Исследование и разработка системы управления ветроэнергетической установкой на эффекте Магнуса тема диссертации и автореферата по ВАК РФ 00.00.00, кандидат наук Лукин Александр Евгеньевич

• Лукин Александр Евгеньевич
• кандидат наук
• 2022

Оглавление диссертации кандидат наук Лукин Александр Евгеньевич

Реферат

Synopsis

Введение

ГЛАВА 1. Конструкция ветроэнергетической установки на

эффекте Магнуса

1.1 Конструктивные особенности ветроэнергоустановки на эффекте Магнуса

1.2 Методы оптимизации конструкции ветроэнергоустановки

1.3 Выводы по Главе

ГЛАВА 2. Математическая модель ветроэнергетической

установки на эффекте Магнуса

2.1 Исследование подъемной силы Магнуса цилиндрических лопастей

2.2 Имитационная модель ветроэнергоустановки на эффекте Магнуса

2.3 Выводы по Главе

ГЛАВА 3. Энергоэффективное управление

ветроэнергетической установкой на эффекте Магнуса

3.1 Управление ветроэнергетическими установками на базе алгоритмов поиска максимальной мощности

3.2 Система управления цилиндрической лопасти

3.3 Управление ветроэнергетической установкой на эффекте

Магнуса методом поиска максимальной мощности

3.4 Моделирование алгоритмов поиска точки максимальной мощности

3.5 Разработка альтернативного алгоритма ОТММ

3.6 Выводы по Главе

ГЛАВА 4. Экспериментальное исследование

ветроэнергоустановки на эффекте Магнуса

4.1 Методы экспериментального исследования и сертификации ветроэнергетической установки на эффекте Магнуса

4.2 Экспериментальная оценка системы управления цилиндрической лопастью

4.3 Схема экспериментального исследования цилиндрической

лопасти ветроэнергоустановки на эффекте Магнуса:

4.4 Схема экспериментального ВЭМ на основе интеллектуальных силовых преобразователей

4.5 Экспериментальные исследования ветроэнергетической

установки на основе эффекта Магнуса

4.6 Выводы по Главе

Заключение

Список сокращений и условных обозначений

Список литературы

Приложение 1. Акт внедрения результатов исследования

Тексты публикаций

Реферат

Введение диссертации (часть автореферата) на тему «Исследование и разработка системы управления ветроэнергетической установкой на эффекте Магнуса»

Общая характеристика работы

Актуальность темы. Современная электроэнергетика стремительно преображается: под влиянием изменений в энергополитике и развития новых технологий: мир вошел в этап 4-го энергетического перехода вслед за четвертой индустриальной революцией, а именно - стремится к широкому использованию возобновляемых источников энергии (ВИЭ) и вытеснению ископаемых видов топлива.

Важнейшей частью гибкой электроэнергетической системы являются возобновляемые источники электрической энергии, в частности, ветроэнергоуста-новки. На сегодняшний день можно говорить о том, что сформировался ряд запросов к системам электроснабжения, который подразумевает необходимость более глубокой проработки синтеза и систем управления генерацией энергии из возобновляемых источников, определение критериев масштабируемости таких систем, исследование энергоэффективности генерации. Современные широко известные и получившие распространение в производстве трех- и пяти-лопастные ветроэнергоустановки применяются, в основном, в офшорных и в промышленных масштабах. Однако в современной парадигме развития распределенных энергосетей, исследования должны быть сосредоточены на так называемых smart micro grid (умных микроэнергосетях) и включать в себя генераторы электроэнергии, доступные для частных хозяйств. Необходима последовательная разработка как конструкции малых ветроэнергоустановок нового типа, в связи с тем, что трех-и пяти-лопастные ветроэнергоустановки малой мощности обладают низкой энергоэффективностью, так и разработка алгоритмов управления, способных решать задачи получения максимума электроэнергии.

Ветроэнергоустановка на эффекте Магнуса является синерге-тическим объединением узлов точной механики с электронными,

электротехническими и компьютерными компонентами, обеспечивающими производство электроэнергии посредством интеллектуального управления её функциональными движениями.В разработку теории проектирования технических систем на эффекте Магнуса внесли свой вклад четыре команды под руководством Бычкова, Sedaghat, Maro Jinbo и Shimizu. Однако промышленного образца достигла только команда под руководством Shimizu и в настоящее время разработки в этой области являются ноу хау исследователей этой фирмы. В настоящее время не синтезирована математическая модель такого класса ветроэнергоустановок, содержащих в своей основе несколько контуров регулирования, а именно контур управления цилиндрами и ветроколеса и не синтезированы программные алгоритмы управления. Применительно к классу лопастных ветроэнергоустановок известны подходы для проектирования алгоритмов поиска максимальной мощности, каталогизированные Kumar и Thongam, однако применительно к ветроэнергоустановкам, функционирующим на эффекте Магнуса, требуется разработка систем управления мощностью.

В связи с вышеизложенным разработка малых ВЭУ является актуальной задачей в рамках современной парадигмы развития энергосетей. Помимо разработки механической составляющей, ветроэнергоустановка на эффекте Магнуса, как робототехническое устройство, требует проектирования системы управления с целью получить максимум выходной мощности.

Степень разработанности темы.

Целью диссертационной работы является проектирование энергоэффективной системы управления малой ветроэнергоустановкой основанной на эффекте Магнуса. Достижение поставленной цели потребовало решения следующих задач:

— доработка конструкции ветроэнергоустановки на основе эффекта Магнуса (ВЭМ) в части проектирования цилиндрической лопасти, дизайна конструкции, проектирование редуктора для получения максимального значения крутящего момента на валу;

— проектирование системы управления цилиндрической лопастью;

— разработка математической модели ВЭМ с учетом коэффициента мощности;

— обзор существующих решений методов управления лопастными ветроэнер-гоустановками;

— синтез и исследование системы управления ветроэнергоустановкой на основе эффекта Магнуса с целью поиска и отслеживания максимальной мощности;

— анализ возможности оценивания и сертификации полученной конструкции ВЭМ с точки зрения существующих стандартов;

— разработка экспериментального стенда и проведение экспериментальных исследований.

Научная новизна:

Научная новизна работы определяется разработкой новых подходов к синтезу конструкции, математическому описанию и алгоритма управления малой ветроэнергоустановкой на эффекте Магнуса.

— Разработана методика оптимизационного синтеза конструкции энергоэффективной малой ветроэнергоустановки на эффекте Магнуса в части расширения диапазона допустимых скоростей цилиндров, а также уменьшения массы и момента инерции аэродинамических поверхностей. Предлагаемая методика охватывает все основные аспекты, которые нужно учесть при проектировании ветроэнергоустановки на эффекте Магнуса с точки зрения механики.

