University of Pisa, Department of Energy, Systems, Territory and Constructions Engineering
Ph.D. Dissertation
Analysis and Control of
Multiple Microgrids
Pietro Ferraro
Supervisors: Prof. Marco Raugi
Prof. Emanuele Crisostomi Reviewed by: Prof. Federico Milano
Contents
1 Introduction 1
2 A High Level Microgrid Model 7
2.1 Introduction . . . 7
2.2 Modeling . . . 8
2.2.1 Simulating the Power Grid . . . 9
2.2.2 Electricity Market Model . . . 9
2.2.3 Microgrid Model . . . 12
2.3 Conclusions . . . 16
3 Impact of Microgrid Penetration 17 3.1 Introduction . . . 17
3.2 Market Based Energy Management System of the Mi-crogrid . . . 18
3.2.1 Example . . . 20
3.3 Case Study . . . 23
3.3.1 Microgrid Scenario . . . 25
3.3.2 DERs & Microgrids Scenario . . . 27
3.3.4 Large Capacity Storage Microgrid Scenario . . 31
3.4 Conclusions . . . 33
4 Impact of the Energy Management System 39 4.1 Introduction . . . 39
4.2 Island Based Energy Management System of the Mi-crogrid . . . 40
4.2.1 Simulation results . . . 41
4.3 Conclusions . . . 44
5 Control of Interconnected Microgrids 49 5.1 Introduction . . . 49
5.1.1 Energy Management System of the Microgrid 52 5.2 PI-Like Controller . . . 54
5.2.1 Case Study . . . 55
5.2.2 Simulation results . . . 56
5.2.3 Conclusions on the PI controller . . . 61
5.3 AIMD Algorithm . . . 63 5.3.1 Case Study . . . 67 5.3.2 Simulations Results . . . 69 5.4 Conclusions . . . 77 Appendices 79 A Frequency Divider 81 B Publications 85 Bibliography 87
List of Figures
2.1 Structure of the connection between the MG. . . 12
2.2 Control scheme of an converter-based DER. The blocks
show a first order dynamics expressed in the Laplace
domain. . . 14
2.3 Realization of the process used to simulate the behav-ior of the generation and the loads of a MG. The time
interval after which the value changes is 300 seconds. 15
3.1 IEEE 14-bus system: market clearing price λ. . . 22
3.2 IEEE 14-bus system: active powers Pout of the 4 MGs. 23
3.3 IEEE 14-bus system: states of charge S of the 4 MGs. 24
3.4 Frequency ωCOIwith 2 (upper panel); 6 (middle panel);
and 12 (lower panel) MGs. The grey lines represent each realization, the black thick line represents the av-erage of the process, while the dotted line represents
3 times the standard deviation. . . 36
3.5 Market clearing price λ with 2 (upper panel); 6
3.6 Frequency of the COI of the 39 bus system with 12 MGs. The grey lines represent each realization, the black thick line represents the average of the process, while the dotted line represents 3 times the standard
deviation. . . 38
4.1 Frequency of the COI of the 39-bus system with 12
MGs and small storage units. . . 44
4.2 Frequency of the COI of the 39-bus system with 12
MGs and large storage units. . . 45
5.1 COI of the frequency of the 39 bus system. All the
realizations of each stochastic process are plotted. . . 59
5.2 COI of the frequency of the 39 bus system. All the
realizations of each stochastic process are plotted. . . 60
5.3 COI of the frequency of the 39 bus system. All the
realizations of each stochastic process are plotted. . . 61
5.4 Frequencies of the 39 bus system. All the realizations
of each stochastic process are plotted. . . 71
5.5 Frequencies of the 39 bus system. All the realizations
of each stochastic process are plotted. . . 72
5.6 Plots of the switching probabilities, averaged across
all the realizations of the stochastic processes. . . 73
5.7 Plots of the switching probabilities, averaged across
all the realizations of the stochastic processes. . . 74
5.8 Plots of the switching probabilities, averaged across
List of Tables
3.1 Microgrid EMS rules for the seller mode . . . 20
3.3 Microgrid EMS rules for the buyer mode . . . 21
3.5 Microgrid parameters for the IEEE 14-bus system . . 21
3.6 Microgrid parameters . . . 27
3.7 Standard deviation of the frequency COI as a function
of the total MG installed capacity . . . 28
3.8 DER parameters . . . 29
3.9 Standard Deviation of the frequency of the COI as the number of DERs decreases and the number of MGs
increases . . . 30
3.10 Standard deviation of the frequency of the COI as a
function of the granularity k of MGs . . . 32
3.11 Large storage Microgrid parameters . . . 33
3.12 Standard deviation of the frequency COI as a function
of the total MG installed capacity. . . 34
4.1 Microgrid EMS rules . . . 42
4.3 Standard deviation of the frequency COI as a function
5.1 Microgrid parameters. The parameters σnet are cho-sen randomly in order to take into account different variations of the load and the DERs energy
produc-tion of each MG. . . 57 5.2 Controller parameters. . . 58 5.3 Controller performance. . . 62 5.4 Controller parameters . . . 69 5.5 Microgrid parameters . . . 70 5.6 Controller performance . . . 70
Chapter 1
Introduction
In the past, the transmission system was a passive delivery infras-tructure with a one directional flow of energy (from source to the consumer) designed and built to serve peak demand and to provide energy to the consumers according to their needs. Recently, the ad-vent of power system deregulation and the drift from a vertically integrated utility business model lead customers to increasingly use the grid as means to balance their own generation. Moreover, they expect to deliver the surplus of energy back to the grid and to be paid for it without restrictions on their production [1]. In this shift of paradigm, that substantially changed the development of the elec-trical grid from the one directional, centralized power system to a bidirectional distributed one, often called smart grid [2], the concept of Microgrid (MG) [3] has received particular attention from the sci-entific community. The rationale behind this interest is that MGs are generally considered the building block of the smart grid [4]:
MGs are independent flexible cells operating at low voltage distri-bution network with Distributed Energy Resources (DERs), control-lable and non controlcontrol-lable loads, storage devices and smart electric vehicles. MG operations are handled by an Energy Management System (EMS) and include, among other tasks, two-way commu-nication and the ability to operate, both, interconnected with the main grid or isolated (islanded) [5] [6]. Due to the spreading of this new paradigm, that is expected to bring economic advantages to the users and to increase the efficiency of the power grid as a whole [7], the scientific community mainly focused its attention on the analy-sis of a single, often islanded MG: main topics include, but are not limited to, the optimization of the scheduling of generation units and loads [8] [9] [10] and the analysis of the internal stability and dynamics of a MG [11] [12] [13]. MGs, however, present some tech-nical challenges related to the very low rotational inertia introduced, that need to be assessed to actually accommodate a large penetra-tion of units in the power system [14]. Among the studies on the impact of a low rotational inertia, an overview regarding the impact of the power system frequency stability has been provided in [15] and, in [16] and [17], the influence of high penetration of wind based DERs is assessed. Angle and voltage stability, as the MG penetra-tion level increases, is analysed in [18] and an analytical approach to analyse the effects of reducing inertia is proposed in [19].
In the aforementioned works, the ability of a MG to conduct poli-cies of Demand & Response and its effect on the frequency control of the transmission system is not taken into consideration. Frequency
deviations, in fact, are a measure of the active power imbalance and should remain within the operational limits in order to avoid trans-mission line overloads and the triggering of protection devices [20].
It is the authors’ opinion that the MGs interaction with the elec-tricity market is a crucial element that can not be neglected: a MG adopts a greedy behavior with respect to the electrical grid, selling or buying energy whenever it is convenient from an economical and operational point of view not necessarily taking into consideration the effects on the stability of the system. This behavior is accept-able only if the penetration of MGs is small with respect to the total system capacity and, due to their small size, MGs can be reason-ably modelled as price takers. This situation, however, might not be acceptable if such a penetration increases.
