Dante Inga Narv´
aez
MODELING, DESIGN AND CONTROL OF A
BIDIRECTIONAL DC-AC CONVERTER FOR POWER
SUPPLYING OF AUTONOMOUS MICROGRIDS
MODELAGEM, CONTROLE E PROJETO DE UM
CONVERSOR CC-CA BIDIRECIONAL PARA A
ALIMENTA ¸
C ˜
AO DE MICRORREDES AUT ˆ
ONOMAS
Campinas
2016
MODELING, DESIGN AND CONTROL OF A
BIDIRECTIONAL DC-AC CONVERTER FOR POWER
SUPPLYING OF AUTONOMOUS MICROGRIDS
MODELAGEM, CONTROLE E PROJETO DE UM
CONVERSOR CC-CA BIDIRECIONAL PARA A
ALIMENTA ¸
C ˜
AO DE MICRORREDES AUT ˆ
ONOMAS
Thesis presented to the School of Electrical and Computer Engineering of the University of Campinas in partial fulfillment of the
requirements for the degree of Master, in the area of Electrical Energy.
Disserta¸c˜ao apresentada `a Faculdade de Engenharia El´etrica e de Computa¸c˜ao da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obten¸c˜ao do t´ıtulo de Mestre em Engenharia El´etrica, na ´Area de Energia El´etrica.
Supervisor: Prof. Dr. Marcelo Gradella Villalva Orientador: Prof. Dr. Marcelo Gradella Villalva
Este exemplar corresponde `a vers˜ao final da tese defendida pelo aluno Dante Inga Narv´aez, e orientada pelo Prof. Dr. Marcelo Gradella Villalva.
Campinas
2016
Inga Narváez, Dante,
In4m IngModeling, design and control of a bidirectional DC-AC converter for power supplying of autonomous microgrids / Dante Inga Narváez. – Campinas, SP : [s.n.], 2016.
IngOrientador: Marcelo Gradella Villalva.
IngDissertação (mestrado) – Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação.
Ing1. Redes elétricas inteligentes. 2. Conversores eletrônicos. 3. Controladores otimos. I. Villalva, Marcelo Gradella,1978-. II. Universidade Estadual de
Campinas. Faculdade de Engenharia Elétrica e de Computação. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Modelagem, controle e projeto de um conversor CC-CA
bidirecional para a alimentação de microrredes autônomas
Palavras-chave em inglês:
Smart grids
Electronic converters Many great drivers
Área de concentração: Energia Elétrica Titulação: Mestre em Engenharia Elétrica Banca examinadora:
Marcelo Gradella Villalva [Orientador] Flávio Alessandro Serrão Gonçalves Mateus Giesbrecht
Data de defesa: 05-05-2016
Programa de Pós-Graduação: Engenharia Elétrica
Candidato: Dante Inga Narv´aez RA: 153882 Data da Defesa: 5 de maio de 2016
T´ıtulo da Tese: “MODELAGEM, CONTROLE E PROJETO DE UM CONVERSOR CC-CA BIDIRECIONAL PARA A ALIMENTA ¸C ˜AO DE MICRORREDES AUT ˆ ONO-MAS”.
Prof. Dr. Marcelo Gradella Villalva (Presidente, FEEC/UNICAMP) Prof. Dr. Fl´avio Alessandro Serr˜ao Gon¸calves (UNESP)
Prof. Dr. Mateus Giesbrecht (FEEC/UNICAMP)
A ata de defesa, com as respectivas assinaturas dos membros da Comiss˜ao Julgadora, encontra-se no processo de vida acadˆemica do aluno.
This project was supported by CNPq (process 454352/2014-0), PAPDIC/PRP/UNICAMP and CAPES/PROEX.
verter and H-bridge converter. Performances of several types of controllers (Type 3, PI, PI+resonant) are compared in order to choose the proper option for each stage, and the uninterruptible power supply (UPS) functionality is evaluated through simulations and tests under several events: step increase and decrease in reference signal, overload and under-load conditions, battery charge and discharge processes, as well as grid fault and recovery. Besides that, non-linear loads are studied, so multiple PI+resonant and fuzzy controllers are used to reduce the voltage and current distortions in that case. Models for the stages are chosen from some available methods, and the parameters of the controllers are optimized by reducing the settling time, overshoot, absolute peak error or total har-monic distortion. Experimentation is made with a 100W converter prototype, which is modeled, controlled, simulated and verified through experimentation. Performance of the system is measured by power factor (PF) in grid-connected mode and total harmonic distortion (THD) in stand-alone mode. In addition, flexibility of the system proposed is proven by designing and simulating a 1-kW converter with the same methodology.
Keywords: Autonomous microgrid, bidirectional DC-AC converter, controllers optimization.
Este trabalho apresenta a modelagem, o projeto dos controladores e testes de um conversor bidirecional CC-CA para aplicações de alimentação de microrredes autôno-mas. O sistema proposto é constituído principalmente por um conversor bidirecional CC-CA com dois estágios: um conversor CC-CC e um conversor em ponte H. Desempen-hos de vários tipos de controladores (tipo 3, PI, PI + ressonante) são comparados a fim de escolher a opção adequada para cada estágio, e a funcionalidade de fonte de alimen-tação ininterrupta é avaliada através de simulações e testes sob vários eventos: aumento e diminuição do sinal de referência, condições de sobrecarga e sub-carga, processos de carga e descarga da bateria, bem como falta de energia e reconexão da rede. Também, cargas não-lineares são estudadas, de modo que controladores PI + ressonante múltiplos e fuzzy são utilizados para reduzir as distorções na tensão e na corrente nesse caso. Modelos para os estágios do sistema são escolhidos a partir de alguns métodos disponíveis, e os parâmetros dos controladores são otimizados através da redução do tempo de estabiliza-ção, overshoot, erro de pico absoluto ou distorção harmônica total. A experimentação é feita com um conversor de 100 W, que é modelado, controlado, simulado e verificado através de experimentação. O desempenho do sistema é medido pelo fator de potência no modo conectado à rede e pela distorção harmônica total no modo isolado. Além disso, a flexibilidade do sistema proposto é comprovada através do projeto e simulação de um conversor de 1 kW, utilizando a mesma metodologia.
Palabras chave: Redes elétricas inteligentes, conversores eletrônicos, contro-ladores ótimos.
1.2. Modeling methods . . . 15
1.3. Control techniques . . . 21
2. Description and design of the proposed system 29 2.1. Description of the system . . . 29
2.2. Design of the system . . . 32
3. Modeling of the system elements 37 3.1. Battery modeling . . . 37
3.2. Load modeling . . . 40
3.3. Grid modeling . . . 41
3.4. DC-DC converter in boost mode . . . 41
3.5. H-bridge converter in inverter mode . . . 44
3.6. H-bridge converter in rectifier mode . . . 46
3.7. DC-DC converter in buck mode . . . 48
4. Controllers design and verification 51 4.1. Controller design for boost converter . . . 52
4.2. Controller design for H-bridge inverter . . . 56
4.3. Controller design for H-bridge rectifier . . . 59
4.4. Controller design for buck converter . . . 62
4.5. Grid fault detector . . . 65
4.6. The supervisory controller . . . 65
5. Simulation and experimental results 67 5.1. Hardware setup . . . 67
5.2. Powering on the system . . . 70
5.3. Grid-connected and stand-alone operation . . . 73
5.4. Non-linear loads . . . 76
6. Conclusions 83
6.1. General conclusions . . . 83 6.2. Future work . . . 84
A. Sizing of the devices of power converter 85
A.1. Overview . . . 85 A.2. Boost converter sizing . . . 85
B. Modeling and operation of power converter 89
B.1. Overview . . . 89 B.2. Boost converter modeling . . . 90
C. Control and stability of power converter 95
C.1. Overview . . . 95 C.2. Boost converter control . . . 95
D. Developed code and apps 99
D.1. Code-generation sources . . . 99 D.2. Applications to design the controllers . . . 104 D.3. Simulation schematics . . . 104
E. Scientific papers 107
E.1. Accepted papers . . . 107 E.2. Submitted papers . . . 108
1. Introduction
Classical grid systems are formed by the main generator, transmission lines, distribution lines and loads. However, distributed generation (DG) and renewable energy sources such as photovoltaic and wind-energy generators are being included in the recent years, which makes the final user capable of producing and exporting energy. Examples of structures for a classical grid system and a modern grid system are shown in Fig. 1.1. Control and stability analysis of the grid system under this new scenario becomes complicated. With the aim of solving the problem, analysis and control design of the grid system in a microgrid basis is being investigated [1, 2]. When using microgrids, several advantages have been pointed out (in addition to the control simplification): reliability under power outages, arbitrage of energy sources, power quality under voltage sags, possibility of smart grid implementation, inclusion of renewable sources and reduced transmission losses.
