Anteproyectodanilov5concorrecciones.pdf

Passivity-Based Analysis and Control of AC Microgrids

Oscar Danilo Montoya Giraldo

Project proposal submitted in partial fulfillment of the requirements to be candidate for the degree of Ph.D in Engineering

August 2, 2018 ´ UNIVERSIDAD TECNOLOGICA DE PEREIRA Doctoral Engineering Program Area of Electrical Engineering

Request for evaluation

Project proposal submitted in partial fulfillment of the requirements to be candidate for the degree of Doctor in Engineering at the Universidad Tecnol´ogica de Pereira

Oscar Danilo Montoya Giraldo. Universidad Tecnol´ogica de Pereria. Student

Revised by:

Alejandro Garc´ es Ruiz, Ph.D. Universidad Tecnol´ogica de Pereira. Advisor

Cartagena, August 2, 2018

Doctoral project proposal

i/23

Table of contents General index 1 General information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Problem Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Justification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Referential framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Passivity-based control . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Hamilton’s systems . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Lyapunov stability theory . . . . . . . . . . . . . . . . . . . . . . 5.4 Stability analysis of PCH systems . . . . . . . . . . . . . . . . . . 5.5 Interconnected systems . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Dynamical system under analysis . . . . . . . . . . . . . . . . . . 5.6.1 Dynamical model of a PWM-VSC for DERs integration 5.6.2 Dynamical model of a PWM-CSC for SMES integration 6 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 General Objective . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Specific Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Scope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 Expected results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 Available resources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 Activity schedule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References

Oscar Danilo Montoya Giraldo

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i 1 1 3 4 6 6 8 9 11 11 12 13 14 15 15 15 15 16 17 17 18 23

August 2, 2018

UTP

Doctoral project proposal

1

1/23

General information

Title: Passivity-based analysis and control of AC microgrids. Research area: Operation and control of power systems. Courses associated to current research: Control of power systems, optimization, non-linear control and energy storage systems. Participants: Oscar Danilo Montoya Giraldo. Adviser: Alejandro Garc´es Ruiz. External advisers: The following professors have accepted to be collaborators during the developed of this research project: Gerardo Ren´e Espinosa P´erez at National Autonomous University of Mexico (UNAM). Marta Molinas at Norwegian University of Science and Technology (NTNU). Carlos Alberto de Castro Jr. at University of Campinas (UNICAMP). Fed´erico M. Serra at National University of San Luis, Argentina.

2

Problem Statement

Modern electrical networks have changed from classical hydro-thermal electric systems with passive loads to active electrical networks with distributed energy resources (DERs) which includes renewable generation [1], energy storage systems [2–4] and dynamic loads [5–8]. These technologies can be integrated to the electric power systems by using power electronic converters under the concept of microgrids (MGs) [9–11], allowing improvements in voltage regulation, reducing power oscillations caused by renewable energy, performing frequency regulation and supplying energy to the loads during service outages [6]. However, these improvements can be only achieved by using effective control strategies that consider the MG as a whole and not only as individual components [12, 13]. Although there is not a standard definition, the U.S. Department of Energy, defines a MG as “a group of interconnected loads and distributed energy resources with clearly defined electrical boundaries that acts as a single controllable entity with respect to the grid and can connect and disconnect from the grid to enable it to operate in both gridconnected or island modes”. Figure 1 shows schematically this concept. In all MGs there exists power electronic converters based on forced commutation (e.g voltage/current source [14] [12, 15]) or line commutation technologies [16]. These allow advanced control strategies for operation in transient and steady state [5]. Diverse control strategies have been explored in the specialized literature, namely: fuzzy-logic [17, 18], feedback linearization [14, 15, 19], model predictive [20] and passivity-based control [21– 25], among others. Nevertheless, proportional-integral controls [5, 26] are the most used approach. Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

2/23

Main grid P, Q

Microgrid

AC AC

AC

AC

AC

DC

AC

G

DC

Load 1

Microturbine +−

G Load 2

Wind

Solar Energy storage device

Figure 1: Typical configuration of a microgrid These controls are used when the MG operates in interconnection mode; however, they must be modified in island mode in order to support voltage profile and regulate electrical frequency by using a primary and secondary control strategies [27] that in turns, can be centralized or distributed [6]. The former is highly efficient but relies on the communication system [28] while the latter requires fewer communication channels, which implies fewer investment costs and allows scalability [?, 29]. Closed-loop stability must be guaranteed in the increasingly complex MGs. However, in many cases, the optimal tuning of a proportional-integral control in one device, could affect negatively the stability of the grid. This is because the model of the system is oversimplified and the control is locally designed. In this context, we can pose the following research question ¿How to control each component such that the stability of the entire MG is guaranteed under interconnected and islanded operation modes? To answer this question it is necessary to use a generalized theory with the following characteristics: • Applicable to different type of devices (e.g renewable energies, batteries, energy storage devices). • Applicable to the non-linear model of the components. • Easily integrable to the model of the entire system. • Suitable to include the communication system. • Guaranteed stability in different operative conditions. • Scalable. Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

3/23

Passivity-based control can be an appropriate control strategy that fulfills all these requirements. However, a research project is required in order to apply this control paradigm to MGs taking into account all the complexities inherent to power networks. Notice that in specialized literature there exist few references that propose the application of the passivation theory to operate ac power MG considering the natural passive model of the electrical power system. Nevertheless, this approaches are mainly focuses in single-phase electrical networks [30]. In review section 4 a detailed review of the state of art will be presented.

