Distributed Control, Optimization, Coordination of Smart Microgrids
Silani, Amirreza
DOI:
10.33612/diss.156215621
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Publication date: 2021
Link to publication in University of Groningen/UMCG research database
Citation for published version (APA):
Silani, A. (2021). Distributed Control, Optimization, Coordination of Smart Microgrids: Passivity, Output Regulation, Time-Varying and Stochastic Loads. University of Groningen.
https://doi.org/10.33612/diss.156215621
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7
Distributed Optimal Microgrid Energy
Manage-ment with Considering Stochastic Load
This chapter is based on our work presented in [153]. The uncertainty of load is a challenging issue for the power systems. These uncertainties can damage the power systems and even cause blackout. Therefore, it is necessary to handle this problem. In this chapter, a method for energy management in microgrids with stochastic loads is proposed where the design is performed through solving a nonlinear optimization problem. In the centralized optimization techniques, high computational capabilities are needed and the customers’ privacy might be infringed. On the contrary, in distributed techniques, the power network and its constraints are neglected since it is assumed that all loads and generations are connected to one bus. Consequently, in the proposed distributed Energy Management Strategy (EMS), the optimization problem is decomposed into two optimization levels which consist of the MicroGrid Centralized Controller (MGCC) and Local Controllers (LCs) and the distribution network and the corresponding constraints are taken into account. Moreover, the loads are considered stochastic since the precise prediction of the loads is not accessible in practice.
7.1
Introduction
Microgrids are power distribution systems which include controllable loads and Distributed Energy Resources (DERs). Controllable loads can work with or without the main grid and DERs are integrated with Distributed Generations (DG) includes PhotoVoltaics (PV), Wind Turbines (WT), and Distributed Storage (DS) [22]. An Energy Management Strategy (EMS) is required in microgrids to control the power flows among different buses (see Subsection 1.2.2 for more details about the literature review for energy management in microgrids). An EMS should provide the opera-tional goals of the microgrid such as the minimization of costs and supplying the demanded loads. Typically, a nonlinear optimization problem is used to model the microgrid energy management where usually it is assumed that we have a perfect prediction of the loads and renewable sources. Thus, commonly an offline optimiza-tion approach is utilized to address the energy management problem. However, the uncertainties of the loads, renewable sources, and market do not let us have a perfect prediction of them [23]. Moreover, new loads such as EVs insert more uncertainties to the microgrids.
It is well known that electric loads are in practice stochastic thanks to the random and unpredictable diversity of usage patterns. However, most of the recent papers do not consider exact stochastic models for the loads (see for instance [22, 23, 79]) and do not consider the underlying power flow constraints with a distributed structure (see for instance [83–86]). Furthermore, in the stochastic EMS, it is crucial how to simulate the randomness. It is a custom to assume that the randomness has a certain distribution and apply Monte Carlo approach to produce simulation data. However, the stochastic loads have time-series scenarios with self-correlation in time. Hence, the simulation data should be generated based on the transformation process of the randomness over time [87]. Differently form Part I (Chapters 3, 4, 5 and 6), in this chapter we propose an energy management strategy based on stochastic optimization method. Then, we generate stochastic scenarios based on time-homogeneous Markov chain and we solve the optimization problem by considering the stochastic load. Also, differently from [22,23,79], in this chapter we provide an exact stochastic model of the load and introduces an efficient distributed algorithm to formulate and solve the optimal power flow (OPF) problem in the distributed manner. The system model proposed by this chapter is more realistic which considers the detailed constraints of the generations, batteries, and loads. It is also robust to communication failure demonstrated by simulation results, i.e., it is suitable for realistic communication net-works. Differently from [83–86], in this chapter we present an EMS which considers the underlying power flow constraints, and has a distributed structure. Also, we utilize the convexification approach which doesn’t make the problem non-exact, has