— Разработана математическая модель малой ветроэнергоустановки на эффекте Магнуса, основанная на синергетическом объединении аэродинамического, механического и электромеханического узлов. Математическое моделирование ветроэнергоустановки на эффекте Магнуса как мехатрон-ной и робототехнической системы следует производить методом синергии ее аэродинамического, механического и электромеханического узлов .

— Разработан метод управления малой ветроэнергоустановкой на эффекте Магнуса на базе модифицированного поискового алгоритма для поддержа-

ния максимальной мощности ветроэнергоустановки на эффекте Магнуса. Разработан метод управления малой ветроэнергоустановкой на эффекте Магнуса на базе поисковых методов путем синтеза «алгоритма поиска восхождением к вершине», представляющую собой двухконтурную систему управления.

Теоретическая и практическая значимость работы определяется тем, что разработанные методики используются при проектировании ветроэнерго-установок на эффекте Магнуса. Их применение позволяет синтезировать малые ветроэнергоустановки на эффекте Магнуса с высокой энергоэффективностью, в том числе, при малых скоростях ветра.

Методы исследования. Для решения поставленных задач использовались методы теории электрических цепей, теории электрических машин, теории автоматического управления, теории цифровой обработки сигналов, теории идентификации динамических систем; методы математического моделирования сложных динамических систем с использованием пакетов MATLAB, ANSYS и Agros; методы анализа и оптимизационного синтеза мехатронных систем; математическое моделирование мехатронных и робототехнических систем и анализ их характеристик методами компьютерного моделирования; методы расчета и проектирования отдельных компонентов, входящих в состав робототехнических и мехатронных систем и машин, в том числе на основе принципов оптимизации и генеративного дизайна.

Положения, выносимые на защиту:

1. Методика оптимизационного синтеза конструкции энергоэффективной малой ветроэнергоустановки на эффекте Магнуса в части расширения диапазона допустимых скоростей цилиндров, а также уменьшения массы и момента инерции аэродинамических поверхностей.

2. Математическая модель малой ветроэнергоустановки на эффекте Магнуса, основанная на синергетическом объединении аэродинамического, механического и электромеханического узлов.

3. Метод управления малой ветроэнергоустановкой на эффекте Магнуса на базе модифицированного поискового алгоритма для поддержания максимальной мощности ветроэнергоустановки на эффекте Магнуса.

Обоснованность и достоверность научных положений, выводов и результатов диссертационной работы обеспечивается строгостью используемых математических методов, корректностью принятых допущений при теоретическом анализе и численном моделировании физических процессов, согласованностью теоретических выводов с результатами численного моделирования и результатами экспериментальных исследований, доказывается публикациями в рецензируемых отечественных и зарубежных журналах, трудах международных научных конференций.

Реализация результатов работы. Результаты диссертационной работы использованы:

— при разработке, исследованиях и изготовлении части проекта 719202 НИР-ПРИКЛ "Интеллектуальные силовые преобразователи распределенных энергетических систем";

— при разработке новых разделов дисциплин «Моделирование технических систем», «CAD системы» для программы индустриальной магистратуры 13.04.02 «Электроэнергетика и электротехника», а также «Аппаратно-программное обеспечение киберфизических систем» программы 27.04.05 Инноватика.

Апробация результатов работы. Основные положения диссертационной работы и ее результаты докладывались и обсуждались на следующих научно-технических конференциях регионального, федерального и международного уровня: IX, X, XI Всероссийский конгресс молодых ученых; 2022 29th International Workshop on Electric Drives: Advances in Power Electronics for Electric Drives; 2021 28th International Workshop on Electric Drives: Improving Reliability of Electric Drives; 2020 27th International Workshop on Electric Drives: MPEI Department of Electric Drives 90th Anniversary (IWED); 19th International Symposium "TOPICAL PROBLEMS IN THE FIELD OF ELECTRICAL AND POWER ENGINEERING" and "Doctoral School of Energy and Geotechnology

III"; SPEEDAM 2020; 49, 50, 51 научная и учебно-методическая конференция Университета ИТМО; Конференция по Автоматизированному электроприводу АЭП-2022. Опубликовано 9 работ в научных журналах.

Личный вклад автора. Представленные результаты, изложенные в диссертации, получены автором самостоятельно. Автором проведен анализ литературы, подготовлены доклады и публикации, а также реализована испытательная установка и получены экспериментальные данные

Структура и объем диссертации Диссертационная работа состоит из введения, списка сокращений, принятых в работе, четыре главы с выводами, заключениями, списка литературы, включающего 85 источников. Основная часть работы изложена на 140 страницах. В работу включены 102 рисунка и 22 таблицы.

Публикации Основные результаты по теме диссертации изложены в 9 публикациях в изданиях, индексируемых в базе цитирования Scopus и WoS.

Содержание работы

Во введении приведены предпосылки к исследованию и проектированию ветроэнергоустановок на базе эффекта Магнуса, являющимся синергетическим объединением узлов точной механики с электронными, электротехническими и компьютерными компонентами, обеспечивающими производство электроэнергии посредством интеллектуального управления её функциональными движениями. Обоснована актуальность разработки малой ветроэнергоустанов-ки на эффекте Магнуса взамен малых лопастных ветроэнергоустановок для применения в частных хозяйствах в разрезе концепции гибких электроэнергетических сетей Интернета Энергий. Рассмотрены основоположники исследований в области синтеза ветроэнергетических устройств на базе эффекта Магнуса (ВЭМ). Приведена структурная схема функционирования ВЭМ, включающая в себя систему управления и схему подключения устройства к сети. Обоснована актуальность разработки математической модели ВЭМ и необходимость синтеза системы управления с применением поисковых методов.

Приведен обзор поисковых алгоритмов, использующихся для синтеза систем управления таким классом систем, как ветрогенераторы, в частности, с применением отслеживания и поддержания максимальной мощности.

В главе показано, что ВЭМ является механическим робототехническим устройством, являющимся энергоэффективным только при проектировании системы управления его частей - цилиндрических лопастей и ветроколеса в совокупности, а исследования в области ВЭМ являются актуальными и в настоящее время разрабатываются и проводятся коммерческими структурами. Функциональная схема ВЭМ приведена на рис.1

В главе 1 «Конструкция ветроэнергетической установки на эффекте Магнуса» диссертации приведен патентный анализ существующих ВЭМ и разработана методика оптимизационного синтеза конструкции энергоэффективной малой ветроэнергоустановки на эффекте Магнуса в части расширения диапазона допустимых скоростей цилиндров, а также уменьшения массы и момента инерции аэродинамических поверхностей посредством использования методов генеративного дизайна.

а.

<и ь

0J CD

О

► ^

V^7

Мультипликатор

Фильтр

AC/DC

AC/DC

ФИЛЬТР Трансформатор

1 JL j-J-yyy^

} ^ Т Т Т

СГПМ

Сигнал управления

Скорость лопасти^

Ток в обмотках

Ток

Контроллер сПЛК

Г Векторное 1 Г Цифровой 1

1 управление 1 1 двойник 1

Первый контур управления

отмм

Второй контур управления

Задание (скорость)

Мощность

Сеть

Беспроводное соединение

Рисунок 1 — Функциональная схема ВЭМ, включенной в энергосеть.