3

Justification

The electric power system plays an important role in the economic development of any country. Colombia has an interconnected power system with a liberalized market, but there are challenges which require research under the new paradigm of the MGs. The country relies on the hydroelectricity complemented by a minor percentage of thermoelectricity which is required in order to deal with the phenomena of El Ni˜ no. This is a complex whether pattern that results when temperatures in the Pacific Ocean increases form the norm. Although it is a natural phenomenon that typically occurs every two to seven years, the 2015-2016 El Ni˜ no was particularly intense as consequence of the global warming. The level of the reservoirs decreases to less than 20%. This event demonstrated the requirement for a diversified energy matrix. There is a high potential of wind and solar energy in the Country; moreover, these potentials increase when El Ni˜ no occurs. Hence, wind and solar are complementary to hydroelectricity. These new renewable resources should be integrated into the paradigm of MGs since large photovoltaic power stations could create additional environmental impacts and compromise the use of land in agriculture. On the other hand, there are some zones in the country which lack a reliable power service. These non-interconnected zones include more than 1500 small towns totaling more than 1 million inhabitants. MGs are also a promising alternative for the integration of renewable sources and energy storage in these zones. Therefore, a research at a local level is required to deal with the particular characteristics of the Colombian system taking into account an equilibrium between theory and practice. In this context, the strategies classically employed to operate conventional grids need to be improved or changed, in order to consider the impact of variations in the energy delivered by DERs [31]; besides, to improve the speed response of existing control strategies to hold the correct system operation during power outages caused by external failures [6]. This research project focuses to explore the passivity properties of electrical grids based on Hamiltonian formulations, in order to guarantee stability in closed-loop using Lyapunov’s theory for autonomous systems [22, 23]. This control strategy was selected because it allows to design global stable controllers by dividing the electrical network into small subsystems [32]. For all subsystems it is possible to use Hamilton theory to developed passive controllers that preserve their passive nature when interconnected with Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

4/23

other passive systems [33]. There are multiple references that use passivity-based control theory to develop global asymptotically stable controllers to integrated DERs on electrical power systems [21]; notwithstanding, it was not possible to find references that use this theory to operate all devices interconnected at the same time in MG considering the natural passive model of the electrical power system. The focal point of this research project corresponds to the design of passivity-based controllers to integrate DERs on MG. The DERs analyzed are: • Renewable energy resources based on wind or photovoltaic technologies using voltage source converter technologies. • Energy storage systems such as: superconducting coils using current source converter technologies, supercapacitors and batteries employing voltage source converter technologies. A passivity-based control theory was selected because it is an adequate control technique to operate all DERs in MGs, since the dynamical model of these systems have a Hamiltonian structure, which is the main characteristic in the design of stable controllers using passivation theory [34]. Nevertheless, it is important to point out that classical formulations such as Bryton-Moser or Lagrangian can be also employed as an alternative formulations to design passivity-based controllers [35].

4

Literature review

This section presents a detailed literature review in the context of designing passivity based controllers for distributed energy resources and MGs. The analysis is concentrated in superconducting coils, supercapacitors, batteries, and photovoltaic and wind generation. Additionally, are presented general approximations to analyze entire ac MGs under passivation theory approach. In Table 1 the most popular control strategies for energy storage devices are presented. The energy storage technologies presented will be the technologies analyzed in this research project. Notice that in Table 1 it is common to find diverse control strategies for any storage devices technology; although, the passivity-based control it is also common, those articles focuses only of analyzing one specific device, without taking into account the rest of the electrical network, which does not bring the possibility to extend the stability properties to all electrical network. On the other hand, in Table 2 is presented the most important approaches to control wind and photovoltaic generators in power systems. In Table 2 has been presented some papers that explore different control techniques to integrate wind and photovoltaic generators in power MGs, notices that passivity-based control theory appears recurrently in the review of the state of art, which implies that it Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

5/23

Table 1: Summary of investigations related related to energy storage devices Energy storage technology

Superconducting coils

Supercapacitors

Batteries

Control approach Passivity-based control Linear matrix inequalities Feedback linearization Passivity-based control Model predictive control Proportional-integral control Passivity-based control Model predictive control Fuzzy logic control Adaptive Passivity-based control Model predictive control Fuzzy-sliding mode control

Type of converter References PWM-CSC [36] PWM-CSC [37] PWM-CSC and VSC [14, 15] VSC with DC/DC Chopper [38] VSC with DC/DC Chopper [39] VSC with DC/DC Chopper [40] DC/DC [41] VSC with DC/DC [42] VSC with DC/DC [43] Bidirectional Boost converter [44] DC/DC [45–47] DC/DC [48]

Table 2: Summary of investigations related to photovoltaic and wind generation Generation technology

Wind generators

Photovoltaic generators

Control approach Passivity-based control Instantaneous power theory Adaptive fuzzy logic control Model predictive control Adaptive control Control by consensus Backstepping control Passivity-based P control Model predictive control Instantaneous power theory

Type of grid References Single-phase and Three-phase AC grid [49–51] Three-phase AC grid [52] Three-phase AC grid [53] Three-phase AC grid [54] Three-phase AC grid [12] Three-phase AC grid [55] Three-phase AC grid [56] Three-phase AC grids [57] Three-phase AC grid [58] Three-phase AC grid [59]

correspond an actual and powerful control strategy for distributed generators, as well as, energy storage technologies. In the literature related were found two approximations that analyze the whole microgrid as a completed entity. The first case [30] explores the structural properties of MGs via passivation theory; nevertheless, this work focuses particularly in single-phase MG with linear and nonlinear components. In the second case, see [60], are studying the operating conditions that allow remaining stable an inverter-based MG. Additionally, it is important to mention that there exist stability analysis in DC microgrids, as can be consulted in [61–65] Based on the previous state of art, it is clear that there are not approaches that analyze three-phase MGs considering distributed generators and energy storage devices at the same time via passivation theory. This situation is used in this research project as an opportunity for investigation. The passivity theory is chosen as a tool of analysis of electrical networks due to the most of the elements analyzed in this investigation can be modeled by using Hamiltonian formulations [30].

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

5

6/23

Referential framework

This section presents a basic description of the referential framework that is necessary to understand this proposal. First, general concepts of passivity-based control are shown. Next, basic properties of Hamiltonian systems are presented. Third, a general theory to analyze time-invariant dynamical systems using Lyapunov stability theorems is defined. Fourth part shows the stability analysis of Hamiltonian system using Lyapunov’s theorem. The fifth section presents the main characteristic of interconnected systems. Finally, in the sixth part, a nonlinear dynamical formulations for integration of distributed energy resources through power electronic converters is modeled. These models will be the base for this research.

