Table Of Content1
A Novel Algorithm for Optimal Electricity Pricing
in a Smart Microgrid Network
Mosaddek Hossain Kamal Tushar, Member, IEEE and Chadi Assi, Senior Member, IEEE
Abstract—The evolution of smart microgrid and its demand- changingwiththemodernizationofthegridandtheintegration
responsecharacteristicsnotonlywillchangetheparadigmsofthe of new technology like smart grid, renewable sources, electric
century-oldelectricgridbutalsowillshapetheelectricitymarket.
vehicles, and storage systems [3], [4], [5].
In this new market scenario, once always energy consumers,
Hence, current deregulation of energy market permits var-
now may act as sellers due to the excess of energy generated
7 from newly deployed distributed generators (DG). The smart ious entities (i.e., consumers, and producers) to control the
1 microgrid will use the existing electrical transmission network market operations [6], [7]. In 2001, the OECD (Organization
0 andapayperusetransportationcostwithoutimplementingnew for Economic Co-operation Development) countries decided
2 transmission lines which involve a massive capital investment. to open up their electricity market. In order to allow com-
In this paper, we propose a novel algorithm to minimize the
n petition, the OECD identified some key issues such as, (i)
electricity price with the optimal trading of energy between
a sellersandbuyersofthesmartmicrogridnetwork.Thealgorithm Unbundling, which means the separation of generation and
J
is capable of solving the optimal power allocation problem transmission systems, (ii) Empowering the end user to choose
3 (with optimal transmission cost) for a microgrid network in a suppliers, (iii) Meeting security of supply, environment, and
1 polynomial time without modifying the actual marginal costs of
social goals, and (iv) Reforming regulatory institutes where
powergeneration.Wemathematicallyformulatetheproblemasa
] nonlinearnon-convexanddecomposetheproblemtoseparatethe the regulator entity is independent from the regulated entities.
Y optimalmarginalcostmodelfromtheelectricityallocationmodel. Ideal competitive electricity markets support a competitive
S Then, we develop a divide-and-conquer method to minimize operation, by purchasing electricity from suppliers and selling
. the electricity price by jointly solving the optimal marginal to utilities, end-users, and other market players [7]. Each day,
s
cost model and electricity allocation problems. To evaluate the
c market operators forecast the electricity demand and update
[ performance of the solution method, we develop and simulate
the previous forecast for next day and up to the month ahead.
the model with different marginal cost functions and compare it
1 with a first come first serve electricity allocation method. Both sellers and buyers determine the amount of electricity to
v besuppliedandtheprice.Consumersadjusttheirconsumption
8 patternbasedontheforecasttoreducetheelectricitywholesale
5 I. INTRODUCTION
price.Themarketoperatormatchesthedemandwithsuppliers’
5
The future electric system, referred to as the smart grid
offers to determine the wholesale price. It first accepts the
3
or microgrid, uses advanced information and communication
0 lowest price offers and then stacks up to the higher priced
. technologies (ICT) to monitor and control the transport of offer until enough has been accepted to match the customer
1
electricity from distributed generation sources (DG) or grids,
0 demand.Mostmarketsfinalizetheelectricitypriceatleasttwo
to meet the varying demand for a variety of customers. The
7 hours before the actual purchasing time [8]. Others decide
1 holisticvisionoffuturesmartmicrogridspromisestoautomate the price at various time-scales, real-time, and day-ahead
: the coordination of user needs, capabilities of generators, grid
v markets. According to the current electricity market policy
operators,andutilitybusinessstakeholdersandenablethemto
i and operations described above, the system has ample scopes
X
operate the system efficiently, reduce costs and environmental
to optimize or to minimize the electricity price with the
r impacts while increasing system reliability, resilience and
instantaneousdemand,production,andtransportofelectricity.
a
stability [1], [2].
Today, the evolution of smart grid, distributed generation
In a traditional electricity market, products and services
and renewable energy sources (RES), storage and electric
(comprising of production and delivery of electricity to end
vehicles (EVs), and demand response, are gradually changing
users) were bundled together in a vertically integrated supply
the power flow of the grid from unidirectional to bidirectional
system. A regulatory body controls this vertically integrated
[9], [10]. It is envisioned that a microgrid network (MGN)
supply system with an important feature; that is, a reliance
may in the near future contain hundreds or even thousands of
on average-cost pricing rather than the marginal cost prices
microgrids(MG)sharingenergywitheachother[11].Usually,
of the competitive market. Under this controlled scenario, it
aMGproducesandconsumesenergylocally;incaseofshort-
is nearly impossible for a new player with a small investment
age,itpurchaseselectricityfromtheneighboringmicrogridsor
(e.g.,amicrogrid)toenter theenergymarketandsurvive.For
sellswheneverithasasurplus.Insuchscenarios,themicrogrid
both customers’ and providers’ benefits, an open competitive
operatorsmaynotowntransmissionlines,andusetheexisting
electricity market is desirable, a market which accepts new
electrical network which hence requires transmission costs
suppliersandmarginal-costpriceforelectricity.Severalinves-
besides the generation cost. The economic dispatch model of
tigations have shown that the electricity market paradigm is
the MGN, therefore, is more complex than the MCP (market
clearing price) and LMP (locational marginal price) model
M.H.K.TusharandChadiAssiarewithConcordiaUniversity,Montreal,
QCH3G2W1,Canada for the existing one-way energy transmission network [12].
2
In [13], the authors studied the problem based on a game
theoreticalframework.Theyproposedanalgorithmthatforms WindTurbine MG3-EMS
MGscoalitionsandminimizesthepowerlossandpricewithin MGNEMS
MG1-EMS MG2-EMS
a coalition. In [14], the authors introduced an optimization
problem that minimizes the electricity costs and peer-to-peer
energy sharing losses in a distribution network consisting of Residenceload load
MGs. They initially formulated the problem as a non-convex
VirtualConnection
and later relaxed it to a second-order cone programming. For
feeder1
LV/HV
calculating the electricity price, they used TOU (time of use)
price given by a central grid. In [15], the authors discussed load Solar loafededer2 LV/HV feeder1
the energy trading in a hybrid electricity market controlled by
a non-profit or profit oriented local trading center (LTC) that generator load load load loadWindTurbineload
maximizes the benefits for each consumer and seller. They
formulated the trading as two optimization problems which Fig. 1. Smart Microgrid Network (MGN). LV/HV: Low Voltage (LV) and
(i) maximize the benefit of the consumer and seller with non- HighVoltage(HV)bi-directionalinterface(suchastransformer).
profitLTCand(ii)maximizetheprofitoftheLTCbyensuring
thebenefitsoftheconsumerandseller.Ademandmanagement
whichaglobaloptimalsolutionisnormallydifficulttoobtain.
of an electrical network of interconnected MGs formulated as
In [19], the authors showed that a semidefinite programming
a power dispatch optimization problem is given in [16]. Here,
(SDP) method, known as moment-sos, converges towards
a real time price is employed as the motivation for interaction
a global optimal solution to the OPF problem at a higher
between MGs.
runtime cost. The authors of [20] designed a semi-definite
In this paper, we propose to minimize the electricity cost
dual optimization model for solving the OPF problem for the
for the MGN in a deregulated competitive electricity market.