ВЭМ является сложным техническим устройством с большим количеством как степеней свободы, так и нелинейностей. Принцип работы ВЭМ заключает-

ся в том, что цилиндрические лопасти ВЭМ создают подъемную силу, которая в свою очередь приводит во вращение ветроколесо ВЭМ и передает механическую энергию на генератор. Ключевой особенностью данного объекта является необходимость в использовании двигателей с системой управления для вращения цилиндрических лопастей, что, при достижении лопастями определенной скорости, вызывает вращательное механическое движение всего ветроколеса с последующим преобразованием механической энергии в электрическую посредством встроенного генератора, нуждающегося в проектировании системы управления его мощностью.

Одной из задач, поставленных в ходе данной работы, являлась модификация конструкции существующего ВЭМ. В обновленной конструкции (рис.2) цилиндрические лопасти синтезированы раздельными

и произведена замена конструкцион-Рисунок 2 — Новая конструкция цилин- ^

ного профиля полой алюминиевой дра ^ ^

трубой. Цилиндрические лопасти содержат бесколлекторные двигатели постоянного тока, а на алюминиевую трубу устанавливаются радиальные однорядные подшипники и фланцы, изготовленные из ABS пластика, к внешней их части крепится цилиндр из ПВХ. Двигатели устанавливаются на край трубы через алюминиевую втулку. Такие решения позволили заметно снизить массу и, как следствие, момент инерции лопастей. Закрепление цилиндра на оси лопасти сделало возможным установку на торцы цилиндров индукторов. Более технологичный вариант установки подшипников снижает биение цилиндра, что играет существенную роль при большой скорости ветра.

Модификация конструкция гондолы производилась методом генеративного дизайна и оптимизацией, были модифицированы фланцы цилиндрической лопасти ВЭМ (рис.3). В ходе анализа сил, действующих на цилиндр, особое внимание было уделено гироскопическому моменту , так как вращающийся вокруг оси симметрии цилиндр представляет собой гироскоп. Данная сила была оценена и учтена в ходе подготовки задания для среды генеративного дизайна вместе с силой лобового сопротивления, подъемной силой, силой тяжести и центробежной силой, как показано на рис.4.

Уравнение для расчета силы, воздействующей на цилиндр из-за гироскопического эффекта, может быть представлено как (3).

МдгГ = • шс • шг • зт(О) (1)

Рисунок 4 — Диаграмма сил, действующих на цилиндр ВЭМ.

Методика оптимизационного синтеза конструкции энергоэффективной малой ветроэнергоустановки на эффекте Магнуса в части расширения диапазона допустимых скоростей цилиндров, а также уменьшения массы и момента инерции аэродинамических поверхностей , представлена на диаграмме на рис. 5

Таким образом в данной главе представлена методика синтеза энергоэффективных малых ветроэнергоустановок на эффекте Магнуса, позволившая уменьшить момент инерции цилиндра на 20%. Также, благодаря индукторам

была увеличена подъемная сила в точке насыщения на 61%, а при максимальных значениях угловой скорости - на 123%.

В главе 2 приведена методика синтеза математической модели малой ветроэнергоустановки на эффекте Магнуса. Математическое моделирование ветроэнергоустановки на эффекте Магнуса как мехатронной и робототехнической системы следует производить методом синергии ее аэродинамического, механического и электромеханического узлов (рис.6).

Для синтеза корректной имитационной модели ВЭМ исследовалась подъемная сила Магнуса. Для этого была построена модель цилиндра, которая затем была проанализирована в среде гидродинамического

моделирования. Исследовалась за-Рисунок 5 — Алгоритм оптимизацион-

висимость подъемной силы, и, как

ного синтеза

следствие, крутящего момента на валу, от скорости ветра и частоты вращения цилиндра. Разработанная модель цилиндрической лопасти ВЭМ показала соответствие экспериментальным данным, как показано на рис.7. Эффект Картмана наблюдается при обтекании цилиндрических тел воздушным потоком при достижении определенного значения числа Рейнольдса Яе.

Эффект Кармана, приводит к значительным колебаниям подъемной силы и силы лобового сопротивления, что может негативно сказаться на системе управления ВЭМ. Эффект Кармана ослабевает при увеличении скорости цилиндра. Численная оценка возникающих колебаний и зависимости их величины от частоты вращения цилиндра показала, что между подъемной силой и частотой

вращения цилиндра существует прямая зависимость - подъемная сила увеличивается при увеличении скорости ротора. Вихреобразование из-за эффекта Кармана достигает максимальных значений при малых частотах вращения ротора, следовательно, при запуске ротор ВЭМ существует риск возникновения резонанса.

Аэродинамическое моделирование ветряной турбины было направлено на определение механической мощности на валу ветроколеса и основано на МЭЛ и может быть представлено как функция механической мощности, извлекаемой ветровым колесом из потока, как показано в (4).

1 2

р = ^ • р • А • V3 • Ср

(4)

где Р - мощность, извлекаемая из потока, А - рабочая площадь ротора, V - скорость ветра, Ср - коэффициент использования энергии ветра (КИЭВ), показывающий отношение извлекаемой механической мощности Р к мощности потока Р0. Максимально возможное значение Ср, известное как предел Бетца, равняется 0.593. КИЭВ — это функция, которая зависит от множества характеристик ветряной турбины и окружающей среды. Построение функции Ср для выбранной ветряной турбины необходимо для создания точной аэродинамической модели ветроколеса. КИЭВ можно представить как функцию отношения скоростей потока до и после взаимодействия с плоскостью ветроколеса. КИЭВ таким образом может быть представлен в виде формулы (5).

Рисунок 6 — Структура математической модели ВЭМ.

с- - - 1

= Р» = 2

1- ^

ШУ +(

1-i

(5)

где Vi и V2 - скорость ветра до и после взаимодействия с ротором соответственно.

В ходе данной работы были рассмотрены три метода построения функции Ср для ВЭМ. Первый подход основан на теории расчета момента элемента лопасти (МЭЛ), в оригинале - blade element momentum theory (BEM). Эта модель обозначена как «аналитическая модель». Коэффициент мощности при данном подходе представляется как функция трех параметров: жесткости ротора , относительной скорости законцовки лопасти (ОСЗ) Ai и относительной скорости цилиндра (ОСЦ) A2.

wmR wcR 2 Nr

Ai •A •-2Лд (6)

Три модели КИЭВ были синтезированы с целью продемонстрировать зависимость от относительной скорости вращения законцовки Ai и относительной скорости вращения цилиндра A2. Однако данные модели обладают следующими недостатками - при использовании аналитической модели КИЭВ принимает комплексные значения и невозможно исследовать систему в динамике, при остальных двух моделях - происходит расхождение экспериментального КИЭВ

>

с1) m

ф а. о

ё а.