5.1

Passivity-based control

The passivity-based control theory corresponds to a general methodology to develop controllers for linear and nonlinear dynamical systems [34, 66]. This technique is oriented to dynamical systems that have a mathematical model of concentrated parameters [35,67]. It is considered that the dynamical systems are interconnected with other systems for some power port through u and y variables, such that {u, y} ∈ Rn and their product has units of power. In case of electric systems these variables are currents and voltages. A passivity system can be represented in a general form as: Z t
H (x (t)) − H (x (0)) = uT (τ ) y (τ ) dτ + E (x (t)) (1) 0 n

where x(t) ∈ R are state variables of the dynamical system, H(x(t)) : Rn → R is the energy function of the dynamical system, commonly known as Hamilton function and E(x(t)) : Rn → R represents the internal energy generation by the dynamical system. Internal energy, given by E(x(t)), defines the demeanor of the dynamical system. In case of passive systems, they do not generate energy by themselves, which implies that they only store or dissipate the energy received from the external sources. For this reason, a passive system defined by (1) can be rewritten changing E(x(t)) for D(x(t)) as follows: Z t
H (x (t)) − H (x (0)) + D (x (t)) = uT (τ ) y (τ ) dτ (2) 0

d H (x (t)) ≤ u(t)T y(t) dt where D(x(t)) : Rn+ → R takes into account the dissipation effect in the dynamical system, for example, the resistive effects in an electrical systems or friction in case of rotating machines. The second part of (2) shows that a dynamical system is passive when the total energy stored is always less or equal than the input energy. Equality is fulfilled for conservative systems [68] and it has units of power.

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

7/23

Figure 2 shows a typical behavior of a generic energy storage function for a passive system. Notice that the energy storage function of this generic passive system has a minimum in xo (the minimum point can be local or global). H(x)

xo

x

Figure 2: Original energy storage function of a generic passive system The main idea of the classical passivity-based control theory is to incorporate energy principles in the design of the controllers. In this context, it is adopted a control strategy that uses the intrinsic interconnection properties present in the dynamics of the system and the controllers as interconnected energy devices. In this way, the passive structure of the dynamical system is maintained by the closed-loop control by changing the energy storage function in order to achieve a desired set-point [35]. The philosophy of the passivity-based control theory is depicted in the Fig. 3, where the energy storage function is moved from the original minimum xo to a desired operative point x∗ . This goal is achieved transforming H(x) into HD (x) through of the inputs of the system [69].

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

8/23

H(x) HD (x)

H(x)

HD (x∗ ) H(xo ) x∗

xo

x

Figure 3: Original and desired energy storage functions of a passive system Stability properties of the dynamical system are maintained by preserving its passive properties [35, 69].

5.2

Hamilton’s systems

A Hamilton’s system is a term commonly used in specialized literature to refers to systems like port-controlled Hamilton systems (PCH), port-controlled Hamilton systems with dissipation (PCHD) or generalized Hamilton systems, among others [34,66]. In this proposal, this term is used to identify PCH systems. One of the main aspects in the PCH formalism is the strong relation between energy storage, dissipation and interconnection structures with the dynamical model of electrical systems as is the case of electrical energy storage systems (EESS). Remark that the PCH formalism can be extended to mechanical, chemical, thermodynamical, hydrodynamical or electromechanical systems [34]. In general, a time-invariant PCH system might be represented in the standard inputstate-output form as follows: x˙ = [J (x, u) − R (x)] ∇H (x) + G (x) u (3) y = G T (x) ∇H (x) where x ∈ Rn , J (·, ·) = −J T (·, ·) is an antisymmetric interconnection matrix, R(·) = R (·) ∈ R+ n×n is a positive semidefinite symmetric dissipation matrix, ∇H(·) ∈ Rn is the gradient of the Hamilton function (H(·) ∈ R), G(·) ∈ Rn×p is the input matrix, y ∈ Rp and u ∈ Rp are named the port variables and their inner product corresponds to the power supplied by the dynamical system. When the Hamilton’s system has bounded inputs, it is possible to find general conditions to demonstrate passivity properties on the dynamical systems by using Lyapunov’s T

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

9/23

theory as follows: H˙ (x) = ∇H(x)T x˙ (4) H˙ (x) = ∇H(x)T [J (x, u) − R (x)] ∇H (x)+∇H(x)T G (x) u After rearrange some terms in (4) is obtained (5). H˙ (x) = −∇H(x)T R (x) ∇H (x) + y T u

(5)

where y T u is the power supplied rate. Notice that if the dissipation matrix R(·)  0 is positive semidefinite. Equation (5) may be reduced as: H˙ (x) ≤ y T u

(6)

Remark that (6) is a passive system since the change in the total energy stored is less than or equal to the total energy supplied by the inputs ∀t ≥ 0 [35].

5.3

Lyapunov stability theory

This section presents the general conditions to guarantee stability of a nonlinear autonomous dynamical system employing Lyapunov analysis. Most of these results were taken from [32, 33, 68]. Consider the next general nonlinear autonomous dynamical system: x˙ (t) = f (x (t)) ,

x (0) = x0 ,

t ∈ Ix0

(7)

where x(t) ∈ D ⊆ Rn , t ∈ Ix0 represent the state vector, D is an open set with 0 ∈ D, f : D → Rn is continuous on D, and Ix0 = [0, τx0 ), 0 ≤ τx0 ≤ ∞. For every initial condition x(0) ∈ D and every τx0 > 0, the dynamical system (7) possesses a unique solution x : [0, τx0 ) → D on the interval [0, τx0 ). Denote the solution of (7) with and initial condition x(0) = x0 by s(·, x0 ), so that the flow of the dynamical system (7) given by the map s : [0, τx0 )×D → D is continuous in x and continuously differentiable in t and satisfies the consistency property s(0, x0 ) = x0 and the semigroup property s(τ, s(t, x0 )) = s(τ + t, x0 ), for all x0 ∈ D and t, τ ∈ [0, τx0 ) such that t + τ ∈ [0, τx0 ). Unless otherwise stated, it is assumed that f (0) = 0 and f (·) is Lipschitz continuous on D. The following definition introduces several types of stability corresponding to the zero solution x(t) = 0 of (7) for Ix0 = [0, τx0 ). Definition 5.1 i. The zero solution x(t) ≡ 0 to (7) is Lyapunov stable if, for all  > 0, there exists δ = δ() > 0 such that if kx(0)k < δ, then kx(t)k < δ, t≥ 0. ii. The zero solution x(t) ≡ 0 to (7) is (locally) asymptotically stable if it is Lyapunov stable and there exists δ > 0 such that if kx(0)k < δ, then lim x (t) = 0. t→∞