(a) Classical grid system and its components. (b) Modern grid system and its components.
Figure 1.1.: Examples of structures for classical and modern grid systems.
A microgrid consists of the point of common coupling (PCC) with the main grid, a static switch that can connect/isolate the microgrid to/from the grid, an energy storage system, local generators such as photovoltaic and wind-energy devices with power converters, and distributed loads that can be classified as critical or secondary loads [3, 4]. The main grid is susceptible to eventual failure, and the local generators based on renewable energy produce variable power during the day, so some energy storage system
based on batteries or fuel cells is needed in order to keep the critical loads working in a uninterruptible power supply (UPS) fashion.
Microgrids may operate in two modes: grid-connected (grid-tie) and stand-alone (islanded). In grid-connected mode, the main grid is available and healthy, so the energy generated in excess by the microgrid can be exported, or some energy can be imported to charge the storage system or when there is a lack of energy generation. On the other hand, under grid fault events, the microgrid has to be islanded (e.g., disconnected from the main grid), so the energy storage system supplies quality power to the critical loads. Besides that, transitions between the two operation modes have to be as smooth as possible [5, 6]. Parallel connection of energy storage systems, coordinated operation of distributed generators and other smart grid concepts are possible through the use of communication among power converters: line-based, such as the droop method, or external-based, for example, by using a communication area network (CAN) bus.
Loads are traditionally treated as resistances or current sources, which makes simple the analysis of the grid system. However, loads can also be inductive (industrial motors), capacitive (voltage compensators) or non-linear (electronic loads such as rec-tifiers) [3, 7]. Inductive and capacitive loads drain high currents when connected, and produce mismatching between voltage and current phases in steady state. Non-linear loads drain non-sinusoidal current (analysed as a summation of sinusoidal components at the grid frequency and its multiples, called harmonics), and are capable of producing distortion in the voltage waveform of the power supply. Inductive and non-linear loads drain reactive power from the grid or local generator, so they are usually compensated by using active filters or capacitor banks.
Renewable energy sources are a matter of interest in the recent decades due to economic and environmental issues. Most utilized sources are wind energy (wind turbine plus electric generator and power converter), sunlight (photovoltaic panels plus power converter) and marine tidal current energy (similar to wind generator). Those sources are renewable and free as part of the nature, and do not generate pollution during operation. In contrast, installation of renewable energy plants is still expensive and the power pro-duced is highly variable [8, 9]. Therefore, power converters and energy storage systems are required to extract the maximum power from the sources and to supply stable power to the loads, respectively.
UPS systems were created and developed to guarantee power availability for critical loads [2, 6, 10, 11]. UPSs can be classified as offline (connects the grid to the loads in normal operation, acts just in case of grid fault event), online (connects the grid to the loads through rectifier-inverter converters, bypasses the converters in case of UPS fault), line-interactive (connects directly to the load and to the grid with an inductor, can manage power flow from/to the grid) and delta-conversion (variation of online type, rectifies just about 10% of inverted power by using a series transformer). Online UPS allows better
(such as IGBT or SCR), rectifier (or charger), storage devices (such as battery or fuel cell), inverter (PWM converter), filters, and the static bypass switch (manages connection with the grid).
Energy storage systems are a necessity for autonomous operation in microgrids and power compensation for variable-power generators or loads [9, 10]. They can support renewable energy generators to supply continuously islanded loads, or help the main grid to supply highly-variable power to intermittent loads such as electric drives and industrial robots. According to rated power and energy density, energy storage devices can be large-scale capacity (for example, compressed air devices) or small-scale capacity (such as ultra-capacitors, batteries and hydrogen fuel cells). Batteries are the most utilized in renewable energy generators and microgrid applications.
Power converters are electronic circuits capable of transforming one voltage/cur-rent level to another level with almost no losses, by using pulse width modulation (PWM) and control techniques. In this way, power converters are used to transfer power from renewable energy devices (photovoltaic panels, wind energy devices) or energy storage de-vices (batteries, flywheels) to the main grid or to the loads [4, 9, 11, 12]. Power converters are classified according to the nature of voltage/current they transform: AC-DC or rec-tifiers, DC-DC or choppers, DC-AC or inverters, and AC-AC or cyclo-converters. Many topologies of simple and complex power converters have been created, those converters have been modelled and controlled through a variety of methods, and vast applications have been found for them. Control techniques are implemented in software by using Digi-tal Signal Processors (DSP) or other similar, and communication interfaces can be added to allow power converters to operate in a microgrid.
The content of this thesis is divided as follows:
• Chapter 1 gives an overview of microgrids and its related concepts, such as renewable energy sources, UPS systems, energy storage systems and power converters. The objectives and justification of this work are presented in this chapter. Also, a com-pilation of some of the existing modeling methods for power converters is included, and the duty-cycle to ouput-voltage transfer function of the buck converter is cal-culated as an example by using those methods. Besides that, most common control techniques are explained, and applied to voltage control of the buck converter as an example.
• Chapter 2 contains the description of the bidirectional DC-AC converter utilized, as well as its proposed control strategy and a design methodology based on the
current literature. In addition, more detailed explanation of the design process for the DC-DC stage in boost mode is found in Appendix A.
• Chapter 3 describes the experimental procedure to get the main parameters of a dynamic model for the battery bank. It also presents a generic methodology to obtain the mathematical models of the four converters involved in this work: DC-DC stage in boost mode, DC-DC-DC-DC stage in buck mode, DC-DC-AC stage in inverter mode, and DC-AC stage in rectifier mode (e.g., AC-DC converter). Besides that, this chapter contains simple models utilized for linear and non-linear loads, as well as the grid system under fault and recovery events. Furthermore, detailed explanation of the calculations to obtain the mathematical model of the CC-CC stage in boost mode is found in Appendix B.
• Chapter 4 presents a generic and simplified procedure to obtain the parameters of the controllers for the converters of the proposed system. These parameters are compared to the ones calculated by using the complete procedure, and simulations are performed in order to verify a good proximity between them. Complete calcula-tions are utilized to implement and test the digital controllers for each stage. Also, grid fault detector and state diagram of the supervisory controller are explained. More detailed analysis and numeric examples of those complete calculations for the boost converter are explained in Appendix C.
• Chapter 5 shows the simulation and experimental results of the complete system during powering up, shutdown, as well as normal operation in grid-connected and stand-alone modes. Some events are also simulated and tested: step in battery-bank charging current, sudden decrease in load, grid fault and grid recovery. To evaluate the performance of the system, power factor is measured in grid-connected mode, and total harmonic distortion is measured in stand-alone mode. In addition, non-linear loads and extension to higher power are studied and simulated.