practical electrical network and showed that their proposed
Westudyandfindthat,foranon-decreasingmarginalcost,the
model solves a dual of the OPF problem which gives a global
modelisnonlinearandnon-convex.Hence,itdoesnotproduce
optimal solution for most of the practical electrical networks.
an optimal solution. Therefore, we decompose the model to
In summary, this paper makes the following contributions
separate the marginal cost from the transmission cost and de-
to the MGN energy management system:
velop a novel method based on a divide-and-conquer strategy
• WeshowthattheproposedMEPMcandetermineoptimal
which is referred to as MEPM (minimum electricity pricing
electricity costs of an MGN for any non-decreasing
model),tosolveitoptimally.First,wedeterminethemarginal
monotonic cost function or a mix of them.
cost boundary according to the overall demand of the MGN,
• We decompose a nonlinear and non-convex MGN cost
also known as the overall marginal cost problem (OMCP).
model into a unique divide-and-conquer based model
Then, using the proposed MEPM strategy, we interactively
MEPM, which (has two optimization models, namely
determine the optimal electricity price by jointly optimizing
the OMCP and the allocation problem) solves the MGN
theOMCPandtransportationcosts(allocationproblem)ofthe
electricity pricing problem optimally.
system. The proposed MEPM algorithm optimally determines
• We prove that MEPM is a polynomial time algorithm
the energy price of the MGN in a polynomial time [17] and
[17]. Therefore, it is suitable for real-time pricing of
is independent of the marginal cost functions. We assume
MGN.
that each MG internally decides its consumption using a
• We show that the OMCP can determine the optimal
demand response model (such as in [18]) which integrates
overall marginal cost for a product (such as electricity)
local generation from renewable sources, dynamic loads and
which has different production costs in different produc-
storage. The total demand and the amount generated by each
tion plants.
generator and capacity of the generator at the real time are
• The MEPM algorithm is amenable to run in a parallel
known to the MEPM system. The MEPM is a divide-and-
computing environment.
conquer based scheme which is scalable and relies primarily
Theremainderofthispaperisorganizedasfollows.SectionII
on the range of the overall marginal costs rather than the size
discussesthesystemmodelincludingelectricitycostfunctions
of the network. MEPM solves the energy trading problem for
and transportation cost of the MGN. Section III depictes the
MGNtodeterminetheoptimalelectricityprice,generationand
energypricingmodelwhichincludesmathematicalmodelsfor
transmission costs. The MEPM is amenable to multicore and
minimum electricity price. The minimum electricity pricing
parallelprogramming.Besidesdecidingtheoptimalelectricity
model is decomposed, and MEPM method is presented in
price, the proposed method also suggests the amount of
Section IV. FCFS (first come first serve) model is discussed
generation for each generator with minimum cost, the path of
in Section V. We present the simulation results in section VI.
energyflowandcanadapttotheintegrationofnewcustomers.
Finally, we conclude the paper in section VII.
We consider quadratic, linear, piecewise convex, nonlinear
non-convex cost models for the non-renewable and Levelized
cost of electricity for the RES. II. SYSTEMMODEL
At this level, we should note that our problem has some We consider a MGN (Fig. 1), consisting of a set of N
resemblance to the Optimal Power Flow (OPF) problem for smart microgrids N, connected with each other by means of
3
transmission lines. Let the set of transmission lines (feeders) important feature of the cost curve is to be monotonic non-
beLwhichareconnectedwitheachotherusingabidirectional decreasing [7]. The cost of a product (such as electricity) is
LV/HV interfaces (transformers) as shown in Fig 1. The set dependent on various factors such as, quantity, investment,
of feeders represents the virtual connections between MGs, labor, fuel (such as gas, oil, wind, solar radiation etc), market
therefore creating our MGN. Let a set of all pairs (seller, demand,establishments,etc.Therefore,itisnearlydifficultto
buyer)ofMGsbeK (K ⊂N N),eachconnectedthrough express the cost by a regular curve (function) [7], [24], [25],
i i
a feeder l (l ∈L). A feeder l has a capacity limit Γ . The [26]. The truth is that a unit cost (marginal cost) or total cost
i i i i
flow of energy among them is controlled by mutual decisions neverdecreaseswiththeincreaseoftheamountofproduction.
of the management system. At any instance of time, let M The amount of production is determined by the total demand
and B be the set of sellers and buyers, where M∩B = ∅ of a market for an instance of time. Let E be the amount
m,w
and M∪B ⊆ N. Each MG has a set of DGs W (n ∈ N), of electricity generated by a generator w of a seller microgrid
n
primarily to fulfill the local demand. A MG sells energy in m; then the total cost of producing E is [7], [26]:
m,w
case of surplus or buys energy, when the demand is more
C(E )=v(α,E )+cE +d, (1)
than its production. Let c be the cost of transporting one m,w m,w m,w
m,b
unit (1 kWh) of electricity from seller m (m∈M) to buyer where α,c,d∈R+, and c & d are the minimum cost (fixed)
b (b∈B), and E (n∈N,w ∈W ) be the pre-authorized forproducingoneunitofelectricityandproducingnothingre-
n,w n
amount of electricity generation of energy source w with spectively,whichisdependentoncapitalorinitialinvestment.
capacity EC of smart microgrid n. Here, E ≤ EC , v(.) is the variable cost which is a non-decreasing monotonic
n,w n,w n,w
and the pre-authorized generation (E ) of DG w of n is function. The parameter α is dependent on the running cost
n,w
controlled by the energy management system (EMS). The of the system which includes labor, fuel, maintenance and
EMS decides the price (µ), to satisfy the MGN electricity various other costs. For a sufficiently long running system,
demand D˜ (total amount of electricity buyers want to buy), the short running fixed cost (c) becomes a variable because
B
from the prices (monotonic non decreasing) proposed by each investment in facilities, equipment, and basic organization
of the seller MGs. Then, the buyer b (b ∈ B) contacts the cannot be significantly reduced in a short period [27], [28].
sellers(M)tobuyelectricityfromthemtocompensateforits Each time, a utility company determines the marginal cost of
shortage. the product and uses it to determine the selling price of the
product. For the survival of the utility company, the selling
TABLEI
price must be higher than the marginal cost of the product.
NOTATION
1.0 (a) Nonlinear Nonconvex 1.0 (b) Piecewise Convex
Wh) 0.8 Wh) 0.8
A. System Assumption Cost ($/k 0.6 Cost ($/k 0.6
MGWNe.eSmimpilloayr tAoNseSvIerCal84V.o1lt-sVtaAndRarodptivmolitzaagteionrarteinsgearfcohr,twhee Marginal 0.4 Marginal 0.4
assume that the MGN system uses a lower voltage (from 0.2 c 0.2 c
range A of C84.1) as the service voltage to minimize the
losses. Each of the MGs has a linear vector that comprises 0 100 200 300 400 500 0 100 200 300 400 500
Amount Generated (in kW) Amount of generation (in kW)
the cost of electricity transportation from the sellers to the
buyers,andthecapacityofthetransmissionlines.Tominimize 0.7 (c) Marginal Cost (Cubic cost) 0.7 (d) Step function
the thermal losses, we assume the maximum capability of the Wh) 0.6 Wh) 0.6
tdreadniscmatiesdsiotrnanlisnpeoritsatpiorendesyfisnteedm,.wAhlestoh,etrrathnesmyiussseioan schoasrtecdmo,br Cost ($/k 0.5 Cost ($/k 0.5
insetpwaoyrakblwehtiochthiseimowpnoesredobfyththeetrtraannssmmisisssioionnaannddddiissttrriibbuuttiioonn Marginal 0.30.4 Marginal 0.4
operator in a competitive electricity market. The transmission 0.2 c 0.3 c
is equipped with VAR compensation component at the receiv-
0 100 200 300 400 500 0 100 200 300 400 500
ingpointtoraisethepowerfactortounity(ornearunity).We Amount of generation (in kW) Amount of generation (in kW)
also assume that i) bi-directional electricity flow, ii) coupling
Fig. 2. Electricity marginal cost curve for (a) nonlinear noncovex, (b)
between microgrids, and iii) integration of renewable energy
piecewiseconvex,(c)cubic,and(d)stepcostfunctions.
sources to the grid should comply with the standard found
in [21], [22], [23] and IEEE1547A, IEEE1547.4 standard.
