversion in energy hubs on two load islands under the sufficient and insufficient energy supply, respectively. The electricity, natural gas and thermal consumptions of the users on two load islands reach to 119%, 104%and 106%times of their baseline electricity, natural gas and thermal demand under the sufficient and insufficient energy supply, respectively. We can observe that the users have the same energy

Table 1 Simulation setup parameters of lower level. Parameter

Value

Parameter

Value

Gui;k

[0.2,0.3]

[0.3 m3 , 0.5 m3 ]

εui;k kgi;k ðtÞ

gi;k

ðtÞ yg;min i;k yg;max ðtÞ i;k ðtÞ yh;min i;k ðtÞ yh;max i;k

khi;k ðtÞ

[2

$ =m3 ,

4

$ =m3 ]

[2 $/kW, 3 $/kW]

[0.6 m3 , 0.8 m3 ] [0.1 kW, 0.3 kW] [0.4 kW, 0.6 kW]

Table 2 Simulation setup parameters of middle level. Parameter

Value

Parameter

Value

Parameter

Value

hc1;st hc2;st

0.9

εr2

20 kW

0.2

0.95

E1 ðtÞ

r

20 kW

hd1;st hc2;st

1.1

E2 ðtÞ

r

20 kW

he1;MT he2;MT h1;G

1.02

q0i

200 kWh

h2;G

0.95

dd1;st dd2;st Gr1 Gr2 εr1

1.1

qmin i

200 kWh

hg1;MT

0.2

1.2

qmax i

2500 kWh

0.3

30%

V 0i

150 m3

30%

V min i V max i

hg2;MT h1;F h2;F

1500 m3

20 kW

150 m3

0.25 0.9

0.1 0.15

Table 3 Simulation setup parameters of upper level. Parameter

Value

Parameter

Value

Parameter

Value

Emin ij

0 kW

me1

1

ae1

1

Emax ij

1000 kW

me2

1

ae2

1.5

Gmin ij

0 m3

pg1

1.5

be1

2

Gmax ij

800 m3

g

2.5

be2

1.5

G1 G2

p2

30%

0.01

ce1

1

0.01

ce2

2

ε1

0.1*E1

2

ag1

2

ε2

2

g

a2

2

pe1

0.1*E2 2

dg1 dg2 mg1 mg2 s11

3

g

0.5

pe2

b1

1

s12

9

b2

g

1

de1 de2

0.01

s21

8

cg1

1

0.01

s22

4

cg2

2

30%

Table 4 Numerical results under sufficient energy supply. Variable dem Esup 11 ; E11 sup dem E12 ; E12 dem Esup 21 ; E21 dem Esup ; E 22 22

Value

Variable

510 kW

dem Gsup 11 ; G11 sup G12 ; Gdem 12 dem Gsup 21 ; G21 sup G22 ; Gdem 22

175 kW 327 kW 380 kW

Value 499

m3

110 m3 60 m3 449 m3

Variable

Value

Edem 1 Edem 2 Gdem 1 Gdem 2

685 kW

Variable

Value

Edem 1 Edem 2 Gdem 1 Gdem 2

630 kW

707 kW 609 m3 509 m3

Table 5 Numerical results under insufficient energy supply. Variable dem Esup 11 ; E11 sup dem E12 ; E12 dem Esup 21 ; E21 sup E22 ; Edem 22

Value

Variable

586 kW

dem Gsup 11 ; G11 sup G12 ; Gdem 12 dem Gsup 21 ; G21 dem Gsup ; G 22 22

44 kW 93 kW 537 kW

Value 500

m3

0 m3 0 m3 500 m3

630 kW 500 m3 500 m3

demand under the two circumstances because of the best trade-off between the payment and user satisfactions. Note that the insufficient energy resources has the bigger impact on load island 1 than load island 2. The aggregator 1 buys the less natural gas under the insufficient energy supply, it has to set the smaller dispatch factor b1 ðtÞand the larger electricity usage in energy hub than in the case of sufficient energy to satisfy the energy demand of users. Meanwhile, the aggregator 2 just needs to adjust the energy usage and dispatch factors at some time slots to meet the unchanged energy demand of users. Fig. 6 shows the comparison results of different distances between resource islands and load islands under the sufficient energy

M. Hu et al. / Energy 185 (2019) 910e921

919

Fig. 4. Energy demand and energy conversion on load island 1 under the sufficient and insufficient energy supply.

Fig. 5. Energy demand and energy conversion on load island 2 under the sufficient and insufficient energy supply.

Fig. 6. Comparison results of different distances between resource islands and load islands under the sufficient energy supply.

supply. We consider three different situations, where (a) the resource island 1 is closer to the load island 1 and the resource island 2 is closer to the load island 2, (b) both the resource islands have the same distance to the load islands, (c) the resource island 2 is closer to the load island 1 and the resource island 1 is closer to the load island 2, which are corresponding to the (a), (b), (c) in Fig. 6. The number on each line segment denotes the distance between the resource island and load island. The black number and the orange number next to each segment denote the electricity and natural gas traded between the operator and aggregator, respectively. We can observe that the operators prefer to sell more energy resources to the closer aggregator to decrease the transportation expenses. Table 6 shows the comparison of results from different robustness parameters. As expected, the total operational cost of operators gradually increases and the revenues of aggregators gradually

920

M. Hu et al. / Energy 185 (2019) 910e921

Table 6 Numerical results under sufficient energy supply with different robustness parameters. Robustness Parameter Gui;k

[0,0.1]

Robustness Parameters Gri ; Gj Operators Total Cost (  106 $)

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

2.11

2.2

2.41

2.53

2.7

2.86

3.04

3.21

3.4

3.63

Aggregator 1 Revenue (  105 $)