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

10/23

iii. The zero solution x(t) ≡ 0 to (7) is (locally) exponentially stable if it is Lyapunov stable and there exist positive constants α, β and δ such that if kx(0)k < δ, then kx(t)k ≤ α kx(0)k e−βt , t ≥ 0. iv. The zero solution x(t) ≡ 0 to (7) is globally asymptotically stable if it is Lyapunov stable and for all x(0) ∈ Rn , lim x (t) = 0. t→∞

v. The zero solution x(t) ≡ 0 to (7) is globally exponentially stable if there exist positive constants α and β such that kx(t)k ≤ α kx(0)k e−βt , t ≥ 0, for all x(0) ∈ Rn . vi. Finally, the zero solution x(t) ≡ 0 to (7) is unstable if it is not Lyapunov stable. Exponential stability implies asymptotic stability and asymptotic stability implies Lyapunov stability. The following result, known as Lyapunov’s direct method, gives sufficient conditions for Lyapunov, asymptotic, and exponential stability of a nonlinear dynamical system. For this result, let V : D → R be a continuously differentiable ∆ ˙ function with derivative along the trajectories of (7) given by V(x) = ∇V(x)T f (x). ˙ Note that V(x) is dependent of the system dynamics (7). Since, using the chain rule, T d ˙ V (s (t, x))| t=0 = ∇V(x) f (x) it follows that if V(x) is negative, then V(x) decreases dt along the solution s(t, x0 ) of (7) through x0 ∈ D at t = 0. Theorem 5.1 (Lyapunov’s Theorem) Consider the nonlinear dynamical system (7) and assume that there exists a continuously differentiable function V : D → R such that V(0) = 0, V(x) > 0,

(8)

x ∈ D,

∇V(x)T f (x) ≤ 0,

x 6= 0,

(9)

x ∈ D.

(10)

Then the zero solution x(t) ≡ 0 to (7) is Lyapunov stable. If, in addition, ∇V(x)T f (x) < 0,

x ∈ D,

x 6= 0

(11)

then the zero solution x(t) ≡ 0 to (7) is asymptotically stable. Finally, if there exist scalars α, β,  > 0, and p ≥ 1, such that V : D → R satisfies αkxkp ≤ V (x) ≤ βkxkp ,

x ∈ D,

(12)

∇V(x)T f (x) ≤ −εV (x) ,

x ∈ D,

(13)

then the zero solution x(t) ≡ 0 to (7) is exponentially stable. For a complete proof of the Lyapunov’s theorem, please refers to [33].

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

5.4

11/23

Stability analysis of PCH systems

Consider a general perturbed autonomous system given in (14) with conditions on the matrices explained in the section 5.2 [34, 35]. x˙ = [J (x, u) − R (x)] ∇H (x) + G (x) u + E

(14)

Suppose that there exists a control law u ∈ Rp such that the PCH system defined by (14) becomes into: x˙ = [JD (x, u) − RD (x)] ∇HD (x)

(15)

where JD (·, ·) and RD (·) are the interconnection and damping desired matrices and preserve the same properties aforementioned for J (·, ·) and R (·), respectively. Additionally, HD (·) represents the desired Hamiltonian function of the system, such that HD (x∗ ) corresponds to a local or global minimum point. Recall that, a new function V(x) = HD (x) − HD (x∗ ) fulfills the first two conditions of the Lyapunov’s theorem presented by (8) and (9) and it is only necessary to prove the condition defined by (10) to guarantee that (15) is Lyapunov stable. By taking the temporal derivative of V(x) is obtained the system (16). H˙ D (x) = ∇HD (x)T x˙ (16) H˙ D (x) = ∇HD (x)T (JD (x, u) − RD (x)) ∇HD (x) After rearrange some terms in (16) considering the properties assigned to the interconnection and damping desired matrices JD (·, ·) and RD (·), it is obtained (17). H˙ D (x) = −∇HD (x)T RD (x)∇HD (x)

(17)

If RD (x) is a positive semidefinite matrix, then the equilibrium point of (15) is Lyapunov stable, in addition, if RD (x) is a positive definite matrix, then the equilibrium point of (15) is asymptotically stable. As (15) fulfills the Lyapunov’s theorem conditions then, a general control law u that converts (14) into (15), makes the equilibrium point of the closed loop system stable in the sense of Lyapunov [21].

5.5

Interconnected systems

Electric power systems can be considered as multiple interconnected dynamical systems, which are analyzed typically by separated parts. Nonetheless, to guarantee stable operation it is strictly necessary that each part that conforms the whole dynamical system is stable under any operating condition. This section analyses the interconnection of two passive systems. This analysis can be applied recursively for large interconnected systems without loss of generality. Consider the dynamical interconnection defined in Fig. 4 [32, 67]. Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal u1

y2

e2

+

12/23

S1 –

y2

e2 S2

+

+

u2

Figure 4: Feedback connection where S1 and S2 correspond to an arbitrarily dynamical systems. Theorem 5.2 The feedback connection of two passive system is passive. Proof: Let H1 (x) and H2 (x) be the storage functions of S1 and S2 , respectively. If either component is a memoryless function, take Hi = 0. Then, eTi yi ≥ H˙ i . From the feedback connection in Fig. 4, it is possible to obtain (18): eT1 y1 + eT2 y2 = (u1 − y2 )T y1 + (u2 + y1 )T y2 = uT1 y1 + uT2 y2

(18)

After rearrange some terms, it is obtained the next expression: uT y = uT1 y1 + uT2 y2 ≥ H˙ 1 + H˙ 2

(19)

where u and y are defined as [u1 u2 ] and [y1 y2 ], respectively. If H(x) = H1 (x)+H2 (x), the total energy storage function for the feedback connection, ˙ which completes the proof. it is easy to verify that: uT y ≥ H, The most important consequence of this theorem is that it can be extended for large interconnected dynamical system guaranteeing that, if each part of the system is passive, then the feedback interconnection is also passive. Observe that the electric power system corresponds a large interconnected dynamical system, which implies that the passivity-based control theory is a natural strategy to analyze this system verifying stability conditions.