Definition 1 (Marginal Cost). The marginal cost function
Theenergygeneratedfromtherenewablesourcesinpredicted
µ (E ) of generator w of operator m, represents the
using the model described in [18]. m,w m,w
price of electricity for producing one more unit [7], [29]. For
B. Marginal Cost & Cost Function any cost function in eq. (1), it is defined as,
It is widely accepted that the cost functions are cubic µ (E )= ∂C(Em,w) (2)
in nature (approximated) but in reality the only and most m,w m,w ∂E
m,w
4
The marginal cost is a non-decreasing monotonic function, where Lli is the inductance (in henries) of the transmission
m,b
and it can be determined by the first order derivative of the line l from m to b, f is the frequency in Hz. There-
i
total cost function. Hence, the marginal cost function can be fore, the power factor of the transmission line is cosθ =
expressed by a linear, quadratic, step, piecewise convex or Rli
m,b . Here, θ is the angle between apparent
(cid:113)
nonlinearnon-convexcost-function(seeFig.2).Forsimplicity, (Rli )2+(Xli )2
m,b m,b
most researchers assume the marginal cost or cost function to power and active power. Now, to reduce the losses, let the
be convex or approximate it near to a convex function [7], power factor of the transmission be cosφ (near to unity and
[29], [30], [31], [32]. In reality, however, this is not accurate; θ >> φ); that is θ is reduced to φ. To do so, a reactive
the marginal cost can not be expressed by a regular function compensation equipment (such as, a shunt capacitor bank)
suchaslinear,nonlinear,convex.Indeed,itisafunctionwhich is added at the input of the buyer microgrid (b). Let the
is irregular in nature and non-decreasing with the increase of capacitance of the capacitor bank be πm,b and the amount
the production quantity. Also, the marginal cost function of ofenergytransferred(withoutloss)frommtobbexm,b then,
a company having more than one generator is more complex (according to [33]),
andcertainlynonlinearnon-convex,eveniftheindividualcost
(x +xd )cosθ(tanθ−tanφ)
function is linear [29], [24]. The operation and maintenance π = m,b m,b , (5)
m,b 2πfV2
cost of the non-renewable energy sources increase with the b
amount of production. Therefore, the marginal cost described
where V is the voltage at a buyer microgrid b and xd
in the Appendices (from A-1 to A-4) are suitable for the non- b m,b
is the total loss (resistive and reactive) of electricity while
renewable energy sources. Whereas the operation and mainte-
transmitted from m to b. In general the value of the receiving
nance cost of the renewable energy sources barely increases
voltage(V )shouldbewithin±5%(V ±5%)ofvoltage(V )
with the amount of the electricity generation. The LRMC b m m
at m [33]. Here, we assume V is chosen a value between
(long run marginal cost) also known an LCOE (Levelized b
V and (V −1%). Therefore, the energy loss xd due to
cost of electricity) is used to determine the energy cost of m m mb,
transportation of electricity from a seller m to a buyer b is,
the renewable energy sources (see Appendix A-5).
As mentioned, for the survival of a company, the selling xd =I2 (Rli cosφ+Xli sinφ), (6)
price should not be less than the marginal cost. The marginal m,b m,b m,b m,b
cost is a monotonic non-decreasing function [29]. Now, let
andthetotalamountofelectricityneededtobetransportedby
E be the total amount of electricity generated by the MGN a seller m to a buyer b is,
system, then the total cost of the electricity to fulfil excess
demand D˜B is, µ(E)·D˜ (3) x˜m,b =xm,b+xdm,b (7)
B
Then, the total transportation cost of MGN is,
where µ(E) is the overall marginal cost of the MGN system,
µ(E) ≥ µ (E ), and E = (cid:80) (cid:80) E . We use (cid:88)
m,w m,w m,w TC = (c ·x˜ ) (8)
m∈Mw∈Wm m,b m,b
the marginal cost functions from Appendices A-1 to A-4 to b∈B,m∈M
calculate the cost of nonrenewable energy sources and LCOE
A buyer may choose another seller to reduce the amount of
(Appendix A-5) for renewable energy sources. The model is
energy it needs to buy if the transmission (or transportation)
applicable for any of the other marginal cost functions by
loss is lower than the current seller. Let D˜ be the shortage
replacing the marginal cost functions illustrated above with of electricity of buyer b, then (cid:80) x =bD˜
the appropriate non-decreasing marginal cost function. m,b b
m∈M
Definition 2. [Allocation Problem] The optimal matching of
C. Electricity Transportation
sellersandbuyerswhichwillminimizetheoverallenergycosts
In our MGN system, we assume that all the microgrids
to the customers can be formulated as,
are connected with each other using electricity transmission
and distribution lines. Most of the energy losses in electricity
(cid:88)
transportation are due to the resistance of the energy network minTC =min (c ·x˜ ) (9)
m,b m,b
andreactivepowerwhichisinjectedbythereactiveload.The
b∈B,m∈M
T&D (transmission and distribution) losses for a transmission
and distribution line is I2 (Rli +jXli ), where Rli is 1) Transformer thermal limit: Both ends of the transmis-
m,b m,b m,b m,b
the resistance, Xli is the reactance of the transmission line sion line are connected to a delivery (at m) and a receiving
m,b (at b) transformer. The amount of current flow through these
from m to b, and j is the complex variable dependent on
transformers generate heat which is typically resolved by
the phase of the voltage V (voltage at seller m) and current
m
coolant (such as oil). Hence, a transformer has an upper limit
I (amount of current a seller m sends to b). The values of
m,b
Rli and Xli are dependent on the physical characteristics ofenergyhandlingcapacitytosustainandextenditslifetime.
m,b m,b
ofthetransmissionline.Xli (inohm)isthereactanceofthe Beyond this limit, the transformer temperature increases and
m,b mayburnoutorshortenthelifeofatransformer.Therefore,a
transmission line which can be expressed as,
transformer must not handle electricity beyond a rated power.