• Chapter 6 contains the conclusions obtained during the realization of this work, and some interesting topics to be covered in future work.
• Appendix D includes the C-code sources of the multimode controllers utilized, screen printings of the applications developed to calculate the parameters of the controllers, and schematic diagrams of the complete system which was simulated in PSIM soft-ware. Finally, Appendix E mentions the scientific papers produced until the mo-ment, as a result of the development of this work.
generators are not fast enough to supply reliable and quality power, mainly for electronic loads.
Renewable energy systems are being spread over the world due to the increases in fuel price and environmental pollution. Main renewable sources used nowadays are wind and solar. Such systems produce intermittent power, and then need power converters and storage elements to manage the energy collected.
The main objective of this work is to design and implement a bidirectional DC-AC converter for power supplying of autonomous microgrids. The system proposed is able to keep the battery bank charged when the grid is connected, and supply stabilized power to the critical loads when a grid fault occurs. A flexible design methodology for a single-phase bidirectional DC-AC converter is explained, and UPS functionality is verified through simulations and experimental tests. Therefore, the converters are modeled and their controllers are designed, then tested individually and as a system under several events: overload, change in reference of battery charge current, grid fault and grid recovery.
1.2. Modeling methods
Several modeling methods have been developed through the years for power electronic circuits, such as: average model, state-space, circuit model, PWM switch model, approx-imate model and discrete-average.
Average theory [13–16] is the base of all the other methods. It has a rigorous mathematical foundation to prove that some time-varying differential equations, as the ones found in the description of switching converters, can be represented by time-invariant equations that behave in a similar way to the average of the original equations. In practice, average theory says that state variables such as inductor currents and capacitor voltages can be represented by the weighted sum of their mathematical expressions for all possible circuit configurations, provided that frequency of modulating signal to drive the PWM switches is less than the switching frequency. Therefore, this technique can be used to get directly the model of buck-type converters, such as the DC-DC buck converter and the H-bridge inverter. By using this method, transfer function of duty-cycle to output-voltage in the DC-DC buck converter is obtained as an example.
Simple DC-DC buck converter circuit is shown in Fig. 1.2. Vi is a DC voltage
to be less than Vi). Transistor T is driven by PWM modulation with duty cycle db,
while diode D is passively driven by the complementary signal d0b, as it is explained in this section. Periodic signal db(t) with frequency fs = 1/Ts is related to status of the
transistor-based switch (where ¯db is the averaged value of db over the switching period Ts,
so 0 < ¯db < 1): db(t) = 1, t < 0, ¯dbTs > (T : closed) 0, t < ¯dbTs, Ts > (T : open) (1.1)
Figure 1.2.: DC-DC buck converter circuit.
Given values for the parameters of the buck converter, it can operate in con-tinuous conduction mode (CCM), when inductor current iL is always positive, or
discon-tinuous conduction mode (DCM), when iLgoes periodically to zero because of the current
ripple and the fact that D is an unidirectional switch. DCM may occur, for example, due to a light load (high value of Ro). As the CCM is utilized to avoid dependance of the
converter gain on the output load Ro [15], a requirement for ¯db has to be satisfied for the
buck converter: ¯db > 1 − R2LoTs. In this work, operation in CCM is considered for all the
power converters.
By using Kirchhoff’s laws in addition to inductor and capacitor physical laws, averaged expressions for the inductor current ¯iLand capacitor voltage ¯vC can be obtained.
Those expressions are written in Eq. (1.2), where output voltage ¯vo is equal to ¯vC. To
solve the two differential equations for ¯iL and ¯vC, it is necessary to obtain an expression
for the transistor voltage ¯vT.
¯ vL(t) = Ldtdi¯L(t) = Vi− ¯vT(t) − ¯vC(t) ¯ iC(t) = Cdtdv¯C(t) = ¯iL(t) − R1 ov¯C(t) (1.2)
is reversely biased. On the other hand, when db is set to zero, see Fig. 1.3b, T is open
and, due to the stored current in inductor L, D conducts. Thus, transistor voltage vT is
calculated for each state, every switching period Ts , and those equations are weighted to
get the averaged value ¯vT:
vT(t) = 0, t < 0, ¯dbTs> vT(t) = Vi, t < ¯dbTs, Ts > ⇒ ¯vT(t) ∼= ¯db× 0 + (1 − ¯db) × Vi = (1 − ¯db)Vi (1.3)
(a) Equivalent circuit when T conducts. (b) Equivalent circuit when D conducts.
Figure 1.3.: Two possible configurations for buck converter operating in CCM.
Equation (1.2) can be solved by replacing ¯vT from Eq. (1.3). After that, the
small perturbation method (e.g., considering ¯db = Db+ ˜db, where Dbis the operating point
and ˜db is a small variation in ¯db) and Laplace transform are applied to get duty-cycle to
output-voltage transfer function (vo = vC):
˜ vL(s) = Ls ˜iL(s) = Vi− (1 − ˜db(s))Vi− ˜vC(s) ˜ iC(s) = Cs ˜vC(s) = ˜iL(s) − R1 ov˜C(s) ⇒ v˜˜o(s) db(s) = Vi LC s2+ 1 RoCs + 1 LC (1.4)
State-space technique [17–19] is utilized to calculate systematically duty-cycle to output transfer function of any DC-DC switching converter, based on mathematical expressions extracted from all the possible circuit configurations. Those expressions are averaged in the state-space representation based on the first-order linear term approxi-mation of the fundamental matrix eAt
≈ I + At. As a result, duty-cycle to output transfer functions are calculated integrally by algebraic operations. This method is used to obtain the transfer functions of any DC converter such as the DC buck and the DC-DC boost converters. Duty-cycle to output-voltage transfer function of buck converter in
CCM is obtained as an example.
From Fig. 1.3, arranging the equations for each circuit configuration in the state-space representation ( ¯d0b = 1 − ¯db): t < 0, ¯dbTs >: d dtiL d dtvC = A1 iL vC + B1Vi ⇒ A1 = 0 −1 L 1 C − 1 RoC , B1 = 1 L 0 (1.5) t < ¯dbTs, Ts >: d dtiL d dtvC = A2 iL vC + B2Vi ⇒ A2 = 0 −1 L 1 C − 1 RoC , B2 = 0 0 (1.6) Second step implies calculating the averaged matrices A and B, by multiplying the matrices of the first configuration by Db and the second ones by D0b (response in steady
state): A = DbA1+ (1 − Db)A2 B = DbB1+ (1 − Db)B2 ⇒ A = 0 −1 L 1 C − 1 RoC , B = Db L 0 (1.7)
Thus, state-space form of the equations for buck converter is defined and can be solved for steady state, in which C = 0 1 and D = 0 (Vo= VC):
d dtX = AX + BVi Vo = CX + DVi ⇒ X = IL VC = −A −1 BVi = DbVi Ro DbVi (1.8)
Finally, the duty-cycle to output transfer function is determined by a formula, provided a small perturbation in duty cycle ( ¯db = Db+ ˜db) and the state variables (for
example, ¯vC = VC+ ˜vC): ˜ x(s) ˜ db(s) = (sI − A)−1[(A1− A2)X + (B1− B2)Vi] ⇒ ˜ vo(s) ˜ db(s) = Vi LC s2+ 1 RoCs + 1 LC (1.9)
Circuit model [13, 15, 17, 18] is a linear circuit representation of power con-verters, and is dual to the state-space technique. In addition, it can be extended in order to obtain the circuit representation of three-phase DC-AC, AC-DC and AC-AC
From Eq. (1.1), Eq. (1.2) and Eq. (1.3), two equations that represent the buck converter can be written, provided that input source Vi is constant:
¯ db(t)Vi = ¯vL(t) + ¯vC(t) ¯ iL(t) = ¯ic(t) + ¯io(t) (1.10)
Equation (1.10) has an equivalent circuit, as it is shown in Fig. 1.4a. In order to get the transfer functions, small perturbations are added to the average values of the variables in the circuit:
¯ db(t) = Db+ ˜db(t) ¯ iL(t) = IL+ ˜iL(t) ¯ vC(t) = VC+ ˜vC(t) (1.11)
(a) Averaged form of circuit model. (b) Static form of circuit model.