Examples of works that take into account both MGs and the electricity market are [21] and [22]. In these works, however, the effect of the MG penetration level on the transient response of the power system are not investigated. On the other hand, a study on how an equivalent dynamic model of the electricity market impacts on power system transients is studied in [23]. The paper highlights potential instabilities that arise when the dynamics of the machines and of the loads are coupled with the dynamics of the energy market. In [24] the model developed in [23] is generalized by taking into account the effects of time delays and market clearing time on the stability of the system.
In this thesis work the aim is to fill this gap by merging to-gether the electricity market model proposed in [23] with a hybrid
dynamic and event-driven MG representation as well as a detailed electromechanical model of the system. The goal is to provide a theoretical framework to study the coupling between the dynamics of the MGs, the power system and the electrical market, with an emphasis on frequency regulation. Furthermore, on the basis of the previous analysis, we try to provide a control strategy to mitigate the negative effects on the transmission system of a large number of MGs, by means of stochastic controller.
Specific contributions of this thesis can be summarized as follows: A high level model of the single MG that takes into account, loads, DERs, storage units and an EMS. This model is suit-able to perform high level analysis that take into account only the dynamics of the power system (thus neglecting the faster dynamics of the internal components of a MG);
An analysis, from the point of view of frequency stability, of the impact of a large number of MGs on the power system. The analysis is performed analysing the effects of different EMSs, size and number of MGs and, finally, the size of the storage units;
On the basis of the conclusions drawn from the aforemen-tioned analysis, a stochastic decentralized control method to reduce the fluctuations of the frequency, caused by the MGs behaviour, is proposed and analysed.
This work is organized as follows: In Chapter 2 we thoroughly describe the MG model employed in this research, that is later used
in Chapters 3 and 4 to analyze the effects of a large number of units in the transmission system. Finally, Chapter 5 focuses on synthesizing a control strategy to mitigate the negative effects, showed in the previous chapters. Lastly, Appendix A describes a local frequency estimator, called frequency divider, used in this while Appendix B presents a list of my publications.
At the moment this thesis is being submitted, the majority of the results obtained in this work have been published in several con-ference and journal papers, [29] [30] [31].
Chapter 2
A High Level Microgrid
Model
2.1
Introduction
This chapter focuses on the theoretical model employed in this re-search. We develop a high level model for MG which focuses on the active power exhanges between the unit and the power grid through a modular stochastic framework that combines different elements (e.g., loads, DERs, storage units, etc.). Simulations on the IEEE 14 and the IEEE 39 bus system are shown to provide an intuitive understanding of the behaviour of these units under different condi-tions.
2.2
Modeling
An MG is, at its core, a cluster of loads and generation units, coordi-nated by an Energy Management System (EMS) that, among other tasks (e.g., load shedding and internal power flow management), allows the MG to operate in island mode (i.e., the MG operates au-tonomously from the power grid) and determines the set point of the active power that the MG sells or buys from the electrical grid [26]. In this work, each MG is modeled using linear stochastic differ-ential equations, taking into account loads, DERs and storage units. These elements are coordinated by an EMS which is responsible, among other tasks (e.g., load shedding, internal power flow man-agement, transition to island mode), to establish the active power set point that the MG sells or buys from the electrical grid. The following assumptions are made:
The dynamics of the MG internal generation units and loads are neglected. This is not a strong assumption as the time constants of the internal MG dynamics are small compared to the ones of the high voltage transmission system [3, 25, 27]. The storage units, the DERs and the loads of each MG are
grouped into an aggregated model. This assumption allows reducing the computational burden of the proposed MG model. Due to their relatively small capacity, MGs are assumed to be price takers. Moreover, MG active power set-points depend on the electricity price. This assumption is consistent with the MG paradigm usually considered in the literature [7].
2.2 Modeling
The remainder of this section describes the power system as well as the electricity market models considered in the simulations. Then, the proposed hybrid dynamic and event-driven MG model is dis-cussed in detail.
2.2.1
Simulating the Power Grid
Let us recall first conventional Differential Algebraic Equation (DAE) models, described by the following equations:
˙x = f (x, y, u) (2.1)
0= g(x, y, u)
where f (f : Rp+q+s 7→ Rp) are the differential equations; g (g :
Rp+q+s 7→ Rq) are the algebraic equations; x (x∈ Rp) are the state
variables; y (y ∈ Rq) are the algebraic variables; and u (u ∈ Rs)
are discrete events, which mostly model MG EMS logic.
The set of equations in (2.1) includes lumped models of the trans-mission system and conventional dynamic models of synchronous machines (e.g., 6th order models) and their controllers, such as, au-tomatic voltage regulators, turbine governors, and power system sta-bilizers, as well as DERs, storage devices and the controllers included in the MGs, which are duly described in Subsection 2.2.3.
2.2.2
Electricity Market Model
The power system model discussed in the previous subsection is as-sumed to be coupled with a real-time – or spot – electricity market
model. This model represents a market for which the price, which is considered a continuous state variable, is computed and adjusted rapidly enough with respect to the dynamic response of the trans-mission system (e.g., PJM, California, etc.) [23]. Note that, while real-time markets are not particularly common at this time, the fast response of electricity price is expected to become a crucial and fun-damental feature of future power systems with large penetration of renewable energy sources and MGs [28].
The model of market dynamics considered in this work follows closely the work by Alvarado et al. [23]. The main assumption of such a model is that price variations are driven by the energy imbalance in the grid. An excess of supply decreases the price of energy while an excess of demand increases it. The following equations describe an ideal market with a single price of energy and with n power suppliers and m power consumers:
˙ E = n X i=1 PGi− m X j=1 PDj− Ploss (2.2) Tλ˙λ = −KEE− λ (2.3)
where PGi are the generated active powers of the n suppliers
con-nected to the grid; PDj are the active power consumption of m loads
connected to the grid; Ploss are the active power losses in the
trans-mission system; E and λ, are the energy imbalance and the
electric-ity price, respectively; and KE and Tλ are parameters that depend
on the design of the market itself. Since in real-world systems it is hard to measure the power imbalance in (2.2), such an imbalance is
2.2 Modeling
deduced in [23] implicitly based on the frequency deviation of the center of inertia (COI). Hence, (2.3) the market clearing price dy-namic is expressed as
Tλ˙λ = KE(1− ωCOI)− λ (2.4)
where ωCOI is the frequency of the COI, defined as
ωCOI = Pr i=1Hiωi Pr i=1Hi (2.5)
where ωi and Hi are, respectively, the frequency and the moment
of inertia of the i-th synchronous machine, and r is the number of
conventional generators in the grid (note that, in general, n 6= r as
not all power plants are equipped with synchronous machines). Finally, generator and load active powers are linked to the mar-ket clearing price λ based on a dynamic version of their bidding functions, as follows:
TGiP˙Gi = λ− cGiPGi− bGi (2.6)
TDiP˙Di=−λ − cDiPDi− bDi
where, cGi, cDi and bGi, bDi are proportional and fixed bid
coeffi-cients, respectively, as in conventional auction models and TGi and
TDi are time constants modeling generator and demand, respectively,
delayed response to variations of the market clearing price λ. Note that in [23] the equations for the load active powers are expressed with different signs. In this work, all coefficients are assumed to be
Microgrid EMS Load DER Storage Power System Electricity Market ωCOI λ λ + − − Pout Pref Pl Pl Pg Pg Ps S
Figure 2.1: Structure of the connection between the MG.
positive.