5.6

Dynamical system under analysis

As aforementioned in the section 3, this research project focuses on the design of controllers using passivity-based control theory to integrate distributed energy resources in microgrids. In this sense, it has been selected energy storage systems based on electrical technology such superconducting magnetic energy storage systems [15] and supercapacitor energy storage systems [5]. Also, it is analyzed the interconnection of distributed generator such Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

13/23

Renewable generation

Renewable generation

ENERGY STORAGE

DC

DC

DC AC

AC

AC

Converter

Main feeder AC DC AC Grid

Controlable loads

Noncontrolable loads

Figure 5: Interconnection of distributed energy resources and loads that conform a typical microgrid scheme as photovoltaic or wind generation [31]. Figure 5 depicts a schematic configuration of a typical microgrid. The most important device that allows to integrate all distributed energy resources and loads corresponds to the power electronic converter. In Fig. 5 the power electronic converter is a generic ac - dc converter [16, 19], which can be constructed using voltage or current technologies [12], as are the cases of pulse-width modulated voltage source converter PWM-VSC [5] or pulse-width modulated current source converter PWM-CSC [12, 15]. 5.6.1

Dynamical model of a PWM-VSC for DERs integration

The typical configuration employed to integrate different distributed energy resources consists of a VSC and a three-phase transformer in the ac side, as was depicted in Fig. 5 [31]; in the dc side it is used a capacitor that has the possibility to stored some energy in order to guarantee a constant voltage level, which allows to interconnect DERs as renewable energy resources, banks of batteries, supercapacitor and controllable loads, among other possible applications. In Fig. 6 it is presented the typical connection of a VSC for distributed energy resources applications [5].

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

AC

LSC

− vdc +

idc

14/23

Main grid LT , RT ek

vk isk ik Ck

DC PWM-CSC

Figure 7: Classical SMES connection using a PWM-CSC Main grid

iDG LT , RT Cdc

+ vdc −

DC

vk

ek

AC VSC

ik

Figure 6: Classical interconnection of a VSC for distributed generation applications The dynamical model of the system can be easily obtained applying Kirchhoff’s laws, energy balance based on the second Tellegen’s theorem and the Park’s invariant power transformation to become abc reference frame into dq reference frame. In (20) it is shown the resulting autonomous Hamiltonian model of this dynamical system, 

       LT x˙ 1 −RL −ωLT md x1 ed diag  LT   x˙ 2  =  ωLT −RL mq   x2  −  eq  CSC x˙ 3 −md −mq 0 x3 −iGD

(20)

where diag(·) denotes a diagonal matrix. Notice that (20) exhibits a Hamiltonian structure, where x1 = id , x2 = iq and x3 = iq . To obtain the general structure presented in (14) is only necessary to redefine the state variables and reorganize some terms in (20). The meaning of the state variables and parameters in the model (20) can be found in [31]. 5.6.2

Dynamical model of a PWM-CSC for SMES integration

A PWM-CSC uses a more elaborate connection [12]. In this case, in the ac side of the converter it is connected a capacitor banks to filter high order harmonics, at this point it is employed a three-phase transformer to interconnect the converter to the main ac grid. In the dc side, is connected a superconducting coil that stored energy in its magnetic field. The typical connection of this devices is depicted in Fig. 7. By employing the same circuit analysis used for the PWM-VSC, is easy to obtain the autonomous Hamiltonian model of this dynamical system as given in (21). Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

   diag   

LT LT C C LSC

     

x˙ 1 x˙ 2 x˙ 3 x˙ 4 x˙ 5





    =    

−RT −ωLT 1 0 0 ωLT −RT 0 1 0 −1 0 0 −ωC md 0 −1 ωC 0 mq 0 0 −md −mq 0

15/23

     

x1 x2 x3 x4 x5





    −    

ed eq 0 0 0

      (21)

These models can be rearranged as a Hamiltonian systems.

6

Objectives

6.1

General Objective

To design passivity-based controllers for distributed energy resources integration in threephase power microgrids.

6.2

Specific Objectives

1. To study the different mathematical models that represent the full dynamical behavior of power electronic converters, that allow to integrate diverse distributed energy resources to the microgrid. 2. To analyze the structural properties of the dynamical models obtained using different references frames. 3. To propose linear and nonlinear controllers using passivity-based control theory to integrate and operate the distributed energy resources in microgrids. 4. To develop a general dynamical model that allows to operate the microgrid using the three-phase representation and to take into account different operating scenarios. 5. To develop passivity-based controllers for non-autonomous systems via dynamics of the error theory with proportional integral actions.

6.3

Scope

This research project is delimited by the following conditions. 1. There are considered only the following distributed generation: X Photovoltaic generation. X Wind generation. 2. There are considered only the following energy storage devices: Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

16/23

X Superconducting coils. X Supercapacitors. X Batteries based on lead-acid or ion-lithium technologies. 3. The main characteristics of the electrical grid are: X Distribution voltage levels. X Three-phase ac connection. X The grid could be unbalanced in loads or impedance. X It is not considered fault analyses inside of the microgrid. X There is not dc feeders.