7.2. Problem Formulation 145 a fast convergence speed proved by [122].
The main contributions of this chapter are as follows: (i) we formulate the EMS problem in microgrids as a nonconvex optimization problem taking into account the loads, power flows, and system operational constraints in a distribution network such that the costs of the DGs, DSs and energy purchased from the main grid are minimized and the customers’ demanded loads are provided where the loads are considered stochastic generated by a time-homogeneous Markov chain; (ii) we relax the nonconvex constraints to obtain a convex optimization problem according to the conditions provided in [110, 111] for the exactness of this convexification; (iii) we decompose the centralized optimization problem into a distributed problem via the Predictor Corrector Proximal Multiplier (PCPM) method proposed by [122] in order to handle the customers’ privacy, communication challenges, and high computational burdens of centralized optimization.
The rest of this chapter is organized as follows. In Section 7.2, the optimal EMS problem is formulated. In Section 7.3, the proposed distributed EMS is introduced. The simulation results are presented and discussed in Section 7.4, while some con-cluding remarks are gathered in Section 7.5.
7.2
Problem Formulation
In this section, firstly we describe the model of the microgrid including the Dis-tributed Storage (DS), DisDis-tributed Generation (DG), and load model. Then, we present the network model and the Optimal Power Flow (OPF) problem formulation of the microgrid.
7.2.1
System model
We assume that our power distributed network has a radial structure [154]. Let Generation = {g1, g2, ..., gG} denote DGs’ set, Load = {l1, l2, ..., lL} denote the loads’
set, and Battery = {b1, b2, ..., bB} denote the DSs’ set. Also, we consider a Microgrid
Central Controller (MGCC) that regulates the operation of the loads and Distributed Energy Resources (DERs). Each load and DER have a Local Controller (LC) which optimizes its objective function locally and can communicate with the MGCC. A discrete-time model is considered as κ = {0, 1, ..., T − 1}, where T is the scheduling horizon.
7.2.2
DS model
The batteries for DS units are considered. For the battery b ∈ Battery, let the complex power sb : R≥0 → C be defined as sb(t) := pb(t) +iqb(t), where pb : R≥0 → R and
qb: R≥0 → R are the active and reactive power, respectively. Then, the constraints
for the battery b ∈ Battery are given by [22, 79]:
pb,min≤ pb(t) ≤pb,max, ∀t ∈ κ (7.1) pb(t)2+ qb(t)2≤s2b, ∀t ∈ κ (7.2) Eb(t + 1) =αbEb(t) + pb(t)∆t, ∀t ∈ κ (7.3) Eb,min≤ Eb(t) ≤ Eb,max, ∀t ∈ κ (7.4) Eb(T ) ≥E f b, (7.5)
where Eb : R≥0 → R is the energy stored in the battery, Eb,max, Eb,min ∈ R are the
maximum and minimum permitted energy stored in the battery, respectively, and Ebf ∈ R is the minimum final energy. Moreover, pb,max∈ R and −pb,min∈ R are the
maximum charging and discharging rate, respectively, sg∈ R is the inverter capacity
and αb ∈ R is a constant parameter.
For battery b ∈ Battery, we model the cost function Cb : R → R such that it
prevents from the charging and discharging damages such as fast charging, deep discharging, and frequent switches. Therefore, the corresponding cost function model is given by [155]: Cb(pb(t)) =ab X t∈κ pb(t)2− bb T −2 X t=0 pb(t + 1)pb(t) + cb X t∈κ min(Eb(t) − σbEb,max, 0) 2 , (7.6) where ab, bb, cb, σb ∈ R>0are positive constants.
7.2.3
DG model
We assume that our microgrid has both conventional and renewable DGs such as WTs and PVs. For g ∈ Generation, let the complex power sg : R≥0 → C be defined
as sg(t) := pg(t) +iqg(t), where pg: R≥0→ R and qg : R≥0→ R are the active, and
reactive power, respectively.
The level of power produced by the renewable DGs depends on the presence of its resources and is not dispatchable. Thus, we need to have a prediction to take account of them in the EMS. The prediction approaches for PV and WT have been represented in [155, 156].