2.2.3
Microgrid Model
Figure 2.1 shows the overall connection of the MG with the power system and the electricity market. The elements that compose the microgrid are the load, the DER, the storage device and the en-ergy management system (EMS) that collects the information of consumed and generated powers by other elements of the microgrid, the state of charge of the storage device, the electricity price λ and
imposes the reference power generation Pref.
The dynamic of the aggregated storage device model is ruled by the following equation, which is the time-continuous equivalent of the model used in [21],
TcS= P˙ s
2.2 Modeling
where S is the state of charge of the MG, Tc is the time constant
of the state of charge of the storage unit, Ps is the power generated
or absorbed by the storage device (Ps > 0 if the storage is
charg-ing); Pout is the power output of the MG; and Pg and Pl are the
produced active power and the local loads, respectively, of the MG. S undergoes saturation hard-limits that model the fully charged and discharged conditions.
The DER dynamic model included in the microgrid is an elabo-ration of the DER models discussed in [32, 33]. The control scheme included in the DER model is shown in Fig. 2.2. This model is suitable for transient and voltage stability analysis and, hence, only current controllers of the VSC included in the DER are modeled. Power injections into the AC bus are:
Pg = vdid+ vqiq (2.8)
Qg = vqid− vdiq
where idand iqare the AC-side dq-frame currents of the VSC,
respec-tively and vd and vq are the dq-frame components of the bus voltage
phasor of the point of connection of the VSC with the AC grid. Fig-ure 2.2 shows the control scheme and the simplified converter model of the DER, which is described by the following equations:
˙id = (irefd − id)/Td (2.9)
˙iq = (irefq − iq)/Tq
The reference currents iref
vd, vq Pg Qg iref d iref q id iq 1 1 Converter Current set point controller 1 + sTd 1 + sTq
Figure 2.2: Control scheme of an converter-based DER. The blocks show a first order dynamics expressed in the Laplace domain.
reactive powers Pg and Qg, as follows:
iref d iref q = vd vq vq −vd −1 Pg Qg (2.10) Both iref
d and irefq are bounded by the converter thermal limits.
Uncertainty and volatility of both generation units and loads are accounted for by modeling the net power produced by the MG as a stochastic process according to
Pnet= Pg − Pl (2.11)
= ¯PgT − ¯PlT + ηM
where ηM has a stochastic process as in [35] with standard deviation
σM, and ¯PgT and ¯PlT are piece-wise constant functions that account
for uncertainty and change randomly with a period T as discussed in [36]. The noise is modeled as a single stochastic state variable as
2.2 Modeling 0.0 200.0 400.0 600.0 800.0 1000.0 1200.0 Time [s] 0.5 0.6 0.7 0.8 0.9 1.0 pg − Pl [p u ]
Figure 2.3: Realization of the process used to simulate the behavior of the generation and the loads of a MG. The time interval after which the value changes is 300 seconds.
and not on their absolute value. For illustration, Fig. 2.3 shows 1,200 seconds of a realization of such a process, in which T is set to 300 seconds.
The EMS is based on a set of if-then rules to decide the most
convenient active power set point, Pref, according to what kind of
internal objective function the user aims to maximize. In the
sim-ulations we assume that Ps, e.g., the power produced or absorbed
by the aggregated storage device included in the MG is the slack
variable that allows imposing the desired Pref, as follows:
TsP˙s= Pout− Pref
where Ts is the time constant of the storage active power controller.
Remark: Clearly the choice of a particular set of if-then rules is
arbitrary. Hence, in the next chapters we perform an analysis of the effects, on the transmission system, of different choices of EMS.
2.3
Conclusions
In this chapter a modular stochastic model that combines a simpli-fied model for a MG, the the electrical market and the transmisision system is presented in detail. In the next chapters several case stud-ies are presented that show the effects of a large penetration of MGs on the transmission system as different aspects and behaviours are taken in consideration. In particular we are interested in investigat-ing what happens when the number of MGs in the system increases, what happens as the MG EMS and storage size are varied and what possible forms of control can be employed to solve the issues that arise in these scenarios.
Chapter 3
Impact of Microgrid
Penetration
3.1
Introduction
In the previous chapter we introduced a stochastic modular frame-work to analyse the effects of the dynamic coupling of the trans-mission electrical system, the electricity market and the MGs. We are now interested in using that model to analyze the behaviour of several units connected to the transmission system as they adopt a greedy behaviour with respect to the energy price.
In particular, this chapter provides the following contributions: An analysis of the dynamic impact of an increasing penetration
level of MGs on power systems, coupled with market dynamics. A realistic time-domain analysis of the power system that takes
into account detailed non linear electro-mechanical models, MGs, storage units and DERs.
An analysis of the effects of different storage capacity sizes and different granularity of MGs on the stability of the power system.
In the reminder of this chapter we describe in detail the EMS employed, an example based on the IEEE 14 bus system and a case study based on the IEEE 39 bus system in which extensive Monte Carlo simulations are performed to draw meaningful conclusions.
3.2
Market Based Energy Management
System of the Microgrid
EMS input quantities are the produced power Pg, the load Pl, the
price λ and the state of charge of the storage units, S. The rules are
divided into two sets: the seller state, for which Pnet ≥ 0 (i.e., the
MG is producing more than it is consuming and it will most likely sell
energy) and the buyer state, for which Pnet< 0 (i.e., the MG is
con-suming more than it is producing and it will most likely buy energy). EMS parameters, l, h, ks, kb, Pbuy, Kbuy, Pch, Kch are specific of each MG, i.e., they depend on the market strategy of the MG owner or on its marginal cost (all the EMS parameters are positive). The meaning and purpose of these parameters is given in Tables 3.1 and
3.3. In particular, l and h indicate two thresholds under and above
3.2 Market Based Energy Management System of the Microgrid
These thresholds are time-varying as the convenience of buying and selling energy depends on the current energy price. For example, if,
in given period, the electricity price is around a value ¯λ1, and the
next period is around ¯λ2, then one can set l1 6= l2 and h16= h2 to
reflect the variation of market conditions. In the following, l and h
are calculated as:
h = (1 + ρ)¯λ(t) (3.1)
l= (1− ρ)¯λ(t)
where ¯λ(t) is the average value of the price, computed as ¯λ(t) =
1/tRt
0λ(τ )dτ (we drop the time dependence for readability purposes,
from now on), and ρ is a threshold that accounts for the fluctuation of the price in a given period. In the simulations discussed in Section 3.3, it is assumed that ρ = 0.0025.
The aforementioned rules are listed and explained in Tables 3.1 and 3.3. The rules are expressed hierarchically; this means that a rule is evaluated only if the conditions on the previous ones are not satisfied. Note that, even if the MG production is greater than its consumption, the EMS can, in some cases, impose to buy energy, e.g., if the storage is empty and the electricity price is low. Similarly, even if the MG production is lower than its consumption, the EMS can impose to sell energy if the storage is full and the price is high. The EMS rules utilized in this work are only a possible choice. There are several proposed EMS schemes in the literature – see, for example, recent works [37–41] and references therein. A comparison of different MG management schemes, however, is beyond of the
scope of this work.