Xli =ωLli =>2πfLli (4) The hot-spot temperature of the transformer can be computed
m,b m,b m,b
5
for any load by using the following standard relations which xd =I2 (Rli cosφ+Xli sinφ) ∀m,∀b (19)
m,b m,b m,b m,b
are given in [34],
ωHS =ωTO+∆ωHR(cid:32)IImR,b(cid:33)2e (10) Im,b = x˜Vmm,b, ∀m∈M,∀b∈B (20)
m,b
x˜ cosθ(tanθ−tanφ)
where ωHS, ωTO, and ∆ωHR are the hot-spot temperature, πm,b = m,b 2πfV2 , ∀m,∀b (21)
top-oil temperature, and rated hot-spot temperature rise above b
top-oil respectively. I , IR , and e are the load current, the
rated current and themw,bindmin,gb exponent accordingly. Now, if Vm =Vb+Im,b(Rmli,bcosφ+Xmli,bsinφ) (22)
thevoltageattheinputofthetransformerisV then,therated
b
power of the transformer is: xm,b ≤Pmr,b, and x˜m,b ≤Pmd,b, ∀m,∀b; (23)
Pm,b =ImR,bVb (11) (cid:88)x˜ ≤E˜ , ∀m∈M (24)
m,b m
whereP istheratedpowerofthetransformerandtherefore
m,b b∈B
it limits the input energy of the transformer as, E˜ = (cid:88) E −D , ∀m∈M (25)
m m,w m
x ≤Pr , for receiving transformer (12)
m,b m,b w∈Wm
and (cid:88) E˜ ≥D˜ , where D˜ =(cid:88)D˜ (26)
m B B b
x˜ ≤Pd , for delivering transformer (13)
m,b m,b m∈M b∈B
where Pr and Pd are the receiving and delivering rated
m,b m,b µ(E) ≥µ (E ), ∀m∈M, ∀w ∈W (27)
power of transformers at m and b. m,w m,w m
(cid:88) (cid:88)
E ≥ E ≥D (28)
m,w B
III. ELECTRICITYPRICINGMODEL m∈Mw∈Wm
It is clear that the electricity price a buyer has to pay is where variable E˜ is the excess generation of electricity by
m
dependent on both the overall marginal cost and the trans- seller m, and D , D represent the total actual demand
B m
portation cost of electricity. For simplicity, we assume that of the MGN buyers and the seller demand respectively. Γ
i
the profit of the seller for selling electricity is included within (l : (m,b) ∈ K ) is the transmission capacity of the line
i i
the marginal costs. Therefore, the price (paid by the buyer) of (l ) which connects a pair of MGs in K . The solution to the
i i
electricity can thus be expressed as:
above optimization model (from eqs. (15) to (28)) will yield
µ(E)·D˜ +(cid:88) (cid:88) (c ·x˜ ) (14) the joint optimal overall marginal cost and transportation cost
B m,b m,b
for the MGN. Eq. (16) states that the electricity transmitted
b∈Bm∈M
fromsellers(M)mustbeequaltothetotalshortage(orexcess
Here, the overall marginal cost (µ(E)) depends on the
demand: D ) of the buyer microgrid (b). The constraint in eq.
b
marginalcostsofthesellers.Hence,thereareamplescopesto
(17) limits the total flows of electricity from m to b through a
determine the optimal electricity price by jointly considering
transmission or distribution line which should not surpass the
overall marginal cost and transmission losses, in an optimiza-
capacity of the line. The second part of eq. (18) ensures that
tion model, and subsequently solve it optimally. Therefore,
theamountofelectricitytransportedfrommiszerowhenx
m,b
a model needs to be developed which will concurrently
is zero. The current flow through l from m to b is calculated
i
minimize the overall electricity price (14).
in eq. (20). Constraint (23) ensures that the delivered and
receivedelectricityshouldbelessthanthepowerrating(Pd
m,b
A. Minimum Electricity Pricing Model (MEPM) and Pr ) of the transformers attached. The selling amount
m,b
Eq. (14) illustrates the total payment of electricity of the of power should satisfy the amount of electricity buyers (B)
MGN buyers. Therefore, the minimum total payment for wants to buy from a seller m which is outlined in eq. (24).
electricity can be determined by solving the following model, Eq. (25) asserts that the total excess amount of generation
is equivalent to the sum of excess electricity generated by
(cid:32) (cid:33)
min µ(E)·D˜ +(cid:88) (cid:88) (c ·x˜ ) (15) all sellers. The MGN total excess production must satisfy the
B m,b m,b
shortageofelectricitywhichismanifestedineq.(26).Finally,
b∈Bm∈M
the overall marginal cost of the electricity for the MGN is
Subject to,
(cid:88) xm,b =D˜b, ∀b∈B (16) detHeremrei,neifdablyl ethqe. (m27a)r.ginal cost functions µm,w(Em,w) are
m∈M convex,thentheMEPMproblemremainsanonlinearandnon-
convex problem due to the overall marginal cost which is the
(cid:88)
x˜m,b ≤Γi,∀li ∈L (17) superimposition [35] of all the marginal costs. Therefore, the
li:(m,b)∈Ki above MEPM problem is a difficult (NP-Hard) problem, and
no polynomial solution exists [36]. Moreover, in practice, the
x˜ =x +xd ; xd ≤x , ∀m,∀b (18) marginal cost function does not need to be convex; rather,
m,b m,b m,b m,b m,b
6
the more accurate property of the marginal cost function is Proof: The proof is provided in Appendix B
monotonicnon-decreasing[7].Themonotonicnon-decreasing From Lemma 1, we find that each overall marginal cost, µi
function will increase or remain the same by increasing the (µl ≤ µi ≤ µu) has a minimum transportation cost (TCj) or
productionofelectricity.Toobtainapolynomialtimesolution minimum average transportation cost Pˆj, where, Pˆj = TCj.
of the MEPM problem, we decompose into two subproblems, LettheoptimalsolutionoftheMEPMpBroblembeµBo×D˜D˜B+
B
the overall marginal cost problem (OMCP) and an optimal Pˆq ×D˜ , then, µo ×D˜ +Pˆq ×D˜ ≤ µi ×D˜ +Pˆj ×
B B B B B B B
electricityallocation.Thesetwosub-problemsarebothusedas D˜ , ∀µi ∈{µl,µu}\µo and ∀Pˆj ∈{Pˆu,Pˆl}\Pˆq, and Pˆq
B B B B B B
modules in our solution methodology of MEPM to determine
istheminimumaveragetransportationcostwhenthemarginal
the optimal solution to our original problem. OMCP is used cost is set to a value, µo.
to determine a feasible interval for the optimal marginal cost,
with a lower bound (µl) and an upper bound (µu). Now, the
problem reduces to a search for the optimal marginal cost A. Algorithmic Solution for MEPM
that yields optimal overall price of electricity (µc+PˆBc). The The MEPM algorithm chooses a value for µi between µu
MEPMperformsasearchfortheoptimalmarginalcostandat andµl,anddeterminestheminimumtransportationcost(Pˆj).