Figure 1.4.: Circuit model in averaged and static form for buck converter.
Therefore, DC and AC circuits are derived by separating the adding variables from the circuit model. DC circuit is represented in Fig. 1.4b, while AC circuit has the same structure as Fig. 1.4a. Thus, transfer function of duty-cycle to output-voltage is obtained by simple algebraic calculations from the AC circuit model:
˜ vo(s) = (Ro//Cs1 ) (Ro//Cs1 ) + Ls ˜ db(s)Vi ⇒ ˜ vo(s) ˜ db(s) = Vi LC s2+ 1 RoCs + 1 LC (1.12)
PWM switch model [21] provides an equivalent circuit for the transistor-diode set that is present in all DC-DC converters, by doing an analysis of its invariant properties.
As a result, DC-DC converters are studied in the same way as linear amplifiers, where transistors are replaced by their circuit models for DC and AC analysis. DC and AC equivalent circuits for the simple buck converter are shown in Fig. 1.5, which is also used to get the duty-cycle to output transfer function.
(a) DC model of PWM switch. (b) AC model of PWM switch.
Figure 1.5.: PWM switch model equivalent buck circuits for DC and AC analysis.
From Fig. 1.5a, inductor behaves as short-circuit (its average voltage is zero) and capacitor is open in steady state. Thus, DC voltage between active and passive terminals (Vap) is easily obtained, as well as the static transfer function of the converter:
Vap = Vi ⇒
Vo
Vi
= Db (1.13)
Hence, from Fig. 1.5b, duty-cycle to output-voltage transfer function is ob-tained by considering short-circuit in the DC input source (Vi):
˜ vo(s) = (Ro//Cs1 ) (Ro//Cs1 ) + Ls × Vap Db ˜ db(s)Db ⇒ ˜ vo(s) ˜ db(s) = Vi LC s2+ 1 RoCs + 1 LC (1.14)
Approximate model [22] is intended to solve modeling difficulties of the AC-DC converters. However, it can be used to simplify the transfer functions of all power converters, by neglecting disturbance signals from the block diagram of the converter (which is constructed by using the average model) or by taking into account magnitude simplifications. From Eq. (1.2) and Eq. (1.3), by applying small perturbations and Laplace transform, block diagram of buck converter can be constructed, as it is shown in Fig. 1.6.
In the first kind of approximation, capacitor voltage (vC) is considered a
Figure 1.6.: Block diagram of the buck converter. is simplified: w = vC(s) ⇒ ˜ iL(s) ˜ db(s) ≈ Vi L 1 s (1.15)
In the second kind of approximation, for voltage controller design, crossover frequency (fc) selected is usually small. Therefore, duty-cycle to output-voltage transfer function is simplified: |s| = wc 1 RoC ⇒ Hp(s) = ˜ vo(s) ˜ db(s) ≈ Vi LC 1 1 RoCs + 1 LC (1.16)
Discrete-average technique [23–25] consists in representing the non-linear DC-DC power converters by a linear, shift-invariant matrix difference equation. The advantage of this method compared to the average or state-space is its accuracy at high frequencies, e.g., more than a half the switching frequency of the converter. However, a disadvantage is the resultant cumbersome equations which do not tell much about the physic nature of the converters, so they are computationally expensive to process. For that reason, discrete-average technique is not considered in this work.
1.3. Control techniques
Several control techniques were developed (applied) for (to) power converters, such as: type 3, PI/PID, PI+r, digital PID, fuzzy and artificial neural networks.
Type 3 amplifier [26, 27] is a simple but powerful compensator, intended for opamp-based control of DC-DC converters. However, generic formulas can be derived to design digitized controllers as well. An example is shown for the buck converter, that is modeled by the approximate method in order to obtain analytic expressions for the parameters of the controller.
Voltage control loop for the buck converter is shown in Fig. 1.7. Hp(s) is the
to be designed. PWM modulator (Hpwm) and voltage sensor (Hs) transfer functions are
considered constants. The output voltage (vo) is measured, and that measured value
(vs = Hsvo) is usually digitized to be subtracted from the reference signal (vr) in order
to obtain the error signal (ve). The error is amplified by the controller, resulting the
control signal (vd) that is scaled by the PWM modulator to obtain the duty cycle (db),
which is applied to the IGBT transistor of the converter. For the sake of simplicity, small-signal variables (for example, ˜db) are represented by normal notation (e.g., db) in all the
controller analyses and calculations.
Figure 1.7.: Block diagram for voltage control design of buck converter.
Given the resistor and capacitor formulas for the analog opamp-based con-troller and its transfer function calculated for being an inverting amplifier, transfer func-tion of type-3 controller is obtained for a selected crossover frequency (fc = wc/2π):
Hc(s) = wc G0k ( √ k wcs + 1) 2 s(√1 kwcs + 1) 2 (1.17)
Open-loop transfer function without controller (H0
l) is considered for type-3
controller design: Hl0(s) = HpwmHp(s)Hs ≈ HpwmHsViRo L 1 s (1.18)
G’ is the gain of Hl0 evaluated at the selected crossover frequency:
G0 = Gain[Hl0(jwc)] ≈
HpwmHsViRo
Lwc
(1.19) P’ is the phase of the open-loop transfer function without controller, also evaluated at fc:
P0 = P hase[HpwmHp(jwc)Hs] ≈ −
π
k is the factor determined by the angle α needed to increase the phase margin of the plant: k = tan2 α + π 4 ≈ tan2P M + π 4 (1.22) Proportional Integral Derivative (PID) controller [28–33] is a widespread tech-nique used in many fields of research and industry. According to the frequency response approach, gain and phase margins requirements are related to system stability and tran-sient response. Thus, Bode diagrams of open-loop transfer functions are used to design the PID controllers (P, PI, PD or PID). PI controllers are usual for power converters due to their zero steady-state error. As an example, desired crossover frequency (fc) and phase margin (PM) parameters are selected to design a voltage controller for the buck converter.
Fig. 1.8 depicts the block diagram of PI controller. As it can be seen, it is implemented as a summation of proportional and integral components.
Figure 1.8.: Block diagram of analog PI controller.
The approximate model from Eq. (3.8) is used to simplify the analysis. Given the PI controller: Hc(s) = kp+ ki s = kp s + ki/kp s (1.23)
As the voltage control loop is depicted in Fig. 1.7, the open-loop transfer func-tion (Hl) is: Hl(s) = Hc(s)HpwmHp(s)Hs ≈ HpwmHsViRokp L s + ki/kp s2 (1.24)
Phase requirement at f c is needed for stability. Selecting desired crossover frequency and phase margin:
P hase[Hl(jwc)] = −π + P M ⇒
ki
kp
≈ wc
tan(P M ) (1.25)
Unity gain requirement is also needed for stability of the control loop:
Gain[Hl(jwc)] = 1 ⇒ kp =
Lwc
HpwmHsViRo
sin(P M ) (1.26)
Resonant PI (PI+r) controller [34, 35] is intended to achieve zero steady-state error for AC reference signals. Therefore, it is used in current control for single and three-phase inverters and rectifiers, as well as voltage control of inverters. PI+r controller is gotten from simple PI controller through synchronous frame transformation, and has the same effect as applying abc-dq reference frame transformation to AC signals in order to use PI controllers for three-phase inverters and rectifiers.