Table 3.1: Microgrid EMS rules for the seller mode Seller mode, Pnet≥ 0
Rule Action Rationale
if S ≥ 80% and λ ≥ h Pref= Pnet(1 + ks) The price of energy is high and the battery is fully charged. It is con-venient to sell more en-ergy
if S ≥ 80% or
(50% ≤ S < 80% and λ ≥ h) Pref= Pnet Sell the surplus
if l≤ λ < hor (50% ≤ S < 80% and λ < l) Pref= 0 Charge the storage with
the production surplus if λ < land S < 50% Pref= −Pbuy− Kbuyλ Storage is low on charge
and, despite the sur-plus of production, it is convenient to buy en-ergy proportionally to the price (i.e., the lower the price the more the EMS can buy)
3.2.1
Example
This subsection illustrates the behavior of the MG model through a simple study based on the IEEE 14-bus system. In particular, the study aims at determining the behavior of 4 MGs interacting with the power system and the electricity market as qualitatively shown in Fig. 2.1. Each MG is assigned a random initial value for the state
of charge. System parameters (e.g., ks, kb and ¯λt) are set in such a
way that price variation trigger MG swtiches from the buyer mode to the seller mode. Table 3.5 shows the MG parameters used in the
simulation. In the table, the values for ¯Pg and ¯Pl refer to the average
values of Pg and Pl, respectively, during the simulation. To account
3.2 Market Based Energy Management System of the Microgrid
Table 3.3: Microgrid EMS rules for the buyer mode Buyer mode, Pnet< 0
Rule Action Rationale
if S ≤ 20% Pref= Pnet− Pch Storage is very low on charge, buy the
deficit of energy plus an extra amount to charge the storage
if λ ≥ hand S ≥ 80% Pref= Pnet(1 + kb) The price of energy is very high and the
storage is full, sell energy
if λ ≥ hand 50% ≤ S < 80% Pref= 0 The price of energy is very high and the
storage has a medium charge, use it to compensate the energy deficit if λ ≥ hand S < 50% Pref= Pnet The price of energy is high and the
stor-age is medium-low on charge, buy the energy deficit
if λ ≤ land S < 80% Pref= Pnet− Pch− Kchλ The price of energy is very low and the
storage is not fully charged, buy an ex-tra amount of energy to store, propor-tional to the price (i.e., the lower the price the more the EMS can buy) if λ ≤ land S ≥ 80% Pref= Pnet The price of energy is very low and the
storage is fully charged, buy the energy deficit
if λ > land S ≥ 50% Pref= 0 The price of energy is not very low and the storage are is medium-high on charge, use it to compensate the energy deficit
if λ > land 20% < S < 50% Pref= Pnet The price of energy is not very low and the storage is medium-low on charge, buy the energy deficit
vary according to (2.6).
Table 3.5: Microgrid parameters for the IEEE 14-bus system
# MG Bus P¯g (pu MW) P¯l (pu MW) Ts (s) σ
1 10 0.48 0.24 5.0 0.025
2 11 0.10 0.20 7.0 0.030
3 12 0.36 0.14 6.5 0.010
Figure 3.1, 3.2 and 3.3 show the marlet clearing price λ; MG
active powers Pout; and MG states of charge S, respectively. In this
example, the two price thresholds are l = 39.845 and h = 39.825,
on average. It is interesting to note that states of charge usually
satisfy S > 0.8, but they drop below 0.8 when the λ > h. This
behavior indicates that the MGs have an economic advantage to
set Pout > Pg− Pl in the seller mode and to use the energy in the
storage in the buyer mode. Moreover, Fig. 3.2 shows that MGs 2 and
4 switch from a Pout > 0 to Pout < 0, mostly because Pg− Pl≈ 0.
In order to better understand the consequences of the MG be-havior on the transmission system, simulations in a larger power grid are provided in the next section.
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] 39.6 39.65 39.7 39.75 39.8 39.85 39.9 39.95 λ [$ (p u (M W )h )]
3.3 Case Study 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] −0.5 0.0 0.5 1.0 1.5 2.0 poutMicrogrid 1[pu] poutMicrogrid 2[pu] poutMicrogrid 3[pu] poutMicrogrid 4[pu]
Figure 3.2: IEEE 14-bus system: active powers Pout of the 4 MGs.
3.3
Case Study
As anticipated in the previous section, the active power set points of the MGs depend, among other variables, on the electricity price. This feature is the main difference between a MG and a DER unit: while a renewable resource usually injects into the grid all available power – which often evolves according to a stochastic process – a MG adopts a greedy behavior with respect to the electrical grid. In fact, the EMS will sell or buy energy when it is convenient from an economical and operational point of view, while in other cases it will store the surplus of produced energy in its storage units, thus
effectively islanding the MG from the power grid, i.e., Pout = 0.
This case study discusses whether a high penetration of MGs within the electrical grid, without a proper coordinated control, is
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] 0.7 0.75 0.8 0.85 0.9 0.95 1.0 1.05 1.1 SMicrogrid 1[pu] SMicrogrid 2[pu] SMicrogrid 3[pu] SMicrogrid 4[pu]
Figure 3.3: IEEE 14-bus system: states of charge S of the 4 MGs.
sustainable for the power system. In particular, the case study is aimed at defining the impact of the MG penetration level, correlation and granularity, and of the storage capacity on the frequency of the COI. Four scenarios are considered, as follows.
a. Microgrid Scenario. An increasing number of large MGs is plugged into the system.
b. DER & Microgrid Scenario. A mix of DERs and MGs is plugged into the system.
c. High Granularity Microgrid Scenario. The MGs of the first scenario are split into several smaller MGs.
d. Large Capacity Storage Microgrid Scenario. An increasing
3.3 Case Study
into the system.
Simulations are based on the IEEE 39-bus 10-machine system; this benchmark grid is chosen in order to have both a fairly complex network and reduced state-space dimensions to easily understand the impact of MGs on the system. The state-space of the simplest case with 1 MG includes 150 state variables and 233 algebraic ones; whereas the case with highest granularity includes 108 MGs, 685 state variables and 1,421 algebraic ones. The results for each scenario are obtained based on a Monte Carlo method (500 simulations are solved for each scenario), in order to account for a large number of stochastic strong trajectories and, hence, accurately infer statistical properties. Accordingly, the standard deviation of the frequency of
the COI, σCOI, is computed as the average of the standard deviation
obtained for each realization.
All simulations are obtained using Dome, a Python-based power system software tool [42]. The Dome version utilized in this case study is based on Python 3.4.1. All simulations were executed on a server mounting 40 CPUs and running a 64-bit Linux OS.
3.3.1
Microgrid Scenario
This scenario consists in studying the impact on the grid of an in-creasing number of large MGs. Note that the MGs are not assumed to be large per se. Rather, they model large aggregated networks of MGs whose power generation, consumption and EMSs are assumed to be either centrally coordinated or strongly correlated. Table 3.6 shows the parameters for the considered MGs while Fig. 3.4 shows
all the realizations of the frequency of the COI, ωCOI, for 2, 6 and
12 MGs. In Table 3.6, ¯Pg and ¯Pl indicate average values of Pg and
Pl, respectively, during the simulation, while σnet is the standard
deviation of the net active power production Pnet. In the four cases,
the MG capacity and the average ratio between the active power provided by the MGs and the active power provided by conventional power plants are, respectively, 1.81, 3.13, 4.52 and 5.84 (pu MW); and 2.1%, 3.8%, 1.44% and 2.4%. The ratio does not increase with the number of MGs because each unit is able to buy or sell power from the grid (i.e., some of the units are, on average, loads). It is clear, even from visual inspection, that the standard deviation of
ωCOI increases as the number of units plugged into the system gets
higher, increasing, approximately, from 0.0005 to 0.0018. In
partic-ular, Table 3.7 shows the σCOI, as a function of the number of MGs,
ranging from 1 to 12. This result is consistent with the size of each MG and their reduced number. The increasing number of MGs, in
fact, leads the variations of Pnet to increase, which directly impact
on the standard deviation of ωCOI.
Figure 3.5 shows the fluctuations of the market clearing price λ for the cases with 2, 6 and 12 MGs. Note that both the mean value and the amplitude of the variations of λ are affected by the number of MGs included into the system. The fact that the mar-ket clearing price volatility increases as the number of MG increases
is a consequence of λ being dependent on ωCOI, according to (2.4).