2.22

2.35

2.56

2.74

2.93

3.08

3.32

3.47

3.7

3.94

Aggregator 2 Revenue (  105 $)

1.96

2.06

2.34

2.4

2.6

2.81

3

3.24

3.44

3.67

Users 1 Total Payoff (  103 $)

3.67

3.67

3.67

3.67

3.67

3.67

3.67

3.67

3.67

3.67

Users 2 Total Payoff (  103 $)

3.26

3.26

3.26

3.26

3.26

3.26

3.26

3.26

3.26

3.26

[0.1,0.2]

[0.2,0.3]

[0.3,0.4]

decrease as the variation range of REGs increases, while the payoff of users changes a little. Since the robust optimization performs on the worst-case scenario of the uncertainties, the energy transactions between users and aggregators, aggregators and operators would be larger with a more roughly estimation of the uncertain variables varying in a larger range, then it incurs the additional energy cost of operators and reduces the revenues of aggregators. Thus the more accurate prediction is, the less conservativeness of the proposed multi-energy management system is. From the above three aspects of comparison results, the proposed multi-energy management framework can work well for both the cases of the sufficient and insufficient energy supply. It can also make economical decisions for different scenarios of the distances between resource islands and load islands, as well as different uncertainties of renewable energy generations. In summary, the proposed energy management framework has the flexibility and capability to tackle the variability and uncertainty in the PIMGCs.

[0.4,0.5]

Conflict of interest The authors declare that there is no conflict of interest. Acknowledgements This work is supported by the National Natural Science Foundation of China under Grants 61773172, 61572210, and 51537003, the Natural Science Foundation of Hubei Province of China (2017CFA035), the Fundamental Research Funds for the Central Universities (2018KFYYXJJ119) and the Program for HUST Academic Frontier Youth Team. Appendix. Convexity of revenue maximization problem The revenue function Ri in (25) is a linear function of decision

yi of aggreagtor i, given fj of operator j2M and gi;k of user k2 L i. The feasible strategy space Yi is defined by (19)-(19b), (20)e(24),

[0.6,0.7]

[0.7,0.8]

[0.8,0.9]

[0.9,1]

(41a)-(41c), which are expressed as the linear functions of yi ¼ ½qi ; in dem Ein ; Gdem , except the power balance equations in i i ; Gi ; ai ; bi ; ji ; E i (22). Specifically, from (22), we have

in in e Eout i ðtÞ ¼ E i ðtÞ þ bi ðtÞai ðtÞhi;MT Gi ðtÞ;

(55)

in Gout i ðtÞ ¼ ð1  bi ðtÞÞhi;G Gi ðtÞ;

(56)

   g Gin Hout i ðtÞ: i ðtÞ ¼ bi ðtÞ hi;F þ ai ðtÞ hi;MT  hi;F

(57)

Equations (55)e(57) are the nonlinear functions of bi ðtÞ; ai ðtÞ; in in out Gin i ðtÞ. From (56), we have bi ðtÞGi ðtÞ ¼ Gi ðtÞ  Gi ðtÞ=hi;G . By

substituting bi ðtÞGin i ðtÞinto (55) and (57), we obtain in in e Eout i ðtÞ ¼ E i ðtÞ þ ai ðtÞhi;MT Gi ðtÞ 

5. Conclusion In this paper, we propose a novel multi-energy management framework for PIMGCs, taking the LERs and the multiple sources of multi-energy dispatch into consideration. A tri-level day-ahead distributed algorithm based on hierarchical multi-scale bidirectional interaction mechanism is proposed to obtain the Nash equilibrium strategy for operators, aggregators and users, where the operators minimize their aggregate operational cost, each aggregator maximizes its revenue and each user maximizes its payoff. The simulation results have shown the effectiveness and benefits of the proposed hierarchical distributed multi-energy management framework in a PIMGC. Interesting future work could focus on the other energy resources, such as hydrogen demand in the PIMGCs.

[0.5,0.6]

Hout i ðtÞ



¼ hi;F

From 1 hgi;MT hi;F

(58)

hi;G

!   Gout g in i ðtÞ þ ai ðtÞ hi;MT  hi;F : Gi ðtÞ 

hi;G

(59),

½Hout i ðtÞ

! Gout i ðtÞ ;



we

ai ðtÞðGin i ðtÞ  out Gi ðtÞ=hi;G Þ. By

Gout i ðtÞ=hi;G Þ ¼

have

hi;F ðGin i ðtÞ 

(59)

substituting this

result into (58), we obtain in Eout i ðtÞ ¼ Ei ðtÞ " e

þ

hi;MT

hgi;MT

 hi;F

H out i ðtÞ

 hi;F

Gin i ðtÞ



Gout i ðtÞ

hi;G

!# ;

(60)

in Equation (60) is a linear function of Ein i ðtÞ; Gi ðtÞ. Hence, the nolinear matrix equation (22) can be rewritten as the linear equality constraint (60) and the following linear inequality constraints, which are equivalent to 0  ai ðtÞ  1and 0  bi ðtÞ  1:

0  Gin i ðtÞ 

Gout i ðtÞ

hi;G

 Gin i ðtÞ;

(61)

" !# 1 Gout in out i ðtÞ 0 g H ðtÞ  hi;F Gi ðtÞ  hi;G hi;MT  hi;F i  Gin i ðtÞ 

Gout i ðtÞ :

hi;G

(62)

Thus the revenue maximization problem (42) under Yi is proven to be a convex optimization problem. References [1] Zhao B, Chen J, Zhang L, Zhang X, Qin R, Lin X. Three representative island

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