7.2. Problem Formulation 147 The power produced by the conventional DG g ∈ Generation has the following constraints:
0 ≤ pg(t) ≤ pg,max, ∀t ∈ κ (7.7)
|pg(t) − pg(t − 1)| ≤ ρgpg,max, ∀t ∈ κ (7.8)
where ρg ∈ R is the ramping parameter and pg,max ∈ R is the maximum allowed
power.
For the conventional DG g ∈ Generation, also we have the following constraint [22, 79]
pg(t)2+ qg(t)2≤ s2g, ∀t ∈ κ (7.9)
where sg∈ R is the inverter capacity.
For the conventional DG g ∈ Generation, we model the cost function Cg: R → R
as [80]
Cg(pg(t)) = ag pg(t)∆t 2
+ bgpg(t)∆t + cg, (7.10)
where ag, bg, cg∈ R are constant coefficients.
7.2.4
Load model
We assume that the loads are stochastic. For the load l ∈ Load, let the complex power sl : R≥0 → C be defined as sl(t) := pl(t) +iql(t), where pl : R≥0 → R and
ql: R≥0 → R are the active and reactive power, respectively. Then, the constraints
for the load l ∈ Load are given by
pl,min≤pl(t) ≤ pl,max, ∀t ∈ κ (7.11)
ql,min≤ql(t) ≤ ql,max, ∀t ∈ κ (7.12)
where pl,max, pl,min ∈ R are the maximum and minimum active power, and ql,max,
ql,min∈ R are the maximum and minimum reactive power, respectively.
Now, we define the stochastic load model for l ∈ Load as [87] pl(t) := pgl(t) + p gap l (t, υ) (7.13a) ql(t) := qlg(t) + qlgap(t, υ), (7.13b) where pgl : R≥0→ R, and q g
l : R≥0→ R are the forecasted active and reactive power
and pgapl : R≥0× R → R, and q gap
l : R≥0× R → R are the active and reactive power
which fill the inconsistency of the demand and supply. The randomness of pgapl (t, υ)
For the load l ∈ Load, we model the cost function Cl: R → R as Cl(pl(t)) =αgl El,max− X t∈κ pgl(t)∆t+ minEχ n X t∈κ αgapl .|pgapl (t, υ)|o, (7.14) where αgl, α gap
l ∈ R>0 are positive constants and El,max ∈ R is the energy upper
bound of the load.
The computation of expectation is difficult and even impossible because a high di-mensional integral should be computed. Therefore, the expectation is approximated by using a number of samples υi, i = 1, 2, ..., nof χl(t, υ)from its sample space.
Then, the expectation is calculated according to Bernoulli’s law as [87]
Eχ n X t∈κ αgapl .|pgapl (t, υ)|o= 1 n X t∈κ υ X i=1 αgapl .|pgapl (t, υ)|. (7.15)
Then, we can use expectation in the optimization algorithm.
7.2.5
Stochastic modeling
In this section, we want to present a sample generating method. In this method, a time-homogeneous Markov chain model is used to reproduce better error samples rather than taking the auto-relationship of the stochastic variable in time [157].
Now, let the stochastic sequence of the random variable be represented by χ = {χ1, χ2, . . . , χT} and the stochastic error have Markov property. Then, the
stochastic error sequence can be considered as a Markov chain. Next, we should discover its transition probability matrix. We assume that the transition proba-bilities are independent of time. Then, we discretize the prediction errors into χt∈ {υ1, υ2, ..., υn} and assume that these points are such that the Markov chain
can be provided. Now, let the Euclidean distance be denoted by d(x, y). Then, υi is related to a real error value of ω if d(ω, υi) ≤ ∆, where ∆ = υi − υi−1 for
∀i ∈ {1, 2, ..., n} [87]. Thus, the transition probability ˆψi,j: R → R is given by
ˆ ψi,j= 1 M M X m=1 ϕ(m)i,j , (7.16) where ϕ(m)i,j = ( 1 if d(ωt−1(m), υi) ≤ ∆ and d(ω (m) t−1, υj) ≤ ∆ 0 otherwise.