Furthermore, the more λ fluctuates, the more frequently each MG
fluctua-3.3 Case Study
Table 3.6: Microgrid parameters
MG Bus P¯g (pu MW) P¯l (pu MW) Tc (s) σnet (pu MW)
1 18 0.88 0.54 5.0 0.025 2 3 0.77 0.20 7.0 0.040 3 15 0.80 0.10 6.5 0.030 4 17 0.40 0.20 8.0 0.020 5 21 0.20 0.10 5.0 0.013 6 28 0.20 0.40 7.0 0.040 7 24 0.36 0.84 6.5 0.010 8 17 0.20 0.50 8.0 0.020 9 11 0.20 0.30 9.0 0.010 10 5 0.10 0.80 5.0 0.010 11 7 0.80 0.10 7.4 0.030 12 12 0.40 0.40 6.8 0.025 tions of ωCOI.
For completeness, Fig. ?? shows the trajectories of the power outputs of three MGs for one realization of the case with 12 MGs. As it can be observed, depending on the set points and parameters, an MG can be always in buyer (MG 6) or seller (MG 5) mode or can alternate between the two modes (MG 10).
3.3.2
DERs & Microgrids Scenario
This section discusses how differently MGs and DERs impact on system dynamics. With this aim, DERs are modeled assuming that
Table 3.7: Standard deviation of the frequency COI as a function of the total MG installed capacity
MG Capacity (pu MW) σCOI (pu Hz)
1 0.96 0.000276 2 1.81 0.000514 3 2.69 0.000584 4 3.13 0.000893 5 3.35 0.000934 6 3.57 0.001121 7 3.97 0.001247 8 4.19 0.001352 9 4.41 0.001401 10 4.52 0.001407 11 5.40 0.001602 12 5.84 0.001788
where Pg varies according to the stochastic process described in
Sec-tion 2.2. DERs are assumed not to provide frequency regulaSec-tion. Three configurations are considered in this subsection, as follows. 0 MGs, 6 DERs
3 MGs, 3 DERs 6 MGs, 0 DERs
To allow for a fair and consistent comparison of the behavior of MGs and DERs, the point of connections, the capacities as well as the mean and the standard deviation of active power set points of the
3.3 Case Study
Table 3.8: DER parameters
DER Bus P¯g (pu MW) σg (pu MW)
1 18 0.88 0.013 2 3 0.77 0.040 3 15 0.80 0.020 4 17 0.40 0.040 5 21 0.20 0.020 6 28 0.20 0.020
DERs have same values as those used for the MGs in the previous subsection. Table 3.8 shows the parameters for the considered DER units, while MG parameters are those of the first 6 rows of Table 3.6.
In Table 3.8, ¯Pg and σg are, respectively, the average active power
Pg produced by the DER during the simulation and its standard
deviation.
Table 3.9 shows the standard deviation of ωCOIfor the three
con-sidered configurations. Note that, to allow for a consistent compar-ison, the installed capacity of the DERs and MGs combined is 3.57 (pu MW) for all cases. Results indicate that the standard deviation of the frequency increases as the number of DERs decreases and the number of MGs increases. This is due to both load variations and active power set-point changes due to the EMSs of the MGs. It ap-pears that, such a greedy behaviour, aimed to maximize MG profits, affects negatively the frequency standard deviation more than mere stochastic variations of uncontrolled DERs.
Table 3.9: Standard Deviation of the frequency of the COI as the number of DERs decreases and the number of MGs increases
# of DERs # of MGs σCOI (pu Hz)
6 0 0.000951
3 3 0.000983
0 6 0.001121
3.3.3
High Granularity of Microgrids Scenario
The results of the previous subsections suggest that, if the level of penetration of MGs and of unregulated DERs is increased, the devi-ation of the system frequency also consistently increases. The capac-ities considered in Subsections 5.2.2 and 3.3.2 are relatively high and can be assumed to represent large aggregated MG and DER models. Moreover, in the previous subsection, load and generator variations are strongly correlated for each MG, thus leading to relatively large
steps of Pg and/or Pl for each MG.
This subsection studies the effect of the correlation of loads and generators within each MG. This is achieved by assuming different granularity levels, i.e., an increasing number of smaller MGs with uncorrelated powers connected at the same bus. With this aim, ev-ery each of the 12 MGs defined in Table 3.6 is split into k smaller MGs with capacity 1/k (see Table 3.10). For example, for k = 4, MG
1 is split into 4 MGs with Pg = 0.22, Pl = 0.135 and σ = 0.00625
(pu MW). Note that the total capacity of the MGs is 5.84 (pu MW) independently from the value of k. The random processes used to define the generation and the load of each MG are uncorrelated, i.e.,
3.3 Case Study
fully independent from each other. Note that, to allow for a consis-tent comparison, the average generation and consumption levels are kept equal for each bus, independently from the granularity of the MG model.
Table 3.10 shows that as k increases the standard deviation of
ωCOI initially increases and then decreases. This is due to the
aver-aging effect caused by the increasing number of independent random processes. This averaging effect is negligible below a certain number of units per bus (24 in this case). The standard deviation becomes smaller than the base-case (k = 1) only when the number of MGs reaches k = 60. Moreover, since the market clearing price λ depends on the frequency according to (2.4), a decrease of the standard
de-viation of ωCOI results in smaller oscillations of λ, and, in turn, each
MG will adjust its active power set-point Pref less frequently. It
is thus reasonable to expect that as the granularity increases, the impact of MG EMS controllers on the transmission system dynamic response decreases. This is consistent with the assumption that MGs characterized by small capacities can be modeled as price takers.
3.3.4
Large Capacity Storage Microgrid Scenario
Subsection 3.3.2 shows that as the number of MGs increases with respect to the DER units, the standard deviation of the frequency of the COI increases. This fact suggests that the greedy behaviour of the MGs impacts on the frequency standard deviation more than the mere stochastic variations of uncontrolled DERs. To further investigate this aspect, this subsection compares the results obtained
Table 3.10: Standard deviation of the frequency of the COI as a function of the granularity k of MGs
k # of MGs σCOI (pu Hz) 1 12 0.001788 2 24 0.003300 3 36 0.002390 4 48 0.001901 5 60 0.001654 6 72 0.001632 7 84 0.001624 8 96 0.001582 9 108 0.001546
in Subsection 5.2.2 with MGs equipped with larger capacity storage. Tables 3.11 and 3.12 show, respectively, the new time constants for the MGs considered in the first scenario and the standard de-viation of the frequency of the COI as the MG penetration level increases. For sake of comparison, the third column of Table 3.7 is included in Table 3.12. Figures 3.6a and 3.6b compare the realiza-tions of the frequency of the COI, respectively, for the small capacity and the large capacity scenario.
Results show that the frequency standard deviation increases as the capacity of the storage gets larger. This increase is due to the fact that a large capacity storage increases the flexibility of the MG. If the storage capacity is small, the EMS is forced to buy/sell the deficit/surplus of energy of the MG as the storage device charges and discharges quickly. On the other hand, if the storage capacity
3.4 Conclusions
Table 3.11: Large storage Microgrid parameters
MG P¯g (pu MW) P¯l (pu MW) Tc (Hours) σnet (pu MW)
1 0.88 0.54 1.5 0.025 2 0.77 0.20 0.8 0.040 3 0.80 0.10 2.3 0.030 4 0.40 0.20 1.8 0.020 5 0.20 0.10 4.0 0.013 6 0.20 0.40 2.0 0.040 7 0.36 0.84 1.5 0.010 8 0.20 0.50 0.5 0.020 9 0.20 0.30 1.2 0.010 10 0.10 0.80 2.0 0.010 11 0.80 0.10 1.3 0.030 12 0.40 0.40 1.5 0.025
is large, the EMS is able to take advantage of the electricity price, i.e., the EMS can sell or buy more power than it currently produces or needs, if the price is convenient. As it appears from simulations, greedy MGs can impact negatively on the frequency of the COI.