I- М

1

О)

2 15 О

р 0 а_

j 5.6

4 м/с /

100 200 300 400 500 600 700

8 м/с

\

13.54 •о

6 м/с

2 18%

100 200 300 400 500 600 700

10 м/с

1

11.2%

О 100 200 300 400 500 600 700 0 100 200 ЗОО 400 500 600 700

Частота вращения цилиндра, рад/с

Рисунок 7 — Результаты моделирования крутящего момента на валу ветроко-леса.

и КИЭВ математической модели, то есть появляется ошибка. Показано, что все вышеприведенные модели становятся некорректными при переменном ветре.

В разработанной математической модели отсутствуют указанные выше недостатки. Интерполяция трехмерной поверхности данных крутящего момента, скорости ветра и скорости вращения ротора ветроколеса производилась с помощью полиномиальной модели с 5 степенью по оси X и 3 степенью по оси У. Результат интерполяции представлен на рис. 8. В результате интерполяции была получена функция зависимости крутящего момента Тт от скорости ветра У и угловой скорости вращения цилиндра шс, представленная функцией (7).

Скорость ветра, м/с ^^^^^^ 200

0 о Скорость вращения ротора

рад/с

Рисунок 8 — Интерполяция ВГД-модели

Tt(œcy) =Р00 + Рю • шс + Р0! • V + Р20 • ш;2 + рп • шс • V+

+ Р02 • V2 + Р30 • ш3 + Р21 • ш2 • У + Р12 • шс • У2 +

+ Р03 • У3 + Р40 • ш4 + Р31 • ш3с • У + Р22 • ш2с • У2+ (7)

+ Р13 • шс • у3 + Р50 • шъс + Р41 • ш4 • У + Р32 • ш3 • у2 +

+ Р23 • ш2 • У3

Согласно уравнению (5), Р0 показывает общую механическую мощность, содержащуюся в потоке воздуха, проходящего через ветроколесо. Мощность Р, отобранную из потока ветроколесом, можно оценить по формуле (4). Угловую скорость ветроколеса шс здесь можно получить из динамической модели.

Таким образом, разработанная модель позволяет получить как значение мощности P, так и значение крутящего момента Тт. Наибольшее значение КИЭВ

достигается при скорости ветра в 10 м/с и при частоте вращения цилиндра около 600 рад/с. Значение Ср при этом достигает 0.375. Данный показатель меньше, чем у аналитической модели, однако, такой результат является более реалистичным, поскольку реальные ВЭУ редко достигают предела Бетца. Анализ показывает, что максимальный крутящий момент на валу ветроколеса для ВГД-модели составляет 24 Нм при скорости ветра в 10 м/с и скорости вращения цилиндра 600 рад/с.

Модель ветроколеса

. Р-А-V*

Р =-■ р-А-2

БДПТ

Ii

Ц, = R + L—Ja + Лтшгсо5(0,) at d

Vf =R iß + L—is -Y iic sinfS,) Jcii„ = -Ь„ыс + Àm (sin(e„) iß) - cos(Ö„} - Г. ^_Sj = ¡1C_J

Синхронный генератор с постоянными магнитами

-ПЛГр Ljij в ПЛГ„ g

('1ТГ'—WW»

•О

'cm Hp 2 l^PM^q + L^iaiq]

Рисунок 9 — Математическая модель ВЭМ

Разработанная ВГД-модель обладает точностью, сравнимой с ранее рассмотренной аналитической, однако показывает лучшие результаты в сравнении с корреляционной и регрессионной моделями. Для дальнейшей оценки качества

разработанной модели было проведено динамическое моделирование для переменного ветра, показавшее, что представленная модель позволяет получать данные о генерируемой механической энергии как при резком повышении, так и при резком снижении скорости ветра. Структурная схема математической модели ВЭМ приведена на рис. 9 . Модель ветроколеса ветроэнергоустановки на эффекте Магнуса строится на базе коэффициента использования энергии ветра. Показано, что коэффициент использования ветра зависит от скорости ветрового потока, частоты вращения цилиндрических лопастей и ветроколеса установки. Математическая модель в части механического узла представляет собой математическое описание ветроколеса ветроэнергоустановки на эффекте Магнуса. Математическая модель электромеханического узла включает в себя математические модели цилиндрических лопастей. Разработанная имитационная модель является оптимальной для динамического моделирования поведения ВЭМ при переменном ветре. Эта модель также может быть представлена системой уравнений (8):

К = Ыа + Ь | + ЛТОШ сСО5(0е)(1)

Ув = Яг в + Ь ^ гр + ЛтШ сзт(ве )(2)

3СШс = -Ьсшс + Лто(5т(0е)^р - соз(де)1а) - ТЬ(3)

е е = ШС(4)

^ ^ето Шг (5)

Тг(ысу) =Р00 + Рю • Шс + Р0! • V + Р20 • Ш2с + Рп • Шс • V+

+ Р02 • V2 + Р30 • Ш3 + Р2! • Ш2 • V + Р!2 • Шс • V2 + + Р03 • V3 + Р40 • Ш4 + Р31 • Ш3 • V + Р22 • Ш2С • V2 + + Р13 • Шс • V3 + Р50 • ШЪС + Р41 • Ш4 • V + Р32 • Ш3 • у2 +

+ Р23 • Ш2С • V3(6)

П =2 ТГШС (7) °Р = 2 рЛ-У3 (1)

р = 1 • р • А • V3 • Ср(8)

(8)

В главе 3 «Энергоэффективное управление ветроэнергетической установкой на эффекте Магнуса» приведен метод управления малой ветроэнергоуста-новкой на эффекте Магнуса.

Сущность отслеживания точки максимальной мощности (ОТММ) заключается в поддержании определенных параметров ВЭУ в состоянии, когда они обеспечивают максимальную генерацию энергии при заданной скорости ветра. Это связано с тем, что значение коэффициента использования энергии ветра является нелинейной функцией. Семейство кривых генерации для ВЭМ представлено на рис.10. График показывает, что для каждой скорости ветра существует такое значение скорости цилиндра, при котором значение механической мощности на валу ветроколеса является максимальным. При дальнейшем увеличении угловой скорости цилиндров система проходит точку максимальной генерации (ТМГ) и начинает вырабатывать меньше энергии, при этом потребляя дополнительную энергию для раскручивания цилиндров. Таким образом, для поддержания оптимального баланса генерации и потребления, для ВЭМ требуется применение алгоритмов ОТММ, позволяющих ВЭМ работать в окрестностях ТМГ.

Скорость ветра, м/с

1 2 3 4 56789 10

Угловая скорость цилиндров, рад/с

Рисунок 10 — Кривые генерации для экспериментальной ВЭМ.