7.2. Problem Formulation 149
Figure 7.1: Droop control sheme [158]. Also, when the transition probability matrix
Ψ = ˆ ψ11 ... ψˆ1n . . . ˆ ψn1 ... ψˆnn , (7.17)
is fixed, Monte Carlo can be applied to engender the prediction error [87].
7.2.6
Droop control scheme
In this section, we present a droop control scheme proposed in [158]. Adopting the droop control, the EMS can work efficiently in islanded mode and respond to the quick load or renewable energy variations. Figure 7.1 shows the structure of the droop control which works based on Enhanced Phase Locked-Loop (EPLL). In this structure, vabc : R≥0 → R, veabc : R≥0 → R, eabc : R≥0 → R, r : R≥0 → R,
and pg : R≥0 → R are the three-phase voltage of DERs, input voltage estimate to
the EPLL, three-phase error, phase angle, and active power of DERs, respectively. Also, d and q components of three-phase errors are denoted by ed = 23N (r)Teabc,
and eq = 23M (r)Teabc, where M (r) = col cos(r), cos(r − 23π), cos(r + 23π), and
N (r) = col sin(r), sin(r −23π), sin(r +23π). The frequency references of DERs, i.e., ωref, is calculated based on the filtered real power pgf : R≥0→ R as [158]
ωref = ωn− γdrpgf(t),
where γdr∈ R is the frequency droop constant, and ωn ∈ R is the nominal frequency
has been presented in [158]. We use this structure in our algorithm to handle the problem of the load and renewable energy fluctuations in islanded mode.
7.2.7
Power network model
Let g ∈ (Ξ, η) denote the power network graph where each link in η shows a branch and i ∈ Ξ depicts a bus. For each link (i, j) ∈ η, let the complex power Sij : R≥0→ C
be defined as Sij(t) := Pij(t) +iQij(t), where Pij : R≥0 → R and Qij : R≥0 → R
are the active and reactive power, respectively. Also, for the branch from i to j, let Iij : R≥0→ R be the complex current and the complex impedance zij ∈ C be defined
as zij := rij+ixij, where rij∈ R is the resistance and xij∈ R is the reactance.
For each bus i ∈ Ξ, let Vi : R≥0 → R be the complex voltage and the net load
si : R≥0 → C be defined as si(t) := Pi(t) +iQi(t), where Pi : R≥0 → R and
Qi: R≥0→ R are the active and reactive power, respectively. Then, for each bus i,
we have the following constraint [22, 79]
si(t) = sli(t) + sbi(t) − sgi(t), ∀i ∈ Ξ − {0}, ∀t ∈ κ (7.18)
where sbi(t) = Pb∈Batteryisb(t), sli(t) = Pl∈Loadisl(t), and sgi(t) =
P
l∈Generationi
sg(t). Moreover, for (i, j) ∈ η, and ∀t ∈ κ, we have the following constraints [22, 79]
zijIij =Vi(t) − Vj(t) (7.19) Sij(t) =Vi(t)Iij∗(t) (7.20) sj(t) =Sij(t) − zij|Iij(t)|2− X k:(j,k)∈η sjk(t), (7.21)
Now, by using (7.19)-(7.21), we can model the distribution network for ∀(i, j) ∈ η and ∀t ∈ κ as [111] pj(t) =Pij(t) − rijlij(t) − X k:(j,k)∈η Pjk(t) + χt(υ) (7.22) qj(t) =Qij(t) − xijlij(t) − X k:(j,k)∈η Qjk(t) + χt(υ) (7.23) vj(t) =vi− 2(rijPij(t) − xijQij(t)) + (rij2 + x 2 ij)lij(t) (7.24) lij(t) = Pij(t)2+ Qij(t)2 vi(t) , (7.25)
where lij(t) = |Iij(t)|2, and vi(t) = |Vi|2. Equations (7.22)-(7.25) introduces a system
7.3. Proposed Distributed EMS 151
7.2.8
Problem setting
The goal of the EMS is the minimization of a cost function to provide the reliable power for the customers. Hence, we design an EMS in order to meet the operational objectives of microgrid and provide supply-demand balance.