3.4
Conclusions
The framework proposed in the previous chapter is here utilized to define the impact of MGs on the transient response of the trans-mission systems and in particular, on its frequency; the different behavior of MGs and DERs; and the effect of MG correlation and granularity.
Table 3.12: Standard deviation of the frequency COI as a function of the total MG installed capacity.
Small storage Large storage
Number of MGs Capacity (pu MW) σCOI (pu Hz) σCOI (pu Hz)
1 0.96 0.000276 0.00082 2 1.81 0.000514 0.00124 3 2.69 0.000584 0.00135 4 3.13 0.000893 0.00188 5 3.35 0.000934 0.00194 6 3.57 0.001121 0.00210 7 3.97 0.001247 0.00243 8 4.19 0.001352 0.00269 9 4.41 0.001401 0.00292 10 4.52 0.001407 0.00339 11 5.40 0.001602 0.00414 12 5.84 0.001788 0.00496
Simulation results show that the deviations of the frequency of the COI and, hence, the overall dynamic response of the transmission system, is consistently affected by the number, the size of MGs and by the dimensions of their storage units. Due to its greedy price-taker behavior, the bigger the size of each MG and of its storage unit, the higher its impact on the system. In other words, a configuration with few large or several small but highly correlated MGs may not be feasible with the physical constraints of the electrical system. On the other hand, a high-granularity and uncorrelated configuration with several small MGs is likely more compatible with a proper operation
3.4 Conclusions
of the system.
The fact that a high correlation of the response of several devices is detrimental to the stability of the overall system has been observed in other fields, such as traffic congestion, e.g., [43]. In control theory, this phenomenon is known as flapping. To mitigate such effects, it is possible to design stochastic distributed and/or decentralized con-trollers. Hence, next chapters will focus on the impacts of different EMSs on the transmission system and on the synthesis of appro-priate controllers and/or the design of proper ancillary services to be provided by the microgrids to mitigate their negative impact on the system and, whenever possible, to leverage their penetration and improve the overall dynamic response of the transmission system.
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] 0.99 0.995 1.0 1.005 1.01 ωC O I [p u (H z) ] 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time[s] 0.99 0.995 1.0 1.005 1.01 ωC O I [p u (H z) ] 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time[s] 0.99 0.995 1.0 1.005 1.01 ωC O I [p u (H z) ]
Figure 3.4: Frequency ωCOI with 2 (upper panel); 6 (middle panel);
and 12 (lower panel) MGs. The grey lines represent each realization, the black thick line represents the average of the process, while the dotted line represents 3 times the standard deviation.
3.4 Conclusions 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time[s] 38.0 38.5 39.0 39.5 40.0 λ [$ /M W h ] 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time[s] 38.0 38.5 39.0 39.5 40.0 λ [$ /M W h ] 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time[s] 38.0 38.5 39.0 39.5 40.0 λ [$ /M W h ]
Figure 3.5: Market clearing price λ with 2 (upper panel); 6 (middle panel); and 12 (lower panel) MGs.
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time[s] 0.98 0.985 0.99 0.995 1.0 1.005 1.01 1.015 1.02 ωC O I [p u (H z) ]
(a) Small storage capacity MG
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time[s] 0.98 0.985 0.99 0.995 1.0 1.005 1.01 1.015 1.02 ωC O I [p u (H z) ]
(b) Large storage capacity MG
Figure 3.6: Frequency of the COI of the 39 bus system with 12 MGs. The grey lines represent each realization, the black thick line represents the average of the process, while the dotted line represents 3 times the standard deviation.
Chapter 4
Impact of the Energy
Management System
4.1
Introduction
In the previous chapter we considered a market based EMS, on the basis of which each MG tries to take advantage of the energy price to maximize its revenues, to model and analyse the behaviour of MGs on the transmission system. The main conclusion of the previous chapter is that a configuration with few large or several small coor-dinated MGs might lead the transmission system dangerously close to its physical constraints. On the other hand, a high-granularity and not coordinated configuration with several small MGs is more convenient for a proper operation of the system. It is still unclear, though, whether the obtained results are caused by chosen the mar-ket model. To answer these questions this chapter further elaborates
on the model previously described to evaluate what happens if the MGs do not compete in a free market but rather keep their power exchanges to the minimum. In particular we assume that MGs try to operate in island mode as long as possible (i.e., using their stor-age reserve when possible or charging it when it is not completely charged), buying and selling energy only when it is strictly necessary. Our goal here is to analyse to what extent different control strategies and different storage sizes affect the electrical transmission system, using as comparing parameter the frequency center of inertia (COI) standard deviation. The MG model to which we refer is The MG model employed are the same described in chapter 3
4.2
Island Based Energy Management
System of the Microgrid
In this section, we evaluate a different strategy for the EMS that, as we shall see in the next section, has a opposite impact on the grid with respect to the EMS proposed in the previous chapter, where the price λ plays a critical role and EMS input quantities are the
produced power Pg, the load Pland the state of charge of the storage
units, S. In this case, as before, the rules are divided into two sets:
the seller state, for which Pnet ≥ 0 (i.e., the MG is producing more
than it is consuming and it will most likely sell energy) and the buyer
state, for which Pnet < 0 (i.e., the MG is consuming more than it
is producing and it will most likely buy energy). The objective of the EMS is to make the MG operate in island mode, if possible (i.e.,
4.2 Island Based Energy Management System of the Microgrid
Pref = 0). Thus, in seller mode each MG uses the active power
surplus to charge its storage and sells it when fully charged, whereas in buyer mode, each MG discharges the storage when it has a high level of charge and buys the energy deficit when the state of charge is low.
The aforementioned rules are shown and explained in Table 4.1. The rules are expressed hierarchically, i.e., a rule is evaluated only if the conditions of the previous ones are not satisfied.
The EMS rules employed in this work are only a possible choice, functional to show that to different EMSs correspond various effects on the power system and that, under certain conditions, a large number of MGs can behave in such a way to harm the electrical system. It is to be stressed that, given the general aim of this work, the particular choice of EMS does not affect the conclusions drawn in the next sections.
4.2.1
Simulation results
Figures 4.1 and 4.2 show the realizations of the frequency of the COI
(ωCOI) of the 39-bus system with inclusion of 12 MGs considering
small and large energy storage capacity scenarios and both Market-based EMS (M-EMS), an EMS Market-based on the rules given in Chapter 4, and Island-based EMS (I-EMS), and EMS based on the rules provided in the previous Section. The following remarks are relevant:
A small-capacity storage affects the ωCOI in a similar way for
both M-EMS and I-EMS. A small capacity storage device leads the EMS to buy (sell) the MG deficit (surplus) power when the
Table 4.1: Microgrid EMS rules
Seller mode, Pnet≥ 0
Rule Action Rationale
if S ≤ 80% Pref = 0 The battery is not fully charged, use the
energy surplus to charge it
else Pref = Pnet Sell the surplus
Buyer mode, Pnet< 0
Rule Action Rationale
if S ≥ 20% Pref = 0 The battery has residual charge, use it
to match the internal energy deficit
else Pref = Pnet Storage is low on charge, buy the energy
deficit
storage is empty (full), no matter what EMS implementation is utilized. This result can be observed by comparing Figs. 4.1a
and 4.2a: the oscillations of the ωCOI appear to be similar
despite the use of two very different EMS. Accordingly, it can be inferred that in absence of a storage, the MGs behavior is not affected by the choice of the EMS.