B
each iteration (after solving an allocation problem) it removes The optimal solution is the lowest value of the summation
a segment of the feasible interval that is of no use (i.e., a of µi and Pˆj. An efficient polynomial solution (divide-and-
B
marginalcostinasegmentremovedbythemethodwillalways conquer)oftheMEPMproblemispresentedinFig.3.InFig.
result in higher overall price). Our solution methodology 3, steps (III) to (XIV) are repeated while all the partitions
follows a divide and conquer approach since such method is are deleted and the MEPM scheme terminates with minimum
deterministic (it will always converge and return the optimal (optimal) per kWh price µc +Pc of the MGN network. In
B
result) and enjoys low complexity. In the following sections, short,theMEPMmethoddividesthemarginalcostspace(Fig.
we describe the decomposition and polynomial time solution 3, step (IV)) into two partitions, then determines the average
of MEPM. transmissioncost(Fig.3,step(VI)).Then,theMEPMdiscards
one or both partitions when the minimum possible cost of the
IV. DECOMPOSITIONOFMEPM partition(s) is greater than µc +Pc (Fig. 3, steps (VIII) and
B
The MEPM clearly is a combination of two inter-related (XII)). Otherwise, it updates the µc+Pc (Fig. 3, steps (IX)
B
optimization problems, (i) minimum overall marginal cost and (XIII)) and repeat the same divide-and-conquer method.
problem (OMCP) by setting the value of cm,b = 0 and The MEPM solution contains two sub-problems, hence the
Em,w =EmC,w, and (ii) minimum transportation cost problem solution of the MEPM is derived by combining the solution
(allocation problem) by setting the value of µ(E) = 0 in of OMCP (Fig. 3, step (I)) and the allocation problem (Fig.
the objective function (Def. 2). The overall marginal cost has 3, step (VI)) interactively.
lowerandupperboundvalues.TheMEPMisinfeasiblebelow 1) Algorithm for OMCP : One of the objectives of our
the lower bound overall marginal costs. Beyond the upper MEPM algorithm is to determine the overall marginal cost
bound value, the system will always produce the same cost and E˜ , ∀m∈M, by solving the OMCP which is described
m
for transportation but the overall marginal cost will increase. in Section IV. Suppose, both µ(E) and E˜ are unknown and
m
Definition 3. [Lower Bound Overall Marginal Cost, µl] The the excess demand D˜B, EmC,w are known. The mathematical
lowerboundoverallmarginalcost,µl ≡µ(E)isthecombined model for OMCP can be realized by replacing x˜m,b = 0 in
marginal cost that is calculated while (∀w ∈W , ∀m∈M), eq. (15) (objective) and eqs. (25) – (28) as constraints of the
m
E and c of MEPM are set to EC and 0 respectively. objective. The solution of the OMCP model will determine
m,w m,b m,w
In other words, the overall marginal cost below µl must be the overall marginal cost µ(E) by achieving the minimum
infeasible for the MGN, i.e., E <D˜ , where E ≤EC . total cost of the excess demand D˜B for any configuration of
B m,w m,w the MGN. Then, µ(E) and E˜ can be determined by the
m
Definition4. [UpperBoundMGNOverallMarginalCost,µu] following steps which are also shown in Fig. 4.
Theupperboundoverallmarginalcost,µu ≡µ(E)isthecom- Step 1:Divide the excess demand D˜B among the generators
bined marginal cost that is calculated after the optimization (∀w ∈ Wm) of all sellers (∀m ∈ M), and determine
of the transportation costs. The optimization of transportation the marginal costs µm,w(Em,w) (see, eq. (35), (38), (40),
costs is carried out with an initial setting of MEPM, where (42),and(45)).Taketheµmin =min{µm,w(Em,w)|w∈
µ(E)=0 and E =EC , (∀w ∈W , ∀m∈M). Wm,m∈M}.
m,w m,w m Step 2:AdjustEm,wwithacalculated(step1)marginalcostµmin
The overall marginal cost beyond µu does not have any using eqs’.(36), (39), (41), (43), (46) or (37).
eEffect obnetwtheeentratnhsepovratlauteiondecteorsmti(nTedC)bybetchaeusoeptainmyizvaatiloune ooff Step 3:Dm˜oBd=ify(DD˜˜BB−with(cid:80)theE˜nmew) aanmdoduinsctaordf gwen∈erWatimon,,Esumc,hwa≥s,
m,w m∈M
transportationcostandEmC,w willyieldthesametransportation EmC,w. Repeat Step 1 to Step 3 while D˜B >0.
costTC andtheMGNsystemisinfeasibleforEm,w >EmC,w. Step 4:E˜m = (cid:80) (Em,w−Dm), ∀m∈M and µl =µmin.
w∈Wm
Step 1 (above) and (ii) in Fig. 4, return the marginal costs
Lemma 1. The decrease of the overall marginal cost µ(E) of a generator to produce the excess amount of electricity.
from upper bound µu to lower bound µl will monotonically However, since a generator cannot produce more electricity
increase the value of the minimum transportation cost (TC). than its capacity, the OMCP (ii in Fig 4) will then return a
7
marginalcostofageneratorwithonlyitsmaximumgeneration
capacityiftherequestedexcessamountofgenerationexceeds
the capacity (EC ).
m,w
StartMEPM
Second, let E be known, then the overall marginal
m,w
cost is µi = max{µ (E )|w ∈ W , m ∈ M}, and
m,w m,w m
(mII)inin[p(auµntdiln[iµ+tiaclPµˆ+Blu,)P,µˆ(Bucµ,]uPˆ+←Bl,PˆBPlˆB)u] MivnaAluvegsT(I(r)µaRnlus,nCµOosuMt,tCoPPˆgBealtn,dthPeˆBiuni)tial tµhme,awl(loEcmat,iwo)niosfdeeletecrtrmiciinteyd(x˜bmy,bS)teispd2e.teTrmhiisneisdtbhyescoalsveinwghtehne
allocation problem before solving the OMCP. With maximum
µu(I≤II)Iµsl? Y (X.I)[Fµolr[+µum≤inµ(Plˆ]Bu:[µ≤k+PˆPBˆBk)]]← coavpearacliltymoafrgthineaDl cGoss,ttµheus(oµluut=ionµrie).sults into the upper bound
IsPˆBuO≤RPˆBl ? PˆBkF]or[←PˆBu[m≤inOP(ˆRµBll],:µ[µuk),+PˆBl] genTehriartdio,nwohfenthteheDvGasluies doefteµrimiisnekdnoawtnS,tetphe2nothre(isvu)gignesFtiegd.
N 4. This calculation is repeatedly used in the MEPM algorithm
b(IyVµ)iPartit=ion[µµll,+2µµuu] min((X[µ.IkI)[+µcPˆ+Bk],Pˆ[Bµcc] +←PˆBc]) to determine E˜m (in Fig. 3, V).
Lemma 2. [Optimal Overall Marginal Cost] The solution of
(V)DetermineE˜m,∀m ∈ OMCP (Fig. 4) determines the optimal overall marginal costs
Musingmarginalcostµi (XIV)Returnor
terminatewith for any configuration of MGN.
(VI)SimplexAlgorithmForLPAllocation [µc,PˆBc]
Problem:MinAvgTransCostandgetPˆBj Proof: Let the overall marginal cost determined by the
solution of OMCP presented in Fig. 4 be µ(cid:48)(E), where
(VII)Update[µc +PˆBc] ← N E = DB (total demand of the MGN). Let the optimal
min[(µc+PˆBc),(µi+PˆBj)] overall marginal cost for the MGN configuration be µ∗(E).