7

Methodology

In order to develop this project considering general and specific aims presented in the section 6, we divide this research project into seven main activities as follows: • Activity 1 actualize constantly and continuously the state-of-art related to nonlinear control strategies applied on DERs as well as their optimal integration considering power electronic converters, in order to review different mathematical models employed in the dynamical representation of distribution networks and microgrids. • Activity 2 select the most appropriate dynamical models to represent the dynamical behavior of different power electronic converters, such as voltage source converters and current source converters. In addition of studying passivity-based control theory by using Hamilton models and Lyapunov stability theory. • Activity 3 evaluate the dynamic performance of the dynamical models previously selected using Matlab toolboxes, such as: Ordinary Differential Equation package (ODExx), Simulink environment or PLECS software. • Activity 4 define the distribution system (or microgrid) configuration where will be integrated all distributed energy resources, evaluate the dynamic performance of the proposed controllers using passivity based control theory and compare it with classical control techniques, such as, proportional-integral controllers or linear matrix inequalities, among others. • Activity 5 make physical implementation using laboratory devices when possible or validate through simulation packages the proposed control methodology to integrate distributed energy resources to the electrical network.

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August 2, 2018

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Doctoral project proposal

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• Activity 6 study non-autonomous nonlinear systems analyzing their characteristics, employ the passivity-based control theory to design linear and nonlinear controllers under time domain and compare with the previously design made for autonomous case. • Activity 7 publication and redaction of journal and conference papers and thesis document.

8

Expected results

From a scientific perspective, the main result that is expected to be obtained is to achieve a deep knowledge of the different dynamic representations of microgrids and their components and, from this, to develop novel control schemes, passivity-based, that guarantee an adequate operation of the microgrid. The main result will be constituted by a collection of particular results that are detailed the above section. In a quantitative context, the expected results from this research work are listed • At least (2) international conference papers. • At least (2) international journal papers classified as Q1 or Q2 by Scimago Journal Rank.

9

Available resources

A detailed budget regarding funding sources is presented in Table 3. All values per item are given in Colombian pesos. Table 3: Detailed Budget Element Number Total [COP$] Source Desk at Lab 1B-148 1 2’500.000.00 ICE3 Research group Personal computer 1 2’250.000.00 Own Books — 1’000.000.00 UTP Paper work — 500.000.00 UTP Matlab software 1 License 1’000.000.00 UTP Monthly income 48 3’000.000.00 COLCIENCIAS 727 2015 Conference assistance 1 3’000.000.00 UTP Internship 1 6’000.000.00 COLCIENCIAS 727 2015 Research support 1 50’000.000.00 COLCIENCIAS 727 2015 Enrollment 8 5’000.000.00 COLCIENCIAS 727 2015 External resources 1 3’500.000.00 UNAM

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

10

18/23

Activity schedule Work \ Semester Activity 1 Activity 2 Activity 3 Activity 4 Activity 5 Activity 6 Activity 7

Oscar Danilo Montoya Giraldo

I X X

II X X

III X X X X

IV X

V

VI

VII

X X X

X X X

X

X

X

X X X X X

X X X X X

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Doctoral project proposal

19/23

References [1] P. Sreedharan, J. Farbes, E. Cutter, C. K. Woo, and J. Wang, “Microgrid and renewable generation integration: University of California, San Diego,” Applied Energy, vol. 169, pp. 709–720, 2016. [Online]. Available: http://dx.doi.org/10.1016/j.apenergy.2016.02.053 [2] Q. Fu, A. Hamidi, A. Nasiri, V. Bhavaraju, S. B. Krstic, and P. Theisen, “The role of energy storage in a microgrid concept: Examining the opportunities and promise of microgrids.” IEEE Electrification Magazine, vol. 1, no. 2, pp. 21–29, Dec 2013. [3] J. Quesada, R. Sebasti??n, M. Castro, and J. A. Sainz, “Control of inverters in a low-voltage microgrid with distributed battery energy storage. Part II: Secondary control,” Electric Power Systems Research, vol. 114, pp. 136–145, 2014. [Online]. Available: http: //dx.doi.org/10.1016/j.epsr.2014.03.033 [4] I. Serban and C. Marinescu, “Control strategy of three-phase battery energy storage systems for frequency support in microgrids and with uninterrupted supply of local loads,” IEEE Transactions on Power Electronics, vol. 29, no. 9, pp. 5010–5020, 2014. [5] A. Ortega and F. Milano, “Generalized model of vsc-based energy storage systems for transient stability analysis,” IEEE Trans. Power Syst., vol. 31, no. 5, pp. 3369–3380, Sept 2016. [6] S. Parhizi, H. Lotfi, A. Khodaei, and S. Bahramirad, “State of the art in research on microgrids: A review,” IEEE Access, vol. 3, pp. 890–925, 2015. [7] G. Kyriakarakos, D. D. Piromalis, A. I. Dounis, K. G. Arvanitis, and G. Papadakis, “Intelligent demand side energy management system for autonomous polygeneration microgrids,” Applied Energy, vol. 103, pp. 39–51, 2013. [Online]. Available: http://dx.doi.org/10.1016/j.apenergy.2012.10.011 [8] N. Kinhekar, N. P. Padhy, F. Li, and H. O. Gupta, “Utility Oriented Demand Side Management Using Smart AC and Micro DC Grid Cooperative,” IEEE Transactions on Power Systems, vol. 31, no. 2, pp. 1151–1160, 2016. [9] S. Huang, D. C. Pham, K. Huang, and S. Cheng, “Space vector pwm techniques for current and voltage source converters: A short review,” in 2012 15th International Conference on Electrical Machines and Systems (ICEMS), Oct 2012, pp. 1–6. [10] A. Marzouki, M. Hamouda, and F. Fnaiech, “A review of pwm voltage source converters based industrial applications,” in 2015 International Conference on Electrical Systems for Aircraft, Railway, Ship Propulsion and Road Vehicles (ESARS), March 2015, pp. 1–6. [11] E. Planas, A. Gil-De-Muro, J. Andreu, I. Kortabarria, and I. Mart??nez De Alegr??a, “General aspects, hierarchical controls and droop methods in microgrids: A review,” Renewable and Sustainable Energy Reviews, vol. 17, pp. 147–159, 2013. [12] E. Giraldo and A. Garces, “An adaptive control strategy for a wind energy conversion system based on PWM-CSC and PMSG,” IEEE Trans. Power Syst., vol. 29, no. 3, pp. 1446–1453, 2014. [13] D. E. Olivares, A. Mehrizi-Sani, A. H. Etemadi, C. A. Ca??izares, R. Iravani, M. Kazerani, A. H. Hajimiragha, O. Gomis-Bellmunt, M. Saeedifard, R. Palma-Behnke, G. A. Jim??nez-Est??vez, and N. D. Hatziargyriou, “Trends in microgrid control,” IEEE Transactions on Smart Grid, vol. 5, no. 4, pp. 1905–1919, 2014. [14] J. Shi, Y. Tang, K. Yang, L. Chen, L. Ren, J. Li, and S. Cheng, “SMES based dynamic voltage restorer for voltage fluctuations compensation,” IEEE Trans. Appl. Supercond., vol. 20, no. 3, pp. 1360–1364, 2010.