A large-capacity storage affects negatively the ωCOI when
en-forcing the M-EMS, whereas it smooths frequency oscillations when enforcing the I-EMS. A large-capacity storage device in-creases the flexibility of the EMS: the M-EMS is able to take advantage of the electricity price (i.e., it is able to sell or buy
4.2 Island Based Energy Management System of the Microgrid
more power than it currently produces or needs, if the price is convenient). This behavior leads to increase the variations
of ωCOI (see Fig. 4.1b). The I-EMS, on the other hand, forces
the MGs to operate in island for longer periods of time, thus mitigating the fluctuations of the frequency of the COI (see Fig. 4.2b).
Table 4.3 shows the standard deviation of the frequency of the
COI (σCOI), vs. the number of MGs, the capacity of the storage
devices and the EMS control strategy. In all cases, σCOI increases
monotonically as the number of MGs increases. Consistently with the results depicted in Fig. 4.1 and 4.2 and regardless the number of
MGs, σCOI is larger for the scenario with small storage devices than
for the large storage one for a given EMS.
Table 4.3 also shows that the impact of the EMS control strategy depends on the size of the storage capacity. For small storage devices,
the M-EMS appears to lead to smaller variations of ωCOI than the
I-EMS. This results is due to the fact that I-EMS-driven MGs, while attempting to operate in island mode, have actually to connect often to the grid due to the small capacity of storage devices. On the other hand, for large capacity, the outcome is the opposite, i.e.,
I-EMS leads to lower σCOI than the M-EMS. In this case, the large
capacity of the storage allows the I-EMS-driven MGs to operate most
of the time in island mode. The large values of σCOIobtained for
M-EMS-driven MGs indicate that MGs, operated with benefit-oriented control strategy, are a potential threat for the stability of the system.
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] 0.96 0.97 0.98 0.99 1.0 1.01 1.02 1.03 1.04 ωC O I
(a) Small storage and M-EMS
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] 0.96 0.97 0.98 0.99 1.0 1.01 1.02 1.03 1.04 ωC O I
(b) Small storage and I-EMS
Figure 4.1: Frequency of the COI of the 39-bus system with 12 MGs and small storage units.
4.3
Conclusions
This chapter analyzes and compares the dynamic behavior of power systems with inclusion of MGs operated with two different EMS
4.3 Conclusions 0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] 0.96 0.97 0.98 0.99 1.0 1.01 1.02 1.03 1.04 ωC O I
(a) Large storage and M-EMS
0.0 500.0 1000.0 1500.0 2000.0 2500.0 3000.0 3500.0 Time [s] 0.96 0.97 0.98 0.99 1.0 1.01 1.02 1.03 1.04 ωC O I
(b) Large storage and I-EMS
Figure 4.2: Frequency of the COI of the 39-bus system with 12 MGs and large storage units.
control strategies. An EMS aims at maximizing the benefit of the MGs, while the other one aims at operating the MGs islanded from the system. The effect of the size of the storage devices included in
the MG is also discussed in the work. The impact on the system is evaluated by means of the amplitude of the standard deviation of the frequency of the COI. The main result obtained from simulations is that the dynamic impact of MGs on the system is a combination of the size of the storage and the EMS rules. The minimum impact on the system is obtained for MGs that tend to operate in island mode and include large storage capacities. However, if the storage capacity is low, the impact due to MGs on the system is lower if the MG attempts to maximize their incomes. Thus, results suggest that the optimal size of the storage device, from the point of view of the outer grid stability, might depend on the control strategy of the MGs. The next chapters will focus on the synthesis of ad hoc control strategies to mitigate the negative effects that are highlighted in these last chapters and maintain the frequency of the transmission system within its safe operational boundaries.
4.3 Conclusions
Table 4.3: Standard deviation of the frequency COI as a function of the total installed capacity of MGs.
Small storage
MGs # Capacity (pu MW) M-EMS σCOI(pu Hz) I-EMS σCOI(pu Hz)
1 0.96 0.00246 0.00312 2 1.81 0.00319 0.00454 3 2.69 0.00322 0.00475 4 3.13 0.00391 0.00498 5 3.35 0.00442 0.00517 6 3.57 0.00498 0.00570 7 3.97 0.00517 0.00607 8 4.19 0.00554 0.00625 9 4.41 0.00581 0.00679 10 4.52 0.00597 0.00719 11 5.40 0.00623 0.00772 12 5.84 0.00689 0.00784 Large storage
MGs # Capacity (pu MW) M-EMS σCOI(pu Hz) I-EMS σCOI(pu Hz)
1 0.96 0.00498 0.00049 2 1.81 0.00532 0.00088 3 2.69 0.00641 0.00094 4 3.13 0.00701 0.00117 5 3.35 0.00741 0.00127 6 3.57 0.00775 0.00168 7 3.97 0.00796 0.00193 8 4.19 0.00848 0.00221 9 4.41 0.00863 0.00261 10 4.52 0.00926 0.00282 11 5.40 0.01089 0.00324 12 5.84 0.01259 0.00392
Chapter 5
Control of Interconnected
Microgrids
5.1
Introduction
Conclusions from the previous chapters show that the presence of many interconnected Microgrids (MGs), that opportunistically man-age their power flow to optimize an individual utility function, may cause large frequency deviations of the interconnecting transmission system.
To overcome this issue we propose a stochastic controller whose purpose is to allow each MG to maintain an acceptable level of oper-ational freedom. The roper-ationale for a stochastic approach, as opposed to a deterministic one, is that the latter would require heavy com-munication between the various MGs or to a centralized entity, in order to be effective. This requirement, besides the obvious
dis-advantages of economical costs and robustness [46] (e.g., requiring communication among the agents implies that if for some technical failure two or more agents are unable to communicate among each other the controller might not be able to provide any form of regula-tion), might also incur into non feasible privacy issues [47]. This is a crucial aspect in the context of market deregulation: sharing data of any kind, with other agents, might result into disclosing strategies and informations that the single user wants to keep private. There-fore, it becomes very useful to have a decentralized structure that avoids any communication among its uses. On the other hand, in the presence of a reasonably large number of independent units, a stochastic approach ensures that statistically the system will con-verge to a predetermined, average behaviour without any need for communication. Moreover, the independence among each unit and the introduction of stochasticity in the control actions prevent possi-bly harmful behaviours from occurring, like synchronization among MGs, that might lead to system instability [49].
According to the previous considerations, this chapter explores the extent to which MGs can operate autonomously, according the their individual utility function, and to which extent ancillary ser-vices should be provided to mitigate the frequency deviations on the grid. A distributed stochastic technique is proposed to allow MGs to operate autonomously, as much as possible, keeping the amount of active power used to provide primary frequency regulation to the minimum. The following points summarize the key points of this chapter:
5.1 Introduction
We present a simple algorithm that, on the basis of past and present measurements of the maximum amplitude of the COI of the frequency provides a trade off between frequency grid stability and the operational freedom of the MGs. This algo-rithm represents a very conservative solution to the proposed problem and it is intended to serve as an example;
An application of the unsynchronized Addictive Increase Mul-tiplicative Decrease (AIMD) algorithm [47] to mitigate the negative effects of the penetration of MGs on the frequency deviation of the transmission system. The analysis is car-ried through realistic time-domain simulations that take into account detailed non linear electro-mechanical models, MGs, storage units and DERs;
A comparison between a centralized and a decentralized ver-sion of the AIMD algorithm. A detailed evaluation of their performance is provided in terms of frequency stability, fair-ness, privacy and operational flexibility.