Известны три алгоритма ОТММ, которые возможно модифицировать для применения в ВЭМ - алгоритм восхождения к вершине с фиксированным шагом (ПВВ-НА), адаптивный алгоритм (ПВВ-Л), полушаговый алгоритм (ПВВ/2). В системе с ПВВ/2 при каждом переходе через ТМГ величина шага уменьшается

в 2 раза. ПВВ-Л алгоритм обладает наибольшим быстродействием, однако приводит к генерации энергии на уровне меньшем, чем у аналогов. Применение ПВВ-НА приводит к заметным колебаниям выходной мощности. Разработанный алгоритм ПВВ/2Л представлен на рис.11.

Список литературы диссертационного исследования кандидат наук Лукин Александр Евгеньевич, 2022 год

Список литературы

1. DNV. Future-proofing our power grids. — URL: https://www.dnv.com/ power-renewables/themes/future-proofing-our-power-grids/index.html (online; accessed: 2022-09-10).

2. Gartner. Gartner Predicts Upstream Capital Projects Will Invest More in Alternative Energy Than Hydrocarbons by 2026. — URL: https://www.gartner.com/en/newsroom/ press-releases/2022-01-18-gartner-predicts-upstream-capital-projects-\ will-invest-more-in-alternative-energy-than-hydrocarbons-by-2026 (online; accessed: 2022-09-10).

3. Валландер Сергей Васильевич. Лекции по гидроаэромеханике. — 1978.

4. Magnus effect: physical origins and numerical prediction / Roxan Cayzac, Eric Carette, Pascal Denis, Philippe Guillen // Journal of applied mechanics. — 2011. — Vol. 78, no. 5.

5. University of West Bohemia in Pilsen. Agros suite. — URL: http://www. agros2d.org/ (online; accessed: 2021-03-10).

6. Flettner Anton. Aerodynamical investigations of ship propulsion // Journal of the American Society for Naval Engineers. — 1925. — Vol. 37, no. 1. — Pp. 149-153.

7. Cook Benjamin. Airfoil Alternative: Magnus Effect Flettner Rotorcraft. — 2022.

8. Schmidt Andreas. E-Ship 1-A Wind-Hybrid Commercial Cargo Ship // 4th Conference on Ship Efficiency. — 2013.

9. An assisted propulsion device of vessel utilizing wind energy based on Magnus effect / Boyang Li, Rui Zhang, Baoshou Zhang et al. // Applied Ocean Research. — 2021. — Vol. 114. — P. 102788.

10. Bychkov NM, Dovgal AV, Kozlov Victor V. Magnus wind turbines as an alternative to the blade ones // Journal of Physics: Conference Series. — 2007. — Vol. 75, no. 1. — P. 012004.

11. Sedaghat Ahmad. Magnus type wind turbines: Prospectus and challenges in design and modelling // Renewable Energy. — 2014. — Vol. 62. — Pp. 619-628.

12. MPPT of Magnus wind system with DC servo drive for the cylinders and boost converter / Maro Jinbo, Felix Alberto Farret, Ghendy Cardoso Junior et al. // Journal of Wind Energy. — 2015. — Vol. 2015.

13. Challenergy inc. Collaboration of the World's First Magnus Vertical Axis Wind Turbines and Satellite Communications for Electricity and Communications Services in Remote Island and Mountainous Areas all around the World. — URL: https://www.skyperfectjsat.space/en/news/files/pdf/news_sjc_en_ 20180130_01.pdf (online; accessed: 2021-03-10).

14. Misak Stanislav, Prokop Lukas. Off-grid power systems // 2010 9th International conference on environment and electrical engineering / IEEE. — 2010. — Pp. 14-17.

15. Tuballa Maria Lorena, Abundo Michael Lochinvar. A review of the development of Smart Grid technologies // Renewable and Sustainable Energy Reviews. — 2016. — Vol. 59. — Pp. 710-725.

16. Seifert Jost. A review of the Magnus effect in aeronautics // Progress in Aerospace Sciences. — 2012. — Vol. 55. — Pp. 17-45.

17. A Magnus wind turbine power model based on direct solutions using the Blade Element Momentum Theory and symbolic regression / Gustavo Richmond-Navarro, Williams R Calderon-Munoz, Richard LeBoeuf, Pablo Castillo // IEEE Transactions on Sustainable Energy. — 2016. — Vol. 8, no. 1. — Pp. 425-430.

18. Luo Dahai, Huang Diangui, Wu Guoqing. Analytical solution on Magnus wind turbine power performance based on the blade element momentum theory //

Journal of Renewable and Sustainable Energy. — 2011. — Vol. 3, no. 3. — P. 033104.

19. An overview of horizontal-axis Magnus wind turbines / OF Marzuki, AS Mo-hd Rafie, FI Romli et al. // IOP Conference Series: Materials Science and Engineering. — 2018. — Vol. 405, no. 1. — P. 012011.

20. Wind turbine cylinders with spiral fins / Radomir Gono, Stanislav Rusek, Miroslav HrabCik et al. — 2009. — Pp. 45-48.

21. Lift forces on a circular cylinder in cross flow resulting from heat/mass transfer / Z Travni, F Marsik, Tomas Vit et al. // WIT Transactions on Modelling and Simulation. — 2013. — Vol. 55. — Pp. 149-159.

22. Tokumaru PT, Dimotakis PE. The lift of a cylinder executing rotary motions in a uniform flow // Journal of fluid mechanics. — 1993. — Vol. 255. — Pp. 1-10.

23. Kenyon Kern E et al. On the Magnus effect // Natural Science. — 2016. — Vol. 8, no. 02. — P. 49.

24. Development and validation of a CFD model using ANSYS CFX for aerodynamics simulation of Magnus wind rotor blades / Brian Kieffer Mara, Brian Christopher Mercado, Luigi Andrew Mercado et al. // 2014 International Conference on Humanoid, Nanotechnology, Information Technology, Communication and Control, Environment and Management (HNICEM). — 2014. — Pp. 1-6.

25. Ogretim Egemen Ol, Uygun Durmus, Koklu Mehtap Ozdemir. Analytical evaluation of solar enhanced Magnus effect wind turbine concept // International Journal of Renewable Energy Research (IJRER). — 2016. — Vol. 6, no. 3. — Pp. 1076-1081.

26. Hably Ahmad, Dumon Jonathan, Smith Garrett. Control of an airborne wind energy system with a Magnus effect // 2016 American Control Conference (ACC). — 2016. — Pp. 4978-4983.

27. Shimizu Atsushi. Vertical axis type magnus wind turbine generator. — 2014. — 1. — US Patent App. 14/007,357.

28. Richmond-Navarro Gustavo, Urena-Sandi Noel, Rodriguez Giancarlo. High correlation models for small scale Magnus wind turbines // 2018 5th International Conference on Renewable Energy: Generation and Applications (ICREGA). — 2018. — Pp. 11-15.