Ifµ(cid:48)(E)≤µ∗(E),thenµ(cid:48)(E)istheoptimaloverallmarginal
cost. Now, suppose, the OMCP shown in Fig. 4 is unable to
(VIII) (XII)R:
L(:(µµcl++PPˆˆBjBc))<? N ((µµic++PˆPˆBuBc))<? parcoodnuficeguarnatioopntiomfaMl sGoNlu,tiio.en.,oµf(cid:48)t(hEe)o>veµra∗l(lEm)a.rIgfinthalisciossttrufoer,
Y then there are at least two generators (in the proposed solu-
(IX)Updateµu ←µi,PˆBl ←PˆBj tion),whichgeneratedifferentamountofelectricitycompared
Return Y to the optimal solution (shown below):
(XI)[µc + PˆBc]
Optimal Solution Generation Set (G*)
(XIII)Updateµl ←µi,PˆBu ←PˆBj G∗ ={E1,1,···, Em,w ,···, Em(cid:48),w(cid:48) ···}
and
Fig.3. SchematicdiagramofMEPMAlgorithm;L:leftpartition;R:right
partition µ∗(E) =max(µ (E )···, µ (E ),···,
1,1 1,1 m,w m,w
(29)
µ (E ),···)
m(cid:48),w(cid:48) m(cid:48),w(cid:48)
where
E = (cid:88) ( (cid:88) E −D )=D˜
m,w m B
m∈M w∈Wm
StartOMCP EDm˜B,w;I,nitµialmizein←0 MEPM Generation Set (G(cid:48))
(30)
G(cid:48) ={E ,···, E(cid:48) ,···, E(cid:48) ···}
(ii)Determineµm,w(Em,w) 1,1 m,w m(cid:48),w(cid:48)
foEr(4mCt0h)e,,ewaxntdra:(r4ge2et)nµ;efrmoatrio,Enwme(qE.,w(3mC5),,w>(38)), (i)thDeivgiedneeDr˜atBorse∀qu(amlly,awmo)ng and
µ(cid:48)(E) =max(µ (E(cid:48) )···, µ (E(cid:48) ),···,
(iii)Determine (iv)Adjustthegenerationof(Em,w) 1,1 1,1 m,w m,w (31)
µmin←min{µm,w(Em,w)} allgeneratoraccordingtothecost
∀w ∈ Wm,∀m ∈ M µmin(36),(39),(41),(43),(45)or(37) µ (E(cid:48) ),···)
m(cid:48),w(cid:48) m(cid:48),w(cid:48)
(v)Deletegeneratorsfromthelist
Y (vi)IsD˜B>0 whereEmUp,dwateD˜=BEmC,w where E = (cid:88) ( (cid:88) Em(cid:48) ,w−Dm)=D˜B (32)
N m∈M w∈Wm
rectousrtnµmmargiinnal Stop Lettheindexofthetwogeneratorsbe(m,w)(generatorw of
microgrid m), and (m(cid:48),w(cid:48)) (generator w(cid:48) of microgrid m(cid:48)).
Now, suppose the amount of generation of both generators
Fig.4. SchematicdiagramofOMCPAlgorithm
determined by optimal solution be Em,w and Em(cid:48),w(cid:48) (eq.
(29)) and OMCP be E(cid:48) and E(cid:48) (eq. (30)). Let us also
m,w m(cid:48),w(cid:48)
assume (without loss of generality) that (m,w) is a low-cost
8
generatorand(m(cid:48),w(cid:48))isahigh-costgenerator.Now,theclaim withone(leftorright)orbothpartitions,otherwise(µc+Pˆc)
B
(µ(cid:48)(E) > µ∗(E)) is true if and only if E(cid:48) < E and istheminimumpayment.ThedetailsoftheMEPMalgorithm
m,w m,w
Em(cid:48) (cid:48),w(cid:48) >Em(cid:48),w(cid:48), therefore, the costs which are calculated in arepresentedinAlg.1.Initially,thealgorithm(fromlines5to
theoptimalsolution(µ∗(E))arelessthanthecostsdetermined 7)comparestheupperandlowerboundvaluesofbothoverall
by the OMCP solution (µ(cid:48)(E)). This leads us to a fact that marginal costs and transportation costs. Alg. 1 will terminate,
thelow-costgenerator(m,w)mustproducemoreorequal(at if the upper and lower values of the overall marginal costs or
least) amount of electricity (with a greater overall marginal transportationcostsarefoundsimilar,otherwise,thealgorithm
costµ(cid:48)(E))inOMCPthantheamountofelectricityproduced continues as discussed above.
bythesamegeneratorintheoptimalsolution(see,step(iv)in
Fig. 4)1. This is always true because the marginal cost is non- Algorithm 1 Algorithm for MEPM.
decreasing (see Section II-B). Therefore, the total amount of 1: procedureMINMGNCOST(µl,µu,PˆBl,PˆBu,µc,PBc)
excess electricity produced by the generators in OMCP must (cid:46)µc,PˆBc aretheglobalvariables,functionmin()returnsthevaluepair,sumof
be greater than the total demand of the buyers or E > D˜B 2: whiIcnhitiisalmizien:im[µucm,PˆBc]←min([µl,PˆBu],[µu,PˆBl])
which is a contradiction according to the steps (i) and (vi) 3: 0<(cid:15)<<1 (cid:46)(cid:15)isaverysmallvalue
of the OMCP solution presented in Fig. 4. In general, let E(cid:48) 4: Be(cid:12)gin (cid:12) (cid:104) (cid:105)
5: if(cid:12)µu−µl(cid:12)≤(cid:15)thenreturn µu,min(Pˆl,Pˆu)
be the electricity produced by the OMCP, then we claim that (cid:12) (cid:12) (cid:12) (cid:12) (cid:104) B B (cid:105)
E(cid:48) =E =D˜ , but 6: elseif(cid:12)(cid:12)PˆBu−PˆBl(cid:12)(cid:12)≤(cid:15)thenreturn PˆBu,min(µl,µu)
B 7: endif
8: µi← µl+µu
if µ(cid:48)(E(cid:48)) >µ∗(E) then (33) 9: PˆBj ←MI2NAVGTRANSCOST(B,M)(cid:46)SimplexorInteriorPointmethodfor
solvingtheallocationproblem.
E(cid:48) >E, 10: [µc,Pˆc]←min([µc,Pˆc],[µi,Pˆj])
B B B
11: if(µl+PˆBj)<(µc+PˆBc)then (cid:46)leftpartition
because, some or all generators in OMCP will produce more 12: [µk,PˆBk]←MINMGNCOST(µl,µi,PˆBj,PˆBu,µc,PˆBc)
electricity (without violating the capacity constraint) than the 13: [µc,PˆBc]←min([µc,PˆBc],[µk,PˆBk])
14: endif
amount determined by the optimal solution. This is a direct 15: if(µi+PˆBl)<(µc+PˆBc)then (cid:46)rightpartition
violation of steps (i) and (vi) of the OMCP solution (see Fig. 16: [µk,PˆBk]←MINMGNCOST(µi,µu,PˆBl,PˆBj,µc,PˆBc)
17: [µc,Pˆc]←min([µc,Pˆc],[µk,Pˆk])
4). The steps (i) and (vi) of the OMCP solution controls the B B B
18: endif
amount of total excess production to the total demand of the 19: return[µc,PˆBc]
20: End
buyers. The solution is valid, if only if both or all (in general)
21: endprocedure
generators produce the same amount of electricity which is
determined by the optimal solution. Thus, the OMCP method
always determines the optimal overall marginal cost of any
configuration of the MGN system.