PI+r controller formula is derived by applying the synchronous frame trans-formation to the simple PI controller. That transtrans-formation is defined as:
Hac(s) =
Hdc(s + jwg) + Hdc(s − jwg)
2 (1.27)
where wg = 2πfg is the angular frequency of grid. PI controller transfer
function from Eq. (1.23) is used as the DC controller. Thus, replacing in Eq. (1.27):
Hac(s) = kp+ kis s2+ w2 g = kp s2+ ki kps + w 2 g s2+ w2 g (1.28)
As the implementation of the ideal PI+r controller is not physically realizable, a practical approximation of the PI controller is defined (wr wg):
Hdc(s) ≈ kp+
ki
s + wr
(1.29)
Replacing in Eq. (1.27), and considering ki
kp 2wr: Hac(s) ≈ kp s2+ ki kps + w 2 g s2+ 2w rs + w2g (1.30)
z domain. Digital form of PI controller for voltage regulation of the buck converter is calculated here.
Tustin transform is the most common and is given by:
s = 2fd 1 − z−1 1 + z−1 = 2 Td z − 1 z + 1 (1.31)
where fd = T1d is the sampling frequency. Given the PI controller from Eq.
(1.23), using the Tustin transform:
Hc(z) = vd(z) ve(z) = kp k i kp Td 2 + 1 z +ki kp Td 2 − 1 z − 1 (1.32)
Multiplying Eq. (1.32) by (z − 1) × ve(z), then dividing by z:
vd(z) − z−1vd(z) = kp ki kp Td 2 + 1 ! ve(z) + kp ki kp Td 2 − 1 ! z−1ve(z) (1.33)
Applying the inverse Z transform to obtain the finite difference equation:
vd[n] − vd[n − 1] = ki Td 2 + kp ve[n] + ki Td 2 − kp ve[n − 1] (1.34)
The implementation of Eq. (1.34) in C code is straightforward: #d e f i n e kp 1 #d e f i n e k i 1000 #d e f i n e Td 0 . 0 0 0 5 #d e f i n e v r 24 #d e f i n e Hs 0 . 0 8 3 3 #d e f i n e n 1 s t a t i c d o u b l e ve [ n+1] = { 0 , 0 } ; s t a t i c d o u b l e vd [ n+1] = { 0 , 0 } ; ve [ n ] = Hs ∗ ( v r − x1 ) ; // x1 : d i g i t i z e d vo
vd [ n ] = vd [ n−1] + ( k i ∗Td/2+kp ) ∗ ve [ n ] + ( k i ∗Td/2−kp ) ∗ ve [ n − 1 ] ; y1 = vd [ n ] ; // y1 : d i g i t i z e d vd
ve [ n−1] = ve [ n ] ; vd [ n−1] = vd [ n ] ;
Fuzzy controller [36–39] has been widely used for industrial processes and research in energy systems. It is essentially governed by non-linear fuzzy rules applied to the output error (ve) and change in output error (vce) signals. However, those fuzzy
rules can be simplified under certain conditions in order to obtain linear controllers, such as the PID. Therefore, fuzzy-PI controllers are designed from a previous PI design, by making the control surface non-linear. Fuzzy-based voltage controller for buck converter is explained as an example.
Fig. 1.9a depicts the block diagram of a fuzzy incremental controller, which is the fuzzy form of the linear PI controller shown in Fig. 1.8. Fuzzy rules block is placed after the fuzzy proportional gain (GE) block, and it has signals ve and vce as inputs. In
place of allocating after the input signals, discrete integration is added to the output of the fuzzy block, in order to avoid fast increase in the controller output.
(a) Block diagram of fuzzy controller.
(b) Membership functions of inputs.
Figure 1.9.: Block diagram and membership functions for fuzzy-PI controller.
Error signal (ve) is one of the two inputs for the fuzzy block. For the another
input, change in output error (vce) is a better candidate than the integrated error, and is
defined by:
vce[n] =
ve[n] − ve[n − 1]
Td
(1.35) where Tdis the sampling period of the discrete implementation. From Fig. 1.9a,
Fig. 1.9b depicts the membership functions of negative (n(v)), zero (z(v)) and positive (p(v)) characteristics for ve and vce values. According to fuzzy theory, values
less than a preset negative voltage (−VM) can be considered trully negative, while values
between −VM and 0 can be considered an intermediate between negative and zero. In
similar way, trully positive values are greater than another preset voltage (VM), and zero
value characteristic is also distributed between −VM and VM (where 0 voltage has the zero
characteristic with fully precision). Hence, membership function of z(ve) is expressed as
(analog for the other cases, and for vce):
z(ve) = 1 VMve+ 1 , −VM 6 ve < 0 − 1 VMve+ 1 , 0 6 ve < VM 0 , otherwise (1.37)
Applying the summation as a linear approximation for the fuzzy rules block:
vd[n] ≈ GCE × GCU ( ve[n] + GCE GE n X i=1 ve[i]Td ) (1.38)
Comparing with classical PI controller, the values of proportional and integral constants are obtained:
kp = GCE × GCU ki = GE × GCU (1.39)
Therefore, by selecting a proper value for GE, the other parameters are com-pletely defined with the PI constants previously designed. In this case, the control surface is a plane (e.g., it shows a linear dependence of vdon veand vce). The last step in designing
the fuzzy controller consists on modifying the control surface with non-linear membership functions and fuzzy rules, so the system performance should be at least as good as the system controlled with linear PI.
Artificial Neural Networks [40–42] are a modern technique utilized for model prediction and adaptive control. Back propagation algorithm is the most successful to calculate the network parameters, so as to train the neural network. In this way, the
controlled system is capable of managing variations in parameters of power converters. Control of power converters by using Artificial Neural Networks is an interesting topic to be investigated, but it is not covered in this work.
For the DC/DC converters (buck and boost) of the system proposed, digital type 3 and PI controllers are utilized due to their simplicity and good performance. For the DC/AC converters (inverter and rectifier), digital PI+r and fuzzy controllers are used and their responses are compared.
2. Description and design of the
proposed system
To meet reliability and quality power requirements of the autonomous microgrid, the proposed system operates in two modes: grid-connected (grid-tie or mode 1) and stand-alone (islanded or mode 2). In grid-connected mode (when the main grid is available) the system charges the battery bank or keeps it charged. In stand-alone mode (when there is a grid fault) the system uses the energy stored in the battery bank to supply power to the loads. Furthermore, the system has to change its operational mode under grid fault and recovery events in order to keep the battery bank fully charged when possible, so the autonomy of the microgrid is maximized. Fig. 2.1 shows the power flow in grid-connected (mode 1) and stand-alone (mode 2) operational modes. Critical loads are considered in the following analysis, while secondary and external loads are not considered because they are disconnected (in case of grid fault event) or supplied by the main grid, respectively. Generating units are not considered in this study as well.
Figure 2.1.: Structure of the microgrid and power flow in operational modes.
2.1. Description of the system
Block diagram of the proposed bidirectional DC-AC converter is shown in Fig. 2.2. It is formed by a battery bank, DC-DC and H-bridge bidirectional converters, two switches
for the grid and the battery bank, and the control subsystem. External components such as the grid and the load are also represented in order to show the power flow from / to the system. Signals vCb, iLb, vCd, iLf and vCf are the state variables, which are measured
by the sensors and processed by the controllers, while d1, d2, d3, Bon and Gon are control
signals applied to the converters or to the switches.