29. Blaabjerg Frede, Chen Zhe. Power electronics for modern wind turbines // Synthesis Lectures on Power Electronics. — 2005. — Vol. 1, no. 1. — Pp. 1-68.

30. Selman Bart, Gomes Carla P. Hill-climbing search // Encyclopedia of cognitive science. — 2006. — Vol. 81. — P. 82.

31. Venter Gerhard. Review of optimization techniques. — 2010.

32. Kumar Dipesh, Chatterjee Kalyan. A review of conventional and advanced MPPT algorithms for wind energy systems // Renewable and Sustainable Energy Reviews. — 2016. — Vol. 55. — Pp. 957-970. — URL: https: //www.sciencedirect.com/science/article/pii/S1364032115012654.

33. Thongam Jogendra Singh, Ouhrouche Mohand. MPPT control methods in wind energy conversion systems // Fundamental and advanced topics in wind power. — 2011. — Vol. 15. — Pp. 339-360.

34. Review and critical analysis of the research papers published till date on maximum power point tracking in wind energy conversion system / Syed Muhammad Raza Kazmi, Hiroki Goto, Hai-Jiao Guo, Osamu Ichinokura // 2010 IEEE energy conversion congress and exposition. — 2010. — Pp. 4075-4082.

35. Rathi Reshu, Sandhu KS. Comparative analysis of MPPT algorithms using wind turbines with different dimensions & ratings // 2016 IEEE 1st International Conference on Power Electronics, Intelligent Control and Energy Systems (ICPEICES). — 2016. — Pp. 1-4.

36. A MPPT strategy based on fuzzy control for a wind energy conversion system / Ali El Yaakoubi, Adel Asselman, A Djebli et al. // Procedia Technology. — 2016.

— Vol. 22. — Pp. 697-704.

37. A fuzzy logical MPPT control strategy for PMSG wind generation systems / Xing-Peng Li, Wen-Lu Fu, Qing-Jun Shi et al. // Journal of Electronic Science and Technology. — 2013. — Vol. 11, no. 1. — Pp. 72-77.

38. Mirecki Adam, Roboam Xavier, Richardeau F. Comparative study of maximum power strategy in wind turbines // 2004 IEEE International Symposium on Industrial Electronics. — 2004. — Vol. 2. — Pp. 993-998.

39. Borg John L, Borg Catherine J. Magnus effect power generator. — 1984. — 1.

— US Patent 4,446,379.

40. Wasilewski Jerzy Boleslaw. Vertical-axis wind turbine with flettner rotors. —

2019. — 22. — US Patent 10,184,449.

41. Scarpa Paolo. Magnus effect horizontal axis wind turbine. — 2002. — 23. — US Patent 6,375,424.

42. Hanson Thomas F. Magnus air turbine system. — 1982. — 28. — US Patent 4,366,386.

43. Murakami Nobuhiro, Ito Jun. Magnus type wind power generator. — 2009. — 17. — US Patent 7,504,740.

44. Aoki Yoshio. Compound-type wind power generator. — 2014. — 16. — US Patent 8,836,159.

45. Платонов А.В., Гуревич А.С., Атращенко А.В. Ветроэнергетическая установка. — 2017. — 10. — RU2684068C1.

46. A laboratory platform for identification and control of a synchronous machine / Aleksandr Mamatov, Marina Babayeva, Sergei Lovlin et al. // 2020 XI International Conference on Electrical Power Drive Systems (ICEPDS) / IEEE. —

2020. — Pp. 1-4.

47. Hornby Gregory S, Lipson Hod, Pollack Jordan B. Evolution of generative design systems for modular physical robots // Proceedings 2001 ICRA. IEEE International Conference on Robotics and Automation (Cat. No. 01CH37164).

— 2001. — Vol. 4. — Pp. 4146-4151.

48. Nordin Axel. Challenges in the industrial implementation of generative design systems: An exploratory study // AI EDAM. — 2018. — Vol. 32, no. 1. — Pp. 16-31.

49. McCormack Jon, Dorin Alan, Innocent Troy. Generative Design: A Paradigm for Design Research. — 2004.

50. McKnight Matthew. Generative Design: What it is? How is it being used? Why it'sa game changer // KnE Engineering. — 2017. — Pp. 176-181.

51. Francalanza Emmanuel, Fenech Alec, Cutajar Paul. Generative design in the development of a robotic manipulator // Procedia CIRP. — 2018. — Vol. 67. — Pp. 244-249.

52. Functional generative design: an evolutionary approach to 3D-printing / Cem C Tutum, Supawit Chockchowwat, Etienne Vouga, Risto Miikkulainen // Proceedings of the Genetic and Evolutionary Computation Conference. — 2018.

— Pp. 1379-1386.

53. Poole Sean, Phillips Russell. Rapid prototyping of small wind turbine blades using additive manufacturing // 2015 pattern recognition association of South Africa and robotics and mechatronics international conference (PRASA-Rob-Mech). — 2015. — Pp. 189-194.

54. Design, implementation, and performance analysis of miniature wind turbine / Ehab E Basta, Mehdi Ghommem, Lotfi Romdhane et al. // 2018 11th International Symposium on Mechatronics and its Applications (ISMA). — 2018. — Pp. 1-5.

55. Davis Michael S, Madani Mohammad R. 3D-printing of a functional small-scale wind turbine // 2018 6th International Renewable and Sustainable Energy Conference (IRSEC). — 2018. — Pp. 1-6.

56. GE. GE Renewable Energy, COBOD and LafargeHolcim co-develop record-tall wind turbine towers with 3D-printed concrete bases. — URL: https://www.ge.com/news/press-releases/ ge-renewable-energy-cobod-and-lafargeholcim-co-develop-3D-printed-concrete-bases (online; accessed: 2021-03-10).

57. State of the art of generative design and topology optimization and potential research needs / Evangelos Tyflopoulos, Flem David Tollnes, Martin Steinert et al. // DS 91: Proceedings of NordDesign 2018, Linkoping, Sweden, 14th-17th August 2018. — 2018.

58. FEM based robust design optimization with Agros and Artap / Pavel Kar-ban, David Panek, Tamas Orosz et al. // Computers & Mathematics with Applications. — 2021. — Vol. 81. — Pp. 618-633. — Development and Application of Open-source Software for Problems with Numerical PDEs. URL: https://www.sciencedirect.com/science/article/pii/S0898122120300778.

59. Orosz Tamas, Panek David, Karban Pavel. FEM based preliminary design optimization in case of large power transformers // Applied Sciences. — 2020. — Vol. 10, no. 4. — P. 1361.

60. Hybrid FEA-simulink modelling of permanent magnet assisted synchronous reluctance motor with unbalanced magnet flux / J Pando-Acedo, A Rassolkin, A Lehikoinen et al. // 2019 IEEE 12th International Symposium on Diagnostics for Electrical Machines, Power Electronics and Drives (SDEMPED) / IEEE. — 2019. — Pp. 174-180.