2) Solving the Allocation Problem (MinAvgTransCost): B. Analysis of the MEPM algorithm
Step (VI) of Fig. 3 is the solution for the allocation problem
Lemma 3. [Minimum Payment] Algorithm 1 determines the
(defined by Def. 2) by setting the value of µ(E) = 0 of the
optimal price for electricity.
MEPM objective function in eq. (15) and taking (16) to (26)
as the constraints. The inputs to the allocation problem are Proof: Alg. 1 divides the overall marginal cost space
E˜m, ∀m ∈ M which are determined by the algorithm for [µl,µu] into two, [µl,µi] and [µi,µu], then, it determines the
TOhMeCvPaluaendofthPeˆBjouistptuhteislotwheermbionuimndumvalturaen(sPpˆoBlr)taftoiornelceocstrticPˆiBtjy. mThineimsoulumtiotrnanosfptohretaatiloloncactoisotn,PpˆrBjobfolermov(eQraLlPl mpraorgbilnemal)caolswtaµyis.
transportation, when E˜m = ( (cid:80) EmC,w −Dm), and upper givestheoptimalvaluePˆBj foranoverallmarginalcostµi.Alg.
w∈Wm 1 discards one partition ([µl,µi] or [µi,µu]) or both when the
boundvalue(Pˆu)whenE˜ istheoutputofOMCPforµl.The
B m minimum possible cost of each partition is greater than the
mathematical model of the allocation problem is a Quadratic current global minimum cost (µc +Pˆc). Suppose, partition
Programming (QP) problem with continuous variables x˜ B
m,b [µi,µu]isdiscarded.Weclaimthatthereisanoverallmarginal
(∀m ∈ M, ∀w ∈ Wm). Hence, the problem can be solved cost µk, (i≤k ≤u) and transportation cost Pˆq (j ≥q ≥l)
(using the interior point or the Simplex algorithm) in a whichproduceminimumpayment((µk+Pˆk)×DB )ofMGN.
polOynncoemtihaletmiminei.mum transportation cost (PˆBj) is determined pTohsesicbllaeimbeicnaduicseateµsith≤atµ(kµk≤+µPˆuBk)an<d(Pµˆjc+B≥PˆPBˆc)k,B≥butPˆitl,isthnuost
(for µi), then, we update [µc,PˆBc] ← min([µc,PˆBc],[µi,PˆBj]) (µk+Pˆk)≥(µi+Pˆl) and Alg. 1 discaBrds a pBartitionB(lines
(mseinei,mstuempsp(VayImI)eanntdµ(Xi +II)Pˆofl Faingd.3µ),la+ndPˆcjomofparrieghthte([pµois,sµibul]e) 11and1B5),iff (µi+PˆBBl)>(µc+PˆBc),hencetheclaimisfalse.
B B Further, if the values for any pair (overall marginal costs or
(aµndc+lePfˆtc()[µ(sle,eµ,iA])lpga.r1titfiroonmswlinitehst1h1etcou1rr7e)n.tIfmainnyimourmbopthayomfethnet transportation cost) are the same (µl =µu or PˆBl =PˆBu), then
partitioBns’ possible minimum costs are lower than (µc+Pˆc), Alg. 1 returns the minimum cost by taking the minimum of
B inequal cost pair (overall marginal cost or transportation cost)
then Alg. 1 (or the steps (III) to (XIV) of Fig. 3) is repeated
and value of equal cost pair. Thus, Alg. 1 solves the MEPM
problem correctly. The complexity of the MEPM algorithm is
1At a higher marginal cost, a generator will produce more electricity
comparedtotheamountelectricitygeneratedduetolowermarginalcost. given in Appendix C.
9
C. Handling Uncertainty in Electricity Generation and Load Algorithm 2 First Come First Serve (FCFS) Allocation
1: procedureFCFS(B,M)
In case of intrinsic uncertainty of electricity load and
2: DB←0;t cost←0
generation, we use electricity storage system to store or 3: for∀b∈Bdo
supply electricity. The storage system will not be used as the 4: DB←(DB+D˜b)
5: endfor
regular electricity source or storage. It is included to resolve 6: DetermineµoptforDB byAlg.OMCPofSec.IV-A1(step1tostep4)
the instantaneous variation of load and generation after the 7: Cm,b←{cm,b|∀b∈B,∀m∈M}
8: Randmize(Cm,b)
decision is made by the MEPM method. The price of the 9: sortvaluesinCm,bofeachbinascendingorder
electricitytosupplyisthepricedecidedbytheMEPMsystem 1101:: foricfmD˜,bb∈>C0mthbendo
for the MGN. The storage system is an intrinsic part of the 12: ifD˜b≤(E˜m) & D˜b>0 & (Γi−D˜b)≥0then
demand-response algorithm of a microgrid which we assume 13: xm,b←D˜b,D˜b←0;E˜m←(E˜m−D˜b)
14: calculatextm,bandπm,b(eqs.(20),(21),(19))
to be the internal energy management system of a microgrid 15: x˜m,b←(xm,b+xdm,b)
(whether a buyer or a seller) as we discussed at end of the 16: t cost←(x˜m,b×cm,b)
Section I. 1187:: elseD˜ifbE˜←m(≤D˜bΓi−thE˜emn);xm,b←E˜m;E˜m←0
19: calculatextm,bandπm,b(eqs.(20),(21),(19))
V. FIRSTCOMEFIRSTSERVE(FCFS)ALLOCATION 2210:: tx˜mco,bst←←(x(xm˜m,b,b+)×xdmc,mb),b
22: else
In FCFS method, the EMS first decides the marginal cost
23: d←Γi
according to the buyers’ demand and a series of bid prices 24: D˜b←(D˜b−d);xm,b←d;E˜m←(E˜m−d)
placedbythesellers.Then,itassignstheamountofelectricity 25: calculatextm,bandπm,b(eqs.(20),(21),(19))
to be transported from a seller microgrid to a buyer according 2276:: tx˜mco,bst←←(x(xm˜m,b,b+)×xdmc,mb),b; Γi←0
to the minimum transportation costs and not exceeding the 28: endif
29: endif
capacityoftheconnectedtransmissionline.TheFCFSscheme 30: endfor
first determines the overall marginal costs for total excess return(µopt+t cost)
31: endprocedure
demand of the buyers (line 6), then, assigns each of the
buyers (b) to the available sellers to buy x˜ amount of
m,b
electricity from seller m (from lines 10 to 30). First, in
linearmarginalcostfunction.Also,nonlinearnon-convexcosts
line 9 the transmission costs are sorted in ascending order
are generated with the random values chosen for α with the
accordingtothebuyer.Next,weselectabuyer-sellerpairand
variationofE .Forallthecases,wechoosearandomvalue
allocate electricity to fulfill the demand of buyer b with the m,w
between$0.002to$0.008forc.Thecapacityofeachgenerator
restriction that the capacity of connected transmission line l
i
is chosen randomly from 300kW to 1MW, and demand
is not exceeded. Otherwise, we select next seller-buyer pair
for each MG is also chosen at random between 200kWh
and continue the allocation of electricity from a seller to a
to 2400kWh. We have chosen ANSI/IEEE standard network
buyer accordingly. In every successful allocation, we modify
transformer with 300kVA to 2500kVA power range, primary
the demand of the buyer with the allocation amount. We
voltage up to 34.5 KV and secondary voltage is up to 600V
assume that the transmission lines have sufficient capacity for
[38].Oncethecapacityofthegeneratorsisfixed,aforecasting
the allocation of electricity and at the end, demands of all the
algorithm [18] based on the historical meteorological data is
buyers are fulfilled.