Figure 2.2.: Block diagram of the proposed system.
Power stage of the proposed bidirectional DC-AC converter is shown in Fig. 2.3. Battery bank is represented as a DC voltage source (Vb) in series with losses resistance
(Rib) and transient RC network (Rob and Cob connected in parallel), while grid is
repre-sented as an AC voltage source (vg) in series with distribution-line impendance (Rg and
Lg connected in series). Besides that, load is represented by an equivalent resistance (Ro)
that consumes the nominal power, while grid and battery switches are represented by ideal switches (Sg and Sb). Output LC filter (Lf and Cf) and voltage transformer (T x)
are also shown as part of the bidirectional H-bridge converter. Dynamic model of battery utilized in this work is investigated in sec. 3.1.
Bidirectional DC-DC converter is the combination of buck and boost convert-ers, and consists of two IGBT transistors (T1 and T2), the DC inductor (Lb), the input
capacitor (Cb) and the DC-link capacitor (Cd). On the other side, H-bridge converter
can operate as either inverter or rectifier, and is formed by 4 IGBT transistors (T3, T4, T5 and T6), the AC inductor (Lf), Cd, the output capacitor (Cf), and the voltage
trans-former. Notice that Cd is shared by boost converter and H-bridge rectifier, although they
do not utilize it at the same time. Each IGBT transistor has a built-in diode to make the power switch bidirectional in current. In addition, all the power converters are designed to operate in continuous conduction mode (CCM).
Control scheme for the system is illustrated in Fig. 2.4. Each bidirectional converter operates in two modes, and the controllers are enabled by signals Ebu, Ebo,
Ehi and Ehr, which are generated by the supervisory controller. Grid fault detector
generates Vg(rms) signal for the supervisory controller, so the controllers are changed when
grid RMS-voltage is less than 90% of the nominal value (a grid power outage occurs) or greater than 110% of the nominal value (over-voltage fault). Supervisory controller is a finite-state machine that also takes care of safety operation, battery charging process and
Figure 2.3.: Sc hematic diagram of the battery , complet e p o w er stage, load, switc h and grid.
user commands.
Figure 2.4.: Block diagram of the control subsystem.
2.2. Design of the system
The proposed system is analysed in a stage-by-stage approach. Thus, for each stage, the other elements of the system are represented by an equivalent resistance (or source) that consumes (or supplies) the nominal output (or input) power. In this way, it is possible to simplify the modeling and design of the controllers for the system. As the converter has two stages and two operational modes by stage, four switching converters are analysed (in this section, their components are sized): boost converter, H-bridge inverter, H-bridge rectifier and buck converter.
The first step consists on doing a preliminary design. If the battery bank is composed of two 12V batteries, nominal voltage of battery bank is Vb = 2 × 12 = 24V .
DC-link voltage is established in twice that value, so VCd = 24 × 2 = 48V , in order to
obtain duty cycle of the first stage close to 0.5. After that, peak value of AC inductor voltage was set to a fraction of the DC-link voltage, so VCf (peak) = 48 × 0.625 = 30V , to
get a value slightly higher than 0.5 for modulation index of the second stage. Therefore, an elevator transformer with turns-ratio 1 : N = 6 is used to reach the grid peak voltage
Vg(peak) = 30 × 6 = 180V . According to the nominal battery discharge current selected
Ib(disc)= 5A, nominal output power is obtained, e.g., Po = 24 × 5 = 100W . Besides that,
for nominal charge current of the battery bank Ib(char) = 5A, input power of the converter
is obtained, e.g., Pi = 24 × 5 = 100W .
Second step implies doing a detailed design of the system. Based on the avail-able literature [15, 43] and actual paramaters of the battery bank (calculated through experimentation, see sec. 3.1), following procedure is intended to design a 100W bidirec-tional DC-AC converter.
In stand-alone mode, DC-DC converter operates as boost and H-bridge con-verter operates as incon-verter. In this case, the system needs a boost concon-verter in order to increase the DC voltage of the battery bank to a suitable value, before it is converted to
VCd = 48V, Po = 100W → Req(bo) =
VCd2 Po
= 23.04Ω (2.1)
Actual values of Vb and Rb = Rib+ Rob are found through tests. Because of
Rb, operating point of duty cycle for boost (D2) is a function of Vb, Rb, VCd and Req(bo):
Vb = 25V, Rb = 0.12Ω → D2 = 1 − Vb 2VCd − v u u t V b 2VCd 2 − Rb Req(bo) ≈ 0.49 (2.2)
Then, DC inductance (Lb) depends on Req(bo) and D2, as well as selected DC
switching frequency (fs(dc)) and current ripple factor for boost (CRFbo), which is selected
to get the available inductance:
fs(dc) = 20kHz, CRFbo= 2.67% → Lb =
Req(bo)D2(1 − D2)2 fs(dc)CRFbo
≈ 5.5mH (2.3)
Finally, for the boost converter, DC-link capacitance (Cd) depends on D2, Req(bo), fs(dc)and voltage ripple factor for boost (V RFbo), that is chosen to get the available
capacitance:
V RFbo = 0.03% → Cd=
D2
Req(bo)fs(dc)V RFbo
≈ 3.5mF (2.4)
In case of H-bridge converter operating as inverter, peak voltage on output capacitor (VCf (peak)) depends on the grid peak-voltage (Vgrid(peak)) and turns-ratio (N ) of
the voltage transformer:
Vgrid(peak) = 180V, N = 6 → VCf (peak) =
Vgrid(peak)
N = 30V (2.5)
the transformer, depends on VCf (peak) and Po: VCf (peak) = 30V → Req(hi)= V2 Cf (peak) 2Po = 4.5Ω (2.6)
To calculate de maximum current through the AC inductor (ILf (max)), a
se-curity constant (Ksec) is usually considered. ILf (max) is a function of VCf (peak), Po and
Ksec: Ksec = 25% → ILf (max) = (1 + Ksec)VCf (peak)2 √ 2 2Po ≈ 4.42A (2.7)
Parametric current through the AC inductor ( ˆILf) is known for two-level PWM
modulation [43], so AC inductance (Lf) depends on ˆILf, VCd, chosen AC switching
fre-quency (fs(ac)), ILf (max) and current ripple factor for inverter (CRFhi), which is selected
to get the available inductance:
ˆ
ILf = 0.5, fs(ac) = 15kHz, CRFhi = 3.3% → Lf =
ˆ
ILfVCd
2fs(ac)ILf (max)CRFhi ≈ 5.5mH (2.8)
As DC-link capacitance (Cd) was already calculated, voltage ripple factor for
inverter (V RFhi) can be calculated for that value of Cd, which is a function of Po, grid
frequency (fgrid), VCd and V RFhi:
fgrid = 60Hz, V RFhi= 6.0% → Cd =
Po
(2πfgrid)VCd2 V RFhi ≈ 3.5mF
(2.9)
Finally, for the inverter, output capacitance (Cf) depends on the values of the
filter’s cut frequency (ff) and Lf:
ff = 2.1kHz → Cf =
1
(2πff)2Lf ≈ 1uF
(2.10)
In grid-connected mode, H-bridge converter operates as a rectifier and DC-DC converter as buck. H-bridge converter operating as rectifier uses all the devices already designed for the inverter, except for the DC-link capacitance (Cd), which is utilized in
Load equivalent resistance for the rectifier (Req(hr)) depends on VCd and Pi: VCd = 48V → Req(hr) = V2 Cd Pi ≈ 23.04Ω (2.12)
Peak value of modulation index for rectifier (M3(peak)) depends on VCf (peak)
and VCd:
M3(peak) =
VCf (peak)
VCd
= 0.625 (2.13)
In case of the DC-DC converter in buck mode, values of Lb, Cd and the input
capacitance (Cb) depend on, respectively, current and voltage ripples for the buck
con-verter [15]. As VCd was already chosen, operating point of buck duty cycle (D1) can be
calculated:
Vb = 25V, VCd = 48V → D1 = Vb
VCd
≈ 0.52 (2.14)
Equivalent load resistance for buck (Req(bu)) is the quotient of Vb by Ibat(char):
Ibat(char) = 4A → Req(bu)=
Vb
Ibat(char)
≈ 6.25Ω (2.15)
As Lb was already designed, current ripple factor for buck (CRFbu) can be
calculated as a function of Req(bu), D1, fs(dc), Lb and CRFbu:
fs(dc) = 20kHz, Lb = 5.5mH → CRFbu =
Req(bu)(1 − D1) fs(dc)Lb
= 2.72% (2.16)
Finally, voltage ripple factor for the buck converter (V RFbu) is selected in
order to get the available value of Cb:
V RF = 2.72% → Cb =
1 − D1
8f2
sLbV RF ≈ 1uF
Parameters of the sized converter are listed in Tab. 2.1. They result in a 100W system with two hours of autonomy (andfor a battery bank of two 12V/24Ah lead-acid batteries). Grid voltage is 127 V(rms) or 180 V(peak). A voltage-elevator transformer with turns-ratio of N=6 is used in order to reach the grid peak voltage from the output capacitor voltage. Other parameters are selected, for example, switching frequency for DC-DC converter (fs(dc)) is set to the maximum allowed by the IGBTs
module, and switching frequency for H-bridge converter (fs(ac)) is set to a value close to
the one recommended by the manufacturer.