61. Mitchell William F, McClain Marjorie A. A comparison of hp-adaptive strategies for elliptic partial differential equations // ACM Transactions on Mathematical Software (TOMS). — 2014. — Vol. 41, no. 1. — Pp. 1-39.

62. Performance analysis of fem solvers on practical electromagnetic problems / Gergely Mate Kiss, Jan Kaska, Roberto Andre Henrique De Oliveira et al. // arXiv preprint arXiv:2009.04399. — 2020.

63. A three-dimensional numerical study of the Magnus wind turbine with different blade shapes / Xiaojing Sun, Yueqing Zhuang, Yang Cao et al. // Journal of Renewable and Sustainable Energy. — 2012. — Vol. 4, no. 6. — P. 063139.

64. González-Hernández José Genaro, Salas-Cabrera Rubén. Representation and estimation of the power coefficient in wind energy conversion systems // Revista Facultad de Ingenieréa. — 2019. — Vol. 28, no. 50. — Pp. 77-90.

65. Krause Paul C, Wasynczuk Oleg, Sudhoff Scott D. Brushless dc Motor Drives. — 2002.

66. Real-time model for motor control coursework / Alecksey Anuchin, Dmitriy Savkin, Yulia Khanova, Daria Grishchuk // 2015 IEEE 5th International Conference on Power Engineering, Energy and Electrical Drives (POWERENG). — 2015. — Pp. 427-430.

67. Characterising the Digital Twin: A systematic literature review / David Jones, Chris Snider, Aydin Nassehi et al. // CIRP Journal of Manufacturing Science and Technology. — 2020. — Vol. 29. — Pp. 36-52.

68. MPPT Algorithms for Magnus Effect Wind Turbine Control System / Marina Babayeva, Artur Abdullin, Nikolay Polyakov, Stanislav Aranovskiy // 2020 XI International Conference on Electrical Power Drive Systems (ICEPDS). — 2020. — Pp. 1-4.

69. Fixed and adaptive step HCC algorithms for MPPT of the cylinders of Magnus wind turbines / Maro Jinbo, Ghendy Cardoso, Felix Alberto Farret et al. // 3rd Renewable Power Generation Conference (RPG 2014). — 2014. — Pp. 1-6.

70. MPPT for Magnus wind turbines based on cylinders rotation speed / LC Correa, JM Lenz, CG Ribeiro et al. // IECON 2013-39th Annual Conference of the IEEE Industrial Electronics Society. — 2013. — Pp. 1718-1722.

71. ГОСТ Р. 51237-98. Нетрадиционная энергетика // Ветроэнергетика. Термины и определения. — 1999.

72. Commission IEC-International Electrotechnical et al. IEC 60050-415. — 1999.

73. ГОСТ Р. 51990-2002. Нетрадиционная энергетика // Ветроэнергетика. Установки ветроэнергетические. Классификация. — 2003.

74. ГОСТ Р. 51991-2002. Нетрадиционная энергетика // Ветроэнергетика. Установки ветроэнергетические. Общие технические требования. — 2002.

75. IEC. IEC 61400-2: 2013—Wind Turbines. II. Small Wind Turbines. — 2013.

76. ГОСТ Р. 54418.2-2014. Возобновляемая энергетика // Ветроэнергетика. Установки ветроэнергетические. Часть 2.Технические требования к малым ветроэнергетическим установкам. — 2014.

77. Commission International Electrotechnical et al. IEC 61400-11 Wind Turbine Generator Systems—Part 11: Acoustic Noise Measurement Techniques // Geneva, Switzerland: International Electrotechnical Commission. — 2012.

78. ГОСТ Р. 54418.11-2012. Возобновляемая энергетика // Ветроэнергетика. Установки ветроэнергетические. Часть 11. Методы измерения акустического шума. — 2012.

79. IEC Wind Turbine Power Performance. Tech. Rep.: : IEC 61400-12-1.

80. ГОСТ Р. 54418.12.1-2011. Возобновляемая энергетика // Ветроэнергетика. Установки ветроэнергетические. Часть 12-1. Измерение мощности, вырабатываемой ветроэлектрическими установками. — 2011.

81. Commission International Electrotechnical et al. IEC 61400-13: Wind turbinesPart 13: Measurement of mechanical loads // International Electro-technical Commission (IEC), Geneva. — 2015.

82. Commission International Electrotechnical et al. IEC 61400-21 // Wind Energy Generation Systems—Part. — 2001. — Vol. 21.

83. ГОСТ Р. 54418.21-2019 (МЭК 61400-21: 2008) Возобновляемая энергетика // Ветроэнергетика. Установки ветроэнергетические. Часть 21. Измерение и оценка характеристик, связанных с качеством электрической энергии, ветроэнергетических установок, подключенных к электрической сети. Измерение и оценка характеристик, связанных с качеством электрической энергии, ветроэнергетических установок, подключенных к электрической сети.

84. ГОСТ Р. 54418.27. 1-2019 (МЭК 61400-27-1: 2015) Возобновляемая энергетика // Ветроэнергетика. Установки ветроэнергетические. Часть.

85. Raj M Praful, Joshua Ann Mary. Design, implementation and performance analysis of a LabVIEW based fuzzy logic MPPT controller for stand-alone PV systems // 2017 IEEE International Conference on Power, Control, Signals and Instrumentation Engineering (ICPCSI) / IEEE. — 2017. — Pp. 1012-1017.

205

Приложение 1. Акт внедрения результатов исследования.

>Л л у к и %

УТВЕРЖДАЮ V ьпяш. Проректор по НР

Университета ИТМО

ч- ■ » И " 1.7 ■ , У с О» .

--¿их.н.. профессор

§ " --вЪ. Никифоров

1 чМж*?! , В

20 ^-Хг.

ы А.Е. Лукина

об использовании результатов,

на тему «Исследование и разработка системы управления ветроэнергетической установкой на эффекте Магнуса»

Мы, нижеподписавшиеся, директор Научно-производственного Центра «Прецизионная электромеханика» Университета ИТМО Томасов Валентин Сергеевич, ведущий научный сотрудник Научно-производственного Центра «Прецизионная электромеханика» Университета ИТМО Николай Александрович Поляков и главный конструктор Научно-производственного Центра «Прецизионная электромеханика» Университета ИТМО Константин Михайлович Денисов составили настоящий акт о том, что следующие результаты диссертационной работы Лукина Александра Евгеньевича «Исследование и разработка системы управления ветроэнергетической установкой на эффекте Магнуса» были внедрены в НИР-ПРИКЛ 719202 «Интеллектуальные силовые преобразователи распределенных энергетических систем» выполняемой Университетом ИТМО:

1. Математическая модель ветроэнергетической установки на эффекте Магнуса, состоящая из аэродинамического, механического и электромеханического блоков, разработанная в диссертационном исследовании была применена при разработке экспериментального образца интеллектуального силового преобразователя ветрогенератора (ИСПВ);

2. Созданный в результате выполнения проекта испытательный стенд, востребованный для изучения распределенных энергетических систем и проведения испытаний экспериментальных образцов силовых полупроводниковых преобразователей в режимах интеграции в реальную сеть и изолированно от нее, а также для апробации разрабатываемых алгоритмов управления интеллектуальными силовыми преобразователями содержит в своем составе управляемую методами отслеживания точки максимальной мощности ветроэнергетическую установку на эффекте Магнуса, модифицированную посредством разработанной методики, приведенной в диссертационном исследовании Лукина А.Е.