executed to estimate the amount of electricity generated from
the RESs. For the simulation, we place the capacitor banks
VI. SIMULATION
with a maximum of 600 MVAR and the impedance of the
A. Simulation Setup
transmission/distribution lines (with 11/33KV base voltage)
We consider an MGN system which contains a set of
areconsideredwhichisgivenin[39].Anenergytransportation
MGs, each with a number of energy sources (DGs) ran-
network is set up among the smart microgrids, each of which
domly chosen from a set of renewable and non-renewable costsc ($0.05/kWh∼$0.1/kWh)totransportoneunitof
m,b
energy sources; such as, (i) renewable: we choose the random
electricity.WeimplementedthealgorithmsforMEPMinC++
(given LCOE range in brace) price of the electricity for
programming language used IBM CPLEX concert technology
offshore wind turbine ($0.15 to $0.218/kWh), onshore wind
to resolve the allocation problem.
($0.05 to $0.116/kWh), solar energy ($0.05 to $0.15/kWh),
hydropower($0.030to$0.059/kWh).Theamountofelectric-
B. Numerical results
ityfromtheRESsispredictedforeachhourofadaybyusing
therenewableenergymodelsdescribedinourpreviousworkin We execute MEPM, FCFS algorithms on variable sized
[18],(ii)non-renewable:gasturbinegenerator($0.144/kWh) MGN, which comprise, 50, 100, ···, 1950, 2000 smart
with given unit production costs ($) in [37]. We choose one microgrids. We execute the algorithms more than fifty times
to five generators at random to power each of the MGs. For for each MGN configuration for the targeted time slot and
non-renewable sources, in the cubic cost function α, and β take the average of the outputs to compare. To get the
are given random values from 0.2×10−6 ∼0.8×10−6, and results we have executed the simulation several (50) times for
0.05×10−6 ∼0.2×10−6, respectively. We generate a set of each configuration (with random demand and new predicted
convex functions to simulate the piecewise convex marginal generation of the RESs) and we present the results in the
cost and choose α between 0.2×10−6 ∼0.8×10−6 for the following graphs.
10
Eletricity Price (in $/kWh) 0.40.60.81.01.2 Elec. Price MFMECEPFPMSM Vs FCFS Execution time (in Seconds) 200400600800 MEMPLNPiiMoeanncre gleEaiwinrxneieasacler uc Ntoiosotnn f -oTCri omCneuvbexic cost function
0.2
0.0 0
0 500 1000 1500 2000 0 500 1000 1500 2000
Number of Smart Microgrids Number of Smart Microgrids
Fig.5. OverallElectricitypricein$/kWhforFCFS,andMEPM.
Fig.7. PerformanceoftheMEPMalgorithm
Fig. 5 shows the amount of money (in $/kWh) to spend
for purchasing one unit of electricity in MEPM, and FCFS times on MGN with various configurations, which contain
schemes. Clearly, the electricity unit price is higher under 50,100,···,2000 smart microgrids. For each iteration, the
FCFS in comparison to MEPM. Indeed, MEPM minimizes simulation selects the number of generators, production ca-
the overall electricity price for any configuration of the MGN pacity, and demand of each MGN independently, such that
compared to the price determined by the FCFS scheme. To the set of sellers and buyers are changed dynamically. We
put this into perspective, in 2013, the Quebec annual demand took the average of the resultant costs of the same sized
was 173.3TWh [40], therefore a reduction of $0.01/kWh MGNs and calculate the percentage of average price savings
will save 1.73 billion dollars 2. Hence the use of an efficient by the MEPM with respect to the FCFS scheme. We found
method (MEPM) to determine the electricity price may play that the percentage of savings (i.e., better spending) is higher
a very important economical role. Fig. 5 also shows that the for smaller sized MGN and decreases with the increase of the
MEPM algorithm reduces the electricity price more when the size of the MGN for both MEPM. The percentage of saving
number of microgrids is less. This is because, the MGN has a decreases because, in the large sized MGN, the buyers have
fewer number of suppliers (sellers), which left fewer options moreopportunitiestochoosepotentialsellerstobuyelectricity
for the FCFS algorithm to choose electricity transportation comparedtoasmallsizedMGN.Hence,thepercentageofcost
costs for the lower bound marginal costs. In addition, for all saving of a large MGN system is less than the percentage of
the schemes, the electricity cost reduces while the number of saving in a smaller MGN.
smartmicrogridsincreasesintheMGN.Clearly,thisincreases
the problem space and present more opportunities for buyers 700 MEPM Combined Execution Time
to choose potential sellers to minimize both the price and 600
tsiOhnenellaettrrlhsaaenctksooptomohfreitnrcaiotmhiooarinnzddeicn,oiatsttshti.oepnIenMrastmEohnPeoaMnFlgCelstFehccShetersbimccuhieyteyemcrphsero,tioocaseel,boswuwysheeeirlcrlthechrherseocswouosilsettthss. Execution time (in Seconds) 200300400500
the objective to minimize electricity price of the whole MGN 100
0
community. Moreover, Fig. 5 together with Fig. 6 clearly
0 500 1000 1500 2000
represent the performance (in respect to the electricity price Number of Smart Microgrids
and saving) of the MEPM compared to FCFS.
Fig.8. CombinedExecutionTimeoftheMEPMalgorithm
60 Percentage (%) of Cost Savings by MEPM
Figs. 7 and 8 show the average runtime of MEPM algo-
50
rithms for various MGN configurations and marginal costs.
Cost Saving (in %) 203040 WteoxeeicmduetpevleemltohepenatprasoengqdruaeemnvatilsauelavtdeeirvatilhdeetimapneedrsfocoromnnaqnauceecromomfeptMuhtoeEdrPiMsny.sCtWe+m+e
10 containing Intel Core i7, 2.67GHz clock speed with 6GB
0 RAM. The results presented in Fig. 7 show that the MEPM
0 500 1000 1500 2000 algorithm indeed is a polynomial time algorithm irrespective
Number of Smart Microgrids
of the marginal cost functions. Also, MEPM run times are
Fig.6. PercentageofcostsavingofMEPMcomparedtoFCFS dependentonthesizeoftheMGN;thustheruntimeincreases
near linear (polynomial) while the number of participants
Fig. 6 presents the percentage of electricity price reduc- (MGs) increases. We have considered a large enough elec-
tion of MEPM with respect to FCFS for various instances. tricity market with 2000 MGs. Fig. 7 represent the overall
We run both FCFS and MEPM algorithms more than 50 run time of the algorithm that includes the duration needed
to determine the upper and lower bound of the combined
2This monetary saving is indeed not due to only lowering the electricity
marginal cost. The calculation of lower bound marginal cost
price,butratherdirectlyrelatedtoloweringthecostofgeneration,ultimately
equallybenefitingthesupplierandbuyer. takes at most three (3 Secs for 2000 microgrids) seconds