Parameter Value Parameter Value
Vb 25 V Vg(peak) 180 V Rb 0.12 Ω Rg 0.1 Ω Ibat(char) 5 A Lb 5.5 mH VCd 48 V Cb 1 uF fgrid 60 Hz Cd 3.5 mF N 6 Lf 5.5 mH Po 100 W Cf 1uF fs(dc) 20 kHz fs(ac) 15 kHz
3. Modeling of the system elements
In this chapter, a procedure to obtain the parameters of a dynamic model for stationary battery is explained. Simple models for the loads and the grid system are also described, as well as the method to simulate overload, grid fault and recovery events. Besides that, a generic method to find average and approximate models for the power converters utilized in the system is presented. Duty-cycle to current and duty-cycle to voltage transfer functions are calculated, and their frequency responses are compared to the Bode diagrams of each simulated circuit.
3.1. Battery modeling
Battery used for implementation is a lead-acid stationary Freedom DF300 (12V / 24Ah). As the IGBT and inductance losses were not considered in the design of the system, overload is taken into account in battery modeling and converters simulation. Thus, nominal discharge current of battery (Ibat(disc)) considered in this section is twice the
designed value.
Dynamic model is used to get an equivalent circuit for each battery [44–47]. Therefore, battery is modeled as a charge-dependent DC voltage source (Voc) in series with
internal losses resistance (Ric during charge or Rid during discharge) and a RC network
that represents the short-term transient response of the battery, see Fig. 3.1a. The RC network includes one transient capacitance (Co) and one transient resistance (Roc for
charge or Rod for discharge) connected in parallel. Losses and transient resistances also
vary with Voc and battery state (e.g., if the battery is charging or discharging). Thus,
ideal diodes are included in the schematic diagram of the model to represent battery state. Self-discharge resistance is not considered in this work because of the long time needed to perform the self-discharge experiments. Also, temperature effects are not considered in this analysis. Parameters of the battery are estimated by using the remaining energy approach [44]. The hardware setup for the parameter estimation of the battery utilized is shown in Fig. 3.1b.
At the beginning, the available energy in fully charged battery (Emax) was
measured for six constant discharge currents (Ibat(disc)) by obtaining the total discharge
time and multiplying it by the average battery power (product of Ibat(disc)and the average
(a) Dynamic model of battery [44].
(b) Setup for parameter estimation.
Figure 3.1.: Circuit model and test setup for parameter estimation of battery.
quadratic or exponential curve fitting, see Fig. 3.2a. For limited time issues, the best dataset from 5 experiments were utilized for the parameter estimation. The resultant polynomial and exponential curves are, for Emax in kJ:
Emax = 0.6044Ibat(disc)2 − 41.1279Ibat(disc)+ 1188
Emax = 1137e−0.02826Ibat(disc)
(3.1)
For fully charged battery, nominal discharge current (Ibat(disc) = 10A) is applied
by 10 minutes in a periodic way, each 30 minutes, until the battery is discharged. At the end of each cycle (after 20 minutes of rest), open circuit voltage (Voc) of the battery is
measured, so its variation with the remaining energy (which is calculated with measured voltage and current of battery, as Erem = Emax −
´
VbIbdt) is estimated. Although
remaining-energy method recommends using a look-up table to obtain the estimated value of Voc from the calculated value of Erem, simple quadratic and exponential curve fitting
Voc= 0.3976e + 11.72e 0 5 10 15 20 25 30 35 400 500 600 700 800 900 1000 1100 1200
Discharge current [A]
M a x im u m e n e rg y [ k J ] Experimental data Quadratic curve fitting
(a) Quadratic curve fitting for Emax vs Ibat(disc).
0 100 200 300 400 500 600 700 800 900 12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13
Remaining energy Erem [kJ]
O p e n c ir c u it v o lt a g e V o c [ V ] Experimental data Quadratic curve fitting
(b) Quadratic curve fitting for Voc vs Erem.
Figure 3.2.: Experimental data and polynomial curve fittings.
In addition to Voc, discharge time constants were measured during transient
response of the battery. Battery voltage and current in short-term discharge test are shown in Fig. 3.3. The third discharge according to the remaining energy method is analysed, in which single battery voltage has been stabilized in 12.5 V after second discharge and rest. When applying constant current discharge Ibat(disc) = 10A, battery voltage settles to 11.9 V. Considering losses and transient resistance with the same values (Rid= Rod = Rd) and
Ohm’s law:
Vb(f )= Vb(i)− 2RdIbat(disc)⇒ Rd =
∆Vb
2Ibat(disc)
= 0.03 (3.3)
From Fig. 3.3a, transient capacitor is calculated by considering the equivalent RC network that produces the settling time, that is considered 5 times the RC time constant, so ∆tdisc = 5Req(disc)Co (which is about 0.1 s). Load resistance is Rload =
Vb(i)
Ibat(disc) = 1.25Ω and the equivalent resistance is calculated in the capacitor terminals by
doing a short-circuit in voltage source:
Req(disc) = (Rid+ Rload)//(Rod) ⇒ Co ≈
∆tdisc
5Req(disc)
= 680mF (3.4)
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 11 11.2 11.4 11.6 11.8 12 12.2 12.4 12.6 12.8 13 Time [s] B a tt e ry v o lt a g e [ V ] ∆tdisc ∆Vb
(a) Transient response in battery discharge.
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 0 2 4 6 8 10 12 Time [s] B a tt e ry c u rr e n t [A ]
(b) Discharge current applied each cycle.
Figure 3.3.: Battery voltage and current in transient of discharge.
transient resistances (considered in this study to have the same values), as well as the transient capacitance. As a result, internal resistances were about 30mΩ when the battery is discharging, and 40mΩ when the battery is charging. Average transient capacitance was estimated in 680mF . Actually, internal resistance varies as a function of open-circuit voltage of battery, and some curve fitting can be calculated from the discharge test, as it is illustrated in Fig. Fig. 3.4. A similar procedure was done for charge process of the battery.