We consider the following constraint for the bus voltage as
Vi,min≤ |Vi(t)| ≤ Vi,max, ∀i ∈ Ξ − {0}, ∀t ∈ κ (7.26)
where Vi,max∈ R, and Vi,min∈ R are the maximum and minimum permitted voltage
constraints.
Now, let the injected power from the main grid sG : R≥0 → C be defined as
sG:= pG+iqG, where pG : R≥0→ R, and qG: R≥0→ R are the active and reactive
power injected from the main grid, respectively. In the islanded mode sG(t) = 0and
in the grid-connected mode sG(t)is given by
sG(t) =
X
k:(0,1)∈η
s0k(t), ∀t ∈ κ (7.27)
Then, we model the energy purchased cost CP : R≥0× R → R as
CP t, pG(t) = λ(t)pG(t)∆t, (7.28)
where λ : R≥0→ R is the energy price.
The purpose of the microgrid is the minimization of the supply-demand mis-match, the cost of generation, and power loss in microgrid. Thus, we should minimize the following objective function in the terms of P, Q, v, I, s:
J =βb X b∈Battery Cb(pb) + βl X l∈Load Cl(pl) + βg X g∈Generation Cg(pg) + β0 X t∈κ CP t, pG(t) + βp X t∈κ X (i,j)∈η rijlij(t), s.t. (7.1) − (7.5), (7.7) − (7.9), (7.11) − (7.13), (7.18), (7.22) − (7.27), (7.29)
where β0, βl, βb, βg, βp∈ R are constant parameters.
7.3
Proposed Distributed EMS
The problem (7.29) is a non-convex problem inasmuch as the equality constraint given in (7.25) is not an affine function and NP-hard to solve [111]. Consequently, we relax the constraint (7.25) to
lij ≥
Pij(t)2+ Qij(t)2
vi(t)
Then, we minimize the following convex objective function in the terms of P, Q, v, I, s: J =βg X g∈Generation Cg(pg) + βb X b∈Battery Cb(pb) + βl X l∈Load Cl(pl) + β0 X t∈κ CP t, pG(t) + βp X t∈κ X (i,j)∈η rijlij(t), s.t. (7.1) − (7.5), (7.7) − (7.9), (7.11) − (7.13), (7.18), (7.22) − (7.24), (7.26), (7.27), (7.30). (7.31)
The represented relaxation is exact if the injected power is not large and the bus voltage is near its nominal value [110]. However, the optimization problem (7.31) is a centralized problem. In order to increase the efficiency and privacy in the microgrid, we use the Predictor Corrector Proximal Multiplier (PCPM) algorithm proposed in [22, 122] to solve the optimization problem (7.31) in the distributed manner (see Section 2.4 for more details on the PCPM algorithm).
In this distributed algorithm, firstly, we set r zero and the LCs initialize their schedules randomly and send them to the MGCC. Then, for each bus i ∈ Ξ − {0}, the MGCC initializes the control laws of {ur
i(t)}t∈κ, {vir(t)}t∈κand sri = p r i +iq
r i,
and the LCs set the optimal value of s∗i as s r+1
i . At each iteration of this algorithm,
the MGCC computes the control laws as ˆur
i = uri + ς(prli + pbir − prgi − pri), and
ˆ
vir= vir+ ς(qlir+ qrbi− qr gi− q
r
i), where ς ∈ R>0is a positive constant, then transmits
them to the LCs. We should notice that each DG unit, DS unit, and load have separate LCs. Then, the algorithm works as follows:
I) The LC minimizes the following objective function in the term of sbfor each
DS unit b ∈ Battery: JDS =βbCb(pb) + (ˆuri) Tp b+ (ˆvir) Tq b+ 1 2ς pb− prb 2 + 1 2ς qb− qbr 2 , s.t. (7.1) − (7.5), (7.32)
and the LC updates sr+1b = s∗b.