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

20/23

[15] J. Shi, Y. Tang, L. Ren, J. Li, and S. Cheng, “Discretization-based decoupled state-feedback control for current source power conditioning system of SMES,” IEEE Trans. Power Delivery, vol. 23, no. 4, pp. 2097–2104, 2008. [16] M. H. Ali, B. Wu, and R. A. Dougal, “An overview of SMES applications in power and energy systems,” IEEE Trans. Sustainable Energy, vol. 1, no. 1, pp. 38–47, 2010. [17] S. Wang and J. Jin, “Design and analysis of a fuzzy logic controlled smes system,” IEEE Transactions on Applied Superconductivity, vol. 24, no. 5, pp. 1–5, Oct 2014. [18] M. Mohammedi, O. Kraa, M. Becherif, A. Aboubou, M. Ayad, and M. Bahri, “Fuzzy logic and passivity-based controller applied to electric vehicle using fuel cell and supercapacitors hybrid source,” Energy Procedia, vol. 50, pp. 619 – 626, 2014, technologies and Materials for Renewable Energy, Environment and Sustainability (TMREES14 – EUMISD). [19] J. Shi, L. Zhang, K. Gong, Y. Liu, A. Zhou, X. Zhou, Y. Tang, L. Ren, and J. Li, “Improved discretization-based decoupled feedback control for a series-connected converter of scc,” IEEE Transactions on Applied Superconductivity, vol. 26, no. 7, pp. 1–6, Oct 2016. [20] T. Govindaraj and D. Hemalatha, “Dynamic Reactive Power Control of Islanded Microgrid Using IPFC,” IEEE Trans. on Power Systems, vol. 2, no. 4, pp. 3649–3657, 2014. [21] F. M. Serra and C. H. D. Angelo, “Ida-pbc controller design for grid connected front end converters under non-ideal grid conditions,” Electr. Power Syst. Res., vol. 142, pp. 12 – 19, 2017. [22] T.-S. Lee, “Lagrangian modeling and passivity-based control of three-phase ac/dc voltage-source converters,” IEEE Trans. Ind. Electron., vol. 51, no. 4, pp. 892–902, Aug 2004. [23] M. Perez, R. Ortega, and J. R. Espinoza, “Passivity-based pi control of switched power converters,” IEEE Trans. Control Syst. Technol., vol. 12, no. 6, pp. 881–890, Nov 2004. [24] I. Mart´ınez-P´erez, G. Espinosa-Perez, G. Sandoval-Rodr´ıguez, and A. D`oria-Cerezo, “IDA PassivityBased Control of single phase back-to-back converters,” IEEE International Symposium on Industrial Electronics, no. 2, pp. 74–79, 2008. [25] M. Hilairet, O. B´ethoux, M. Ghanes, V. Tanasa, J. P. Barbot, and M. D. Normand-Cyrot, “Experimental validation of a sampled-data passivity-based controller for coordination of converters in a fuel cell system,” IEEE Transactions on Industrial Electronics, vol. 62, no. 8, pp. 5187–5194, Aug 2015. [26] A. Rahim and E. Nowicki, “Supercapacitor energy storage system for fault ride-through of a {DFIG} wind generation system,” Energy Conversion and Management, vol. 59, pp. 96 – 102, 2012. [27] S. J. Ahn, J. W. Park, I. Y. Chung, S. I. Moon, S. H. Kang, and S. R. Nam, “Power-sharing method of multiple distributed generators considering control modes and configurations of a microgrid,” IEEE Transactions on Power Delivery, vol. 25, no. 3, pp. 2007–2016, July 2010. [28] N. L. D´ıaz, A. C. Luna, J. C. Vasquez, and J. M. Guerrero, “Centralized control architecture for coordination of distributed renewable generation and energy storage in islanded ac microgrids,” IEEE Transactions on Power Electronics, vol. 32, no. 7, pp. 5202–5213, July 2017. [29] W. Gu, G. Lou, W. Tan, and X. Yuan, “A nonlinear state estimator-based decentralized secondary voltage control scheme for autonomous microgrids,” IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1–1, 2017. [30] S. Avila-Becerril, G. Espinosa-P´erez, and P. Fernandez, “Dynamic characterization of typical electrical circuits via structural properties,” Mathematical Problems in Engineering, vol. 2016, pp. 1–13, 2016. [31] F. Serra, C. D. Angelo, and D. Forchetti, “Passivity based control of a three-phase front end converter,” IEEE Latin America Transactions, vol. 11, no. 1, pp. 293–299, Feb 2013.