In what follows we outline the base control strategy and the mathematical formulation for the problem at hand. In the following section a sub optimal algorithm is proposed, as an example to solve it. Finally we introduce the AIMD algorithm, its stochastic counter-part, the unsynchronized AIMD and the simulations that are used to properly evaluate the performance of the control system.
5.1.1
Energy Management System of the
Micro-grid
As stated in Section I, the purpose of this work is to synthesize a strategy that mitigates the impact of MGs on the power grid while allowing them to continue to operate advantageously according to their own policy. To achieve this result we propose a control system such that, whenever the behaviour of the MGs brings the frequencies close to the power grid operational boundaries, each MG switches in a stochastic way between two operational modes:
Market based mode (M-mode): The MG is free to maximize its revenues (e.g., selling and buying energy without limitations); Frequency regulation mode (F-mode): The MG participates to
the primary frequency regulation of the power grid.
The M-mode used in this work follows the same set of if-then rules proposed in Section 3.2, while the F-mode behaves according to the droop control equations classically employed in the primary frequency regulation of the power system [50]. Note that the M-mode is only one of the possible choices: the same control system can be applied without limitations to other strategies (e.g., the island based set of rules employed in Section 4.2).
The switching between the two operational modes of the i-th MG
is controlled by the probability Psi(t). In particular, whenever the
frequency violates the operational boundaries of the system, each
5.1 Introduction
Psi(t). Conversely, every TP seconds, if the control signal lies within
the boundaries, each MG switches from the R-mode to the M-mode with probability Psi(t).
The smaller the value of Psi(t) over time, the larger the chance
for the i-th MG to regulate the frequency. Thus, the i-th MG’s goal
is to maximize Psi(t) (i.e., keeping it as close to 1 as possible). It
is possible to provide a mathematical formulation for the problem at hand: given n MGs, we assign to each of them a convex, strictly
differentiable function, fi(·) : [0, 1] 7−→ R. The proposed situation
can now be expressed in terms of the following optimization problem minimize Ps(t) n X i=1 fi(Psi(t)) subject to |ωi− ω0| 6 ωm,∀i ∈ Θ (5.1)
where Ps(t)∈ Rn is the vector whose i-th component is Psi(t), Θ is
a discrete set of indexes and ωm is the maximum allowed deviation
from the nominal frequency, ω0. The functions fi(·) can be assigned
to each MG by the System Operator responsible for the specific area
(e.g., this could be based on CO2 emissions, size of the MG, its
energy class, etc).
Even though (5.1) presents a unique solution, due to its convexity,
finding an analytical relationship between Ps(t) and all the ωi is a
challenging task. Equation (5.1), in fact, is a coupled optimization problem where the variations of a single switching probability affects every local frequency in the transmission system in a non linear way. To overcome this issue we propose two different solutions:
A simple algorithm, based on the classical PI controller, that
computes the probabilities Psi(t) on the basis of the maximum
allowed frequency deviation σωm, and measurements of the COI
of the frequency in a time window of length Tp;
A stochastic version of the AIMD algorithm, to solve (5.1). Such an algorithm has the advantage of having a decentralized, model-free structure (i.e., it does not require any knowledge of the specific system it is being applied to) that results in a very high level of robustness and low bandwidth communication among the MGs.
The next Section presents the first of the two control strategies and provide some simulations to show its effectiveness.
5.2
PI-Like Controller
A first, possibly conservative solution for (5.1) can be provided in the
form of a controller that regulates the probability Psi(t), at discrete
time intervals Tp, according to the following equation:
Psi(kTp) = 3 ασCOI(kTp) + (1− α) k−1 P n=k−M lnσCOI(nTp) ¯ ωm , (5.2)
where σCOI(kTp) is the standard deviation of the COI of the
fre-quency in the past time window [(k − 1)Tp, kTp], α is a parameter
belonging to [0, 1], M represents the number of past values of σCOI
5.2 PI-Like Controller
such that PM −1
n=1 ln = 1. This control law is similar to a discrete PI
controller, as it takes into account the actual value of the controlled
variable σCOI and its weighted integral. This choice avoids
oscilla-tions in the value of Ps that might occur if only the last value of
σCOI were taken into account (similarly to what would happen to a
system with a purely proportional controller). Finally, the
parame-ters α, M and the normalized weights ln, can be tuned, by means of
a trial and error procedure, to ensure that the MGs do not behave
too conservatively with respect to the boundary value ¯ωm (i.e., given
a certain bound ¯ωm, if the parameters are not properly tuned the
controller might behave too aggressively, forcing the MGs to
regu-late the frequency even when the σCOI is still far from the maximum
allowed deviation) and to achieve a desired trade off between the
past and the present values of σCOI.
5.2.1
Case Study
In order to show the effects of (5.2), three different scenarios are proposed, as follows:
36 MGs without frequency control;
36 MGs with the deterministic version of the controller pro-posed in this section, that switches from M-EMS to F-EMS whenever 3σCOI > σωm;
36 MGs with the control system proposed in this Section. In this scenario we compare four different sets of parameters to
evaluate the controller overall performance as shown in Table 5.2.
In all scenarios above, ¯ωm is set equal to 0.02 pu.
The performance of the stochastic controller is evaluated on the
basis of the revenues of each MG, R(t) and on the quantity Ts that
represents the average percentage of time spent in F-EMS mode.
Ts can be interpreted as an indicator of the operational freedom
granted to each MG. In fact MGs should be interested in minimizing
their own Tsi and, therefore, minimizing the time spent in providing
services to the grid.
The state-space of the each case with 36 MG includes 432 state variables and 773 algebraic ones. The results for each scenario are obtained based on a Monte Carlo method (100 simulations are solved for each scenario). Table 5.1 shows the parameters for the 36 consid-ered MGs. Three MGs are connected to each of the buses indicated in the table.
All simulations are performed using Dome, a Python-based power system software tool [42]. The Dome version utilized in this case study is based on Python 3.4.1; atlas 3.10.1 for dense vector and matrix operations; cvxopt 1.1.9 for sparse matrix operations; and klu 1.3.2 for sparse matrix factorization.
5.2.2
Simulation results
Figures 5.1, 5.2 and 5.3 show the realizations of the COI of the frequency for the different scenarios. The deterministic controllers performances are quite poor as it fails to maintain the COI of the
5.2 PI-Like Controller
Table 5.1: Microgrid parameters. The parameters σnet are chosen
randomly in order to take into account different variations of the load and the DERs energy production of each MG.
MG Bus P¯g (pu MW) P¯l (pu MW) σnet (pu Hz) Ts (s)
1 18 0.88 0.54 0.025 18000.0 2 3 0.77 0.20 0.040 25200.0 3 15 0.80 0.10 0.030 23400.0 4 17 0.40 0.20 0.020 28800.0 5 21 0.20 0.10 0.013 18000.0 6 28 0.20 0.40 0.040 25200.0 7 24 0.36 0.84 0.010 23400.0 8 17 0.20 0.50 0.020 28800.0 9 11 0.20 0.30 0.010 14400.0 10 5 0.10 0.80 0.010 18000.0 11 7 0.80 0.10 0.030 26640.0 12 12 0.40 0.40 0.025 24480.0
frequency inside the desired range (see Fig. 5.1b). On the other hand, the stochastic controller exhibits better performances, effec-tively mitigating the oscillations into the desired range [0.98, 1.02] (see Figs. 5.2 and5.3).
Table 5.3 shows the standard deviations, the revenues (values are normalized with respect to the largest revenue, since the abso-lute values depend on the choice of the market model and not on the proposed control scheme and thus, only the relative values are meaningful) and the switching times, averaged over each MG and all realizations. The switching times and the revenues are the