II) The LC minimizes the following objective function in the term of sgfor each
DG unit g ∈ Generation: JDG=βgCg(pg) − (ˆuri) Tp g− (ˆvir) Tq g+ 1 2ς pg− prg 2 + 1 2ς qg− qgr 2 , s.t. (7.7) − (7.9), (7.33)
and the LC updates sr+1 g = s∗g.
7.3. Proposed Distributed EMS 153
Set r zero and LCs initialize their schedules randomly
The MGCC calculates
ur
i(t), and vir(t)and
send them to LCs The LCs solve the
optimiza-tion problem (7.33)-(7.34), and use the stochastic
mod-eling technique and droop control in islanded mode
The LCs send their op-timal values to MGCC
The MGCC solves the optimization problem (7.35) Calculate ur+1i (t), and vr+1i (t) r=r+1 Convergence achieved? stop no yes
Figure 7.2: Flowchart of the proposed distributed approach.
load l ∈ Load: JLoad=βlCl(pl) + (ˆuri) Tp l+ (ˆvri) Tq l+ 1 2ς pl− prl 2 + 1 2ς ql− qrl 2 , s.t. (7.11) − (7.13), (7.34)
and the LC updates sr+1l = s ∗ l.
IV) The MGCC minimizes the following objective function in the terms of P, Q, v, I, s: JM GCC=β0CP(t, pG(t)) + βp X (i,j)∈η rijlij(t) − (ˆuri)Tp(t) − (ˆvir)Tq(t) + 1 2ς p(t) − pr(t) 2 + 1 2ς q(t) − qr(t) 2 , s.t. (7.22) − (7.24), (7.26), (7.27), (7.30). (7.35) The LCs send sr+1 g , s r+1 b , and s r+1
l to the MGCC at each iteration, and the MGCC sets
the values of ur+1i = uri + ς(p r+1 li + p r+1 bi − p r+1 gi − p r+1 i ), and ˆv r+1 i = vri + ς(q r+1 li +
qr+1bi − qr+1gi − qr+1i ), then this algorithm stops when it achieves the convergence. Figure 2 depicts the flowchart of this algorithm.
Note that the objective functions (7.32), (7.33), (7.34) and (7.35) are defined based on the objective function (7.31) via the PCPM algorithm (see Section 2.4 for more details on the PCPM algorithm). More precisely, since the objective function (7.31) has a detachable structure, the PCPM algorithm can be applied to solve the convex optimization problem in a distributed manner such that the objective functions (7.32), (7.33), (7.34) and (7.35) are iteratively minimized until the convergence is achieved.
Furthermore, In this algorithm, the privacy of customers will be kept because it uses a distributed EMS where the LCs keep the information of the loads and DERs and solve the optimization problem locally. The information communicated between the LCs and MGCC is only the schedules and the control laws and the private information of customers will not be communicated between the MGCC and LCs. Thus, the customers’ privacy will not be infringed.
7.4
Simulation Results
In this section, we apply the proposed algorithm to the microgrid depicted in Fig-ure 7.3, where the maximum power of each load and DER have been represented. In this simulation, we assume that the time interval is 1 hour, and a day is represented by κ = {0, 1, ..., 23}.
The constant parameters of the battery are set to be ab= 1, bb= 0.5, and cb= 0.4.
7.4. Simulation Results 155
Figure 7.3: Topology of the microgrid [79].