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August 2, 2018

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Doctoral project proposal

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August 2, 2018

UTP

Doctoral project proposal

22/23

[46] T. I. Bo and T. A. Johansen, “Battery power smoothing control in a marine electric power plant using nonlinear model predictive control,” IEEE Transactions on Control Systems Technology, vol. 25, no. 4, pp. 1449–1456, July 2017. [47] P. Golchoubian and N. L. Azad, “Real-time nonlinear model predictive control of a batterysupercapacitor hybrid energy storage system in electric vehicles,” IEEE Transactions on Vehicular Technology, vol. PP, no. 99, pp. 1–1, 2017. [48] M. I. Ghiasi, M. A. Golkar, and A. Hajizadeh, “Lyapunov based-distributed fuzzy-sliding mode control for building integrated-dc microgrid with plug-in electric vehicle,” IEEE Access, vol. 5, pp. 7746–7752, 2017. [49] R. Cisneros, F. Mancilla-David, and R. Ortega, “Passivity-based control of a grid-connected smallscale windmill with limited control authority,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 1, no. 4, pp. 247–259, Dec 2013. [50] F. Mancilla-David and R. Ortega, “Adaptive passivity-based control for maximum power extraction of stand-alone windmill systems,” Control Engineering Practice, vol. 20, no. 2, pp. 173 – 181, 2012. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0967066111002139 [51] R. P. na, R. Fern´ andez, and R. Mantz, “Passivity control via power shaping of a wind turbine in a dispersed network,” International Journal of Hydrogen Energy, vol. 39, no. 16, pp. 8846 – 8851, 2014. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S036031991302956X [52] E. Rezaei, M. Ebrahimi, and A. Tabesh, “Control of dfig wind power generators in unbalanced microgrids based on instantaneous power theory,” IEEE Transactions on Smart Grid, vol. 8, no. 5, pp. 2278–2286, Sept 2017. [53] W. Dai, Y. Yu, M. Hua, and C. Cai, “Voltage regulation system of doubly salient electromagnetic generator based on indirect adaptive fuzzy control,” IEEE Access, vol. 5, pp. 14 187–14 194, 2017. [54] L. Guo, X. Zhang, S. Yang, Z. Xie, L. Wang, and R. Cao, “Simplified model predictive direct torque control method without weighting factors for permanent magnet synchronous generator-based wind power system,” IET Electric Power Applications, vol. 11, no. 5, pp. 793–804, 2017. [55] L. Y. Lu and C. C. Chu, “Consensus-based secondary frequency and voltage droop control of virtual synchronous generators for isolated ac micro-grids,” IEEE Journal on Emerging and Selected Topics in Circuits and Systems, vol. 5, no. 3, pp. 443–455, Sept 2015. [56] A. Nemmour, F. Mehazzem, A. Khezzar, M. Hacil, L. Louze, and R. Abdessemed, “Advanced backstepping controller for induction generator using multi-scalar machine model for wind power purposes,” Renewable Energy, vol. 35, no. 10, pp. 2375 – 2380, 2010. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0960148110000868 [57] C. Meza, D. Jeltsema, J. Scherpen, and D. Biel, “Passive P-Control of a Grid-Connected Photovoltaic Inverter,” IFAC Proceedings Volumes, vol. 41, no. 2, pp. 5575 – 5580, 2008, 17th IFAC World Congress. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1474667016398330 [58] W. Qi, J. Liu, and P. D. Christofides, “Supervisory predictive control of an integrated wind/solar energy generation and water desalination system,” IFAC Proceedings Volumes, vol. 43, no. 5, pp. 829 – 834, 2010, 9th IFAC Symposium on Dynamics and Control of Process Systems. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S1474667016301380 [59] S. Adhikari, F. Li, and H. Li, “P-Q and P-V Control of Photovoltaic Generators in Distribution Systems,” IEEE Transactions on Smart Grid, vol. 6, no. 6, pp. 2929–2941, Nov 2015. [60] J. Schiffer, R. Ortega, A. Astolfi, J. Raisch, and T. Sezi, “Conditions for stability of droop-controlled inverter-based microgrids,” Automatica, vol. 50, no. 10, pp. 2457 – 2469, 2014. [Online]. Available: http://www.sciencedirect.com/science/article/pii/S0005109814003100

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP

Doctoral project proposal

23/23

[61] J. Liu, W. Zhang, and G. Rizzoni, “Robust stability analysis of dc microgrids with constant power loads,” IEEE Transactions on Power Systems, vol. PP, no. 99, pp. 1–1, 2017. [62] L. Guo, S. Zhang, X. Li, Y. W. Li, C. Wang, and Y. Feng, “Stability analysis and damping enhancement based on frequency-dependent virtual impedance for dc microgrids,” IEEE Journal of Emerging and Selected Topics in Power Electronics, vol. 5, no. 1, pp. 338–350, March 2017. [63] L. Herrera, W. Zhang, and J. Wang, “Stability analysis and controller design of dc microgrids with constant power loads,” IEEE Transactions on Smart Grid, vol. 8, no. 2, pp. 881–888, March 2017. [64] J. Xiang, Y. Wang, Y. Li, and W. Wei, “Stability and steady-state analysis of distributed cooperative droop controlled dc microgrids,” IET Control Theory Applications, vol. 10, no. 18, pp. 2490–2496, 2016. [65] A. P. N. Tahim, D. J. Pagano, E. Lenz, and V. Stramosk, “Modeling and Stability Analysis of Islanded DC Microgrids Under Droop Control,” IEEE Transactions on Power Electronics, vol. 30, no. 8, pp. 4597–4607, Aug 2015. [66] F. Casta˜ nos and D. Gromov, “Passivity-based control of implicit port-Hamiltonian systems with holonomic constraints,” Systems and Control Letters, vol. 94, pp. 11–18, 2016. [67] H. Khalil, Nonlinear Systems, ser. Always learning. Pearson Education, Limited, 2013. [Online]. Available: https://books.google.com.co/books?id=VZ72nQEACAAJ [68] L. Perko, Differential Equations and Dynamical Systems, ser. Texts in Applied Mathematics. Springer New York, 2013. [Online]. Available: https://books.google.com.co/books?id= VFnSBwAAQBAJ [69] R. Ortega, A. van der Schaft, B. Maschke, and G. Escobar, “Interconnection and damping assignment passivity-based control of port-controlled hamiltonian systems,” Automatica, vol. 38, no. 4, pp. 585 – 596, 2002.

Oscar Danilo Montoya Giraldo

August 2, 2018

UTP