As the long-term discharge behavior is important to to check the state of the battery bank, which is related linearly to the open circuit voltage (Voc), total DC voltage
source in series with average losses resistance (e.g., Thevenin model) is enough to design the system elements. Notice in Fig. 3.2b that open circuit voltage for each battery is about Vb = 12.5V when the battery is in middle charge. In addition, total internal
resistance for each battery is about Rb = 0.06Ω. Validation of the dynamic model and
thevenin model is shown in Fig. Fig. 3.5.
Monitoring battery bank voltage helps to detect if the battery is getting fully charged or discharged. According to the datasheet, the battery is fully discharged when its measured voltage reaches 10.5 V, and is fully charged when the measured voltage reaches 13.8 V. Dynamic model (with internal and transient resistances as well as transient capacitor) is utilized in short-term simulations in order to validate the controllers design. Battery charge and discharge are simulated with a linear-by-pieces DC source, which can be programmed as constant or with linear variations.
3.2. Load modeling
Linear load is modeled as an equivalent resistance that consumes the same power as the nominal power of the system. Overload condition is simulated by connecting (through
12 12.1 12.2 12.3 12.4 12.5 12.6 12.7 12.8 12.9 13 0 20 40 60 80 100 120
Open circuit voltage Voc [V]
D is c h a rg e i n te rn a l re s is ta n
Figure 3.4.: Quadratic curve fitting for Rd vs Voc.
an ideal switch) another resistor in parallel, so the equivalent resistance decreases and the consumed power increases. Non-linear loads are studied as well, by considering an ideal diode-bridge with input inductor and output capacitor that supplies DC voltage to another resistor (thus considered as “non-linear” power), see Fig. 3.6. Non-linear load is a diode bridge with input inductor of 10 mH and output capacitor of 700 µF.
3.3. Grid modeling
Grid is modeled as an AC voltage source connected in series with the transmission-line impedance. The AC voltage source is simulated as the sum of fundamental and two har-monic components (3rd and 5th) that represent distortion of the grid, considered about 2% according to experimental measurements. Impedance of the transmission line has in-ductive and resistive components, being mainly resistive for low-voltage applications [48], as in this work. Grid power outage is simulated by multiplying the value of grid voltage by a linear decreasing function, so a progressive decrease in grid voltage is obtained. Grid power recovery is simulated in a similar way.
3.4. DC-DC converter in boost mode
When the system is in stand-alone mode, DC-DC converter operates as boost and H-bridge converter as inverter. Then, battery switch (Sb) is closed and grid switch (Sg) is open. For
boost converter analysis, H-bridge converter and the load are replaced by an equivalent resistance that consumes the nominal power (Req=
VCd2
2.2 2.4 2.6 2.8 3 3.2 3.4 3.6 3.8 11.6 11.8 12 12.2 12.4 12.6 12.8 Time [s] B a tt e ry v o lt a g e V b [ V ] Experimental data Dynamic model Thevenin model
Figure 3.5.: Model validation in discharge test.
Figure 3.6.: DC-AC converter supplying power for both linear and non-linear loads.
considered charged, so the simplified circuit is shown in Fig. 3.7. Operation in continuous conduction mode (CCM) is considered for all the power converters in this chapter.
By grounding the control signal of T1 (d1 = 0) and applying PWM modulation
to T2, average theory can be used to obtain expressions for T1 reverse current (iT) and T2
voltage (vT) as a function of duty cycle (d2) or its complementary value d02 = 1 − d2:
iT = d2× 0 + (1 − d2) × iLb = d02iLb vT = d2× 0 + (1 − d2) × vCd = d02vCd (3.5)
Applying Kirchhoff’s laws to the circuit of Fig. 3.7, and using physical laws for the inductor and capacitor on Eq. (3.5):
vLb(t) = LbdtdiLb(t) = Vb− RbiLb− vT = Vb− RbiLb(t) − d02(t)vCd(t) iCd(t) = CddtdvCd(t) = iT − io = d02(t)iLb(t) − R1eqvCd(t) (3.6)
Figure 3.7.: Equivalent circuit for the boost converter.
From Eq. (3.6), block diagram of the boost converter can be plot. Transfer functions cannot be directly obtained from Eq. (3.6), due to the presence of d02. In order to get the duty cycle to current and voltage transfer functions, state-space modeling method is used. More extended explaining of that process is in Appendix B. The results are shown, where D2 = 1 − 2VVb Cd − q ( Vb 2VCd) 2− Rb
Req is the operating point of d2:
Hpid(s) = idLb2(s)(s) = Vb(1−D2)/Lb (1−D2)2+ReqRb s+ 2 Req Cd s2+( 1 Req Cd+ Rb Lb)s+ 1 LbCd((1−D2) 2+Rb Req) Hpvd(s) = vdCd2(s)(s) = −Vb/(ReqCd) (1−D2)2+ReqRb s−Req Lb (1−D2) 2+Rb Lb s2+( 1 Req Cd+RbLb)s+ 1 LbCd((1−D2)2+ReqRb ) (3.7)
By using the approximate modeling method, input voltage (Vb) is considered
a disturbance for current control. For voltage control, crossover frequency is small (wc Req
Lb (1 − D2)
2) and battery resistance is small as well (Rb
Lb (1 − D2)
2). Therefore, transfer
functions from Eq. (3.6) are simplified:
Hpid(s) = idLb(s) 2(s) ≈ Vb (1−D2)Lb 1 s+Rb Lb Hpvd(s) = vdCd(s) 2(s) ≈ Vb LbCd 1 ( 1 Req Cd+ Rb Lb)s+(1−D2)2LbCd (3.8)
Transfer functions were verified by simulations in PSIM, see Fig. 3.8. The two models from Eq. (3.7) and Eq. (3.8) are close to the Bode diagrams of simulated circuit’s current transfer function, for frequencies greater than 50 Hz, which is usually
used in current controller design. In addition, Bode diagrams of duty-cycle to capacitor-voltage transfer function for state-space and approximate models are close to the simulated circuit for frequencies less than 10 Hz, which is usual for voltage controller design of boost converter. 101 102 103 -20 0 20 40 60 Frequency [Hz] G a in [ d B ] 101 102 103 -100 -50 0 50 Frequency [Hz] P h a s e [ °] Simulated circuit State-space model Approximated model Simulated circuit State-space model Approximated model
(a) Bode plots of Hpid(s) for boost.
101 102 103 -20 0 20 40 60 80 Frequency [Hz] G a in [ d B ] 101 102 103 -300 -200 -100 0 100 Frequency [Hz] P h a s e [ °] Simulated circuit State-space model Approximated model Simulated circuit State-space model Approximated model
(b) Bode plots of Hpvd(s) for boost.
Figure 3.8.: Frequency response of simulated boost circuit, state-space and approximate
models.
3.5. H-bridge converter in inverter mode
H-bridge converter operates as inverter when the system is in stand-alone mode. Thus, boost converter is replaced by an ideal DC voltage source that supplies the nominal power (VCd), grid switch (Sg) is open and output load is referred (Req = NRo2) to the primary side
of the voltage transformer, so the equivalent circuit is simplified, see Fig. 3.9.
Two-level PWM modulation is applied to transistors T3, T4, T5 and T6.
There-fore, average theory gives expressions for the input current (iT) and bridge voltage (vT)
as a function of the duty cycle (d3) or modulation index (m3 = 2d3− 1):
iT = d3× iLf + (1 − d3) × (−iLf) = m3iLf vT = d3× VCd+ (1 − d3) × (−VCd) = m3VCd (3.9)