MWh, and Eb,min= 0.2MWh. Also, we choose Eb(0) = 1.6MWh, E f
b = 1.0MWh,
and αb= 0.9. The cost function of the diesel is chosen as Cg(pg(t)) = 9(pg(t)∆t)2+
60pg(t)∆t, and its nominal power factor, and ramping parameter are set to be 0.8,
and ρg= 0.4, respectively. The constant parameter of the load cost function is set to
be αgl = α gap l = 10
2. The optimization parameters are set to be β
g= βb= βl= β0=
βp= 1, and ς = 0.8. The droop control coefficients are chosen as γ1= γ2= γ2= 0.2,
and γdr= 0.4.
Figure 7.4 represents the day-ahead price of our microgrid. Figure 7.5 represents the schedule in islanded mode which is provided by the proposed EMS and it shows that the great portion of power is generated by the diesel and the small portion of it is generated by the renewable resources. Also, we can observe that the battery will be discharged in case the renewable generations are low and charged in case the renewable generations are high.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 20 40 60 80 100 120 140 Time (hour) Price ($/MWh)
Figure 7.4: Day-ahead price.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 −1 0 1 2 3 4 5 6 7 Time (hour) Power (MW) Load Diesel Battery WT PV
7.4. Simulation Results 157 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 −1 0 1 2 3 4 5 6 7 Time (hour) Power (MW) Load Diesel Battery WT PV
Figure 7.6: Day-ahead schedule in grid-connected mode.
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 −1 0 1 2 3 4 5 Time (hour) Injected Power (MW)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.94 0.95 0.96 0.97 0.98 0.99 1 1.01 1.02 1.03 1.04 1.05 1.06 Voltage (p.u.)
Maximum bus voltage using EMS without stochastic load Maximum bus voltage using EMS with stochastic load
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0.94 0.95 0.96 0.97 0.98 Time (hour) Voltage (p.u.)
Minimum bus voltage using EMS without stochastic load Minimum bus voltage using EMS with stochastic load
Figure 7.8: Minimum and maximum bus voltage.
Figure 7.6 depicts the schedule in grid-connected mode. In grid-connected mode, the diesel generations are reduced and compensated by the injected power from the main grid. The battery is charged and discharged according to the market price such that it makes financial benefits.
Figure 7.7 shows the injected power from the main grid. We can notice that the import and export of power from the main grid are proportional to the market price such that the financial benefits are maximized. Hence, the proposed EMS is designed such that microgrid can make the financial profit.
In order to compare the algorithm with considering the stochastic load and without it, we investigate the location influence on voltage tolerances. Thus, we rise the line lengths by six times. The minimum and maximum voltage of the EMS with considering the stochastic load and without it are depicted in Figure 7.8. In both algorithms, the bus voltages are maintained within the voltage constraints. However, the minimum and maximum voltages in the EMS with considering stochastic load have greater margin from their lower and upper bound than those in the EMS without considering stochastic load. Indeed, it shows that the EMS with considering stochastic load have better performance in confronting the location effect.
inves-7.4. Simulation Results 159 Table 7.1: Comparison of different cost coefficient of β
β Load Cost Diesel Cost Battery Cost Power Loss Cost Purchase Cost Equal β 312.13 1964.3 3.12 0.33 2916.13 βl=20 282.34 8.2 3.23 0.52 4812.33 βg=200 261.53 2143.16 0.52 0.44 2912.67 βb=20 12.34 2131.83 3.56 0.48 3312.31 βp=20 14561.34 2041.23 4.03 0.31 -8711.43 β0=2000 693.73 2976.34 3.67 0.28 2216.57
tigate the loads’ demand reduction demonstrated in Figure 7.9. In both algorithms, we can note that loads’ demand reduction in bus 11 and 12 is less than other buses because these loads are close to the renewable generations. On the other hand, the loads far from the generations, bus 2 and 3, have greater loads’ demand reduction and the bus voltage should be maintained above the lower bound. However, we can see that loads’ demand reduction in the EMS with considering stochastic load is less than those in the EMS without considering stochastic load. Therefore, the EMS with considering stochastic load can provide better utility benefit for customers.
Now, we want to compare the proposed distributed method with centralized one. We assume that the communication network is not perfect and we have the communication failure. We consider the communication network the same as the physical network and the communication failure probability is chosen as 0.3, i.e., the failure happens with the probability of 0.3. Figure 7.10 illustrates the comparison of the maximum and minimum bus voltage of the distributed and centralized EMS under communication failure. We can observe that the voltages are maintained within their allowed tolerance constraints in the proposed distributed EMS whereas the voltages violate their constraints in the centralized method; therefore, the proposed distributed method is robust with respect to the communication failure and can work efficiently with the imperfect communication network.
We considered βl, βg, βb, β0,and βpas the coefficients of the load, diesel generator,
battery, market, and power loss objective functions. In order to appraise the influence of these coefficients, we compute the cost of each unit with the various amount of β as depicted in Table 7.1. We can notice that the different amounts of β are compared with the manner that all of them are equal. We can note that selecting different β can result in different cost optimization and it can affect the performance of the EMS. If we give a large β to a unit while other units have small equal β, the optimal cost of such unit will be less than the manner that all of the units have equal β. Therefore,
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 Power (MW)
EMS without considering stochastic Load
Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Bus 11 Bus 12 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 Time (hour) Power (MW)
EMS with considering stochastic Load
Bus 2 Bus 3 Bus 4 Bus 5 Bus 6 Bus 11 Bus 12
Figure 7.9: Loads’ demand reduction.
the value of β for each unit is chosen based on the significance of that unit’s cost in the microgrid.
The effect of the different parameters αgl, and α gap
l is evaluated in Table 7.2. We
can observe that both small and large αgl, and α gap
l increase the total cost in
grid-connected and islanded mode. This is because a small αgapl ignores the variations of
the load while a large αgapl increases the cost of uncertainty and ignores the prediction
of the load in the proposed EMS. Also, a small αgapl neglects the prediction of the load
whereas a large αgapl neglects the variations of load. Therefore, there exists a trade-off
between these two parameters where the choice of αgl = α gap l = 10
7.4. Simulation Results 161 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0.94 0.96 0.98 1 1.02 1.04 1.06 Voltage (p.u.) Distributed EMS
Maximum bus voltage Minimum bus voltage Allowed tolerance 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 0.94 0.96 0.98 1 1.02 1.04 1.06 1.08 Time (hour) Voltage (p.u.) Centralized EMS
Maximum bus voltage Minimum bus voltage Allowed tolerance
Figure 7.10: Comparison of the maximum and minimum bus voltage of the dis-tributed and centralized EMS under communication failure.
Table 7.2: Cost comparison with different parameter αgl, and α gap l
αgapl = 102
αgl Cost of islanded mode Cost of grid connected mode αgl=1 6678.10 6138.89 αgl=10 6428.42 5535.64 αgl=102 6289.45 5275.86 αgl=103 6358.90 5468.93 αgl=104 6589.47 5665.15 αgl = 102
αgapl Cost of islanded mode Cost of grid connected mode αgapl =1 6425.15 6014.27 αgapl =10 6331.48 5424.14 αgapl =102 6289.45 5275.86 αgapl =103 6587.41 5595.19 αgapl =104 6610.01 5712.37
7.5. Concluding Remarks 163
7.5
Concluding Remarks
In this chapter, a distributed energy management strategy has been proposed with stochastic loads. In this approach, we have considered the underlying power dis-tribution network and its constraints while the existing methods have ignored the underlying power and assumed that there is a perfect prediction of the loads. More precisely, an OPF problem has been formulated and solved in distributed manner such that the LC of each unit optimizes the cost function of it and sends an optimal schedule to the MGCC and the MGCC optimizes the total cost function of grid based on the optimal schedule received from the LCs and its constraints. To produce stochastic loads, we have generated random data based on statistical analysis and transformation process of the uncertainty over time by using Markov chain rule. In the next chapter, the optimality and social behavior of EVs with vehicle-to-grid options are investigated.