Диссертационная работа А.Е. Лукина на тему «Исследование и разработка системы управления ветроэнергетической установкой на эффекте Магнуса» была представлена на научно-техническом совете Научно-производственного Центра «Прецизионная электромеханика».

Директор Научно-производственного Центра «Прецизионная электромеханика» Университета ИТМО

Ведущий научный сотрудник Научно-производственного Центра «Прецизионная электромеханика» Университета ИТМО

Главный конструктор Научно-производственного Центра «Прецизионная электромеханика» Университета ИТМО

В.С. Томасов

Н.А. Поляков

К.М. Денисов

206

Тексты публикаций

ImdpiJ

Article

Small Magnus Wind Turbine: Modeling Approaches

Aleksandr Lukin ^ * , Galina Demidova , Anton Rassölkin 1,2(3, Dmitry Lukichev 1 , Toomas Vaimann 1,2 and Alecksey Anuchin 1,3

1 Faculty of Control Systems and Robotics, ITMO University, 197101 Saint Petersburg, Russia; [e-mail] (A.R.); [e-mail] (D.L.); [e-mail] (T.V.); [e-mail] (A.A.)

2 Department of Electrical Power Engineering and Mechatronics, School of Engineering, Tallinn University of Technology, Ehitajate tee 5,19086 Tallinn, Estonia

3 Department of Electric Drives, Moscow Power Engineering Institute, Krasnokazarmennaya 14, 111250 Moscow, Russia

* Correspondence: [e-mail] (A.L.); [e-mail] (G.D.)

Featured Application: This manuscript makes a significant contribution to the simulation modeling of innovative Wind Turbines based on the Magnus effect, which is becoming a promising technology in private houses. This research will assist the researchers to create correct mathematical modeling of the Magnus Wind Turbine for further realization control system based on maximum power tracking.

©

check for updates

Citation: Lukin, A.; Demidova, G.; Rassolkin, A.; Lukichev, D.; Vaimann, T.; Anuchin, A. Small Magnus Wind Turbine: Modeling Approaches. Appl. Sci. 2022, 12, 1884. https://doi.org/ 10.3390/app12041884

Academic Editors: Junji Tamura, Masaki Yagami, S. M. Muyeen, Kenta Koiwa and Rifat Hazari

Abstract: Renewables have passed the peak of the inflated expectation hype cycle for emerging technologies, but interest in the design of new energy conversion devices is still high due to widespread distributed energy systems for private households. Magnus effect-based wind turbine combines mechanical and electronic engineering that provides a broader wind speed range and potential maximum power point tracking for deeper grid integration. This paper provides a comparative analysis of Magnus effect-based wind turbine simulation models and the development of the numerical model for the maximum power point tracking algorithm. The advanced model contributes to the reduction of the number of actual tests required for the mechatronics system tuning and deals with sustainability-related challenges, such as climate change and the development of new renewable sources of energy.

Keywords: wind turbine; simulation; intelligent control; mechatronics system control

Received: 5 January 2022 Accepted: 8 February 2022 Published: 11 February 2022

Publisher's Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Copyright: © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

1. Introduction

According to the Global Wind Report 2021 of GWEC [1], the wind industry in 2020 was growing by 13% when compared to the previous year. Nowadays, 743 GW of installed wind power helps avoid over 1.1 billion tons of CO2 produced by fossils. This is happening because climate change requires increased wind energy exploitation for decarbonization procedures, and wind energy is still one of the most popular renewable energy systems, despite the increasing usage of solar energy. However, traditional wind generators are not suitable for places with an annual average wind speed lower than 5 m/s, and the studies show [2] that in such cases, the needed power of the wind generator will grow too high, which makes the generator economically not feasible. Notwithstanding the previous, Russia already has ambitious projects [3] to design wind power plants in areas with low wind speed, such as in Saint Petersburg. Furthermore, similar solutions may be implemented in neighboring Estonia, where the annual average wind speed further, away from the coast and islands, is not reaching the required levels [4]. Furthermore, the rapid development and massive implementation of distributed energy solutions are based on an alternative paradigm, which offers the closeness of generators to the consumers, changes

Appl. Sci. 2022,12,1884. https://doi.org/10.3390/app12041884

https://www.mdpi.com/journal/applsci

in consumer roles, and the development of new types of generators in power systems, making small wind turbines a popular choice. However, the traditional blades of small wind turbines have not proven to tie; energy efficient enough, especially in areas with low wind speeds. That is why developing a new type of small wind turbine with maximum energy efficiency at low wind speed is a crucial challenge for modern technology. Magnus effect-baned Wind Turbines (MWT) described in [5] could be energy efffcient ire a wide wind speed range ef 2-40 m/s, in comparison with 5-25 m/s for conventional Maded small wind tuhbines [d|.

The Magnus effect is a sideways force that influences the rotating cylinder, immersed m the surrounding moSion between ihe rotation cylinder and dhe environment. As a whole, MWT is a complex de-vice needing an extra subsystem that rotates the turbine's cylinders. The ntructuae scheme of MWT is presented in Figurel. It consis ts ot rwo cylindrical blades w[th diameSers 5575 x 125 mm, with a maximum power density of 2 kW. Cylinder blades are installed on an aluminum rack connected to the shaft on MWT via ths hub. °he shaft rotates inside the nacelfe, nupported by two bearings. The cylinders aire caused to rodate by two motors, and connected to nhe shaft on the MWT via the hub. Rotating aylinders provide lift force. The shaft rotates insiia the naeelle and rotates all the nacelle, laading to the rotation of the generator. As a generator in this realization of MWT, we used a Permanent Magnet Synchronous Generator (PMSMG). However, it could lee also bee built using a Flux-Swdrhing hermanent Magnet Synchronous Generator as described in [7]. This protolype of MWT is defigned to operale at e 100 rpm external wheel that eatisfies low wind speed (2-8 m/s).

Figure 1. Render drawing (a) and laboratory prototype (b) of Magnus effect-based wind turbine used in current research (c) drivetrain of the MWT.

A research survey in this area revealed that four investigators' teams were developing different types of MWTs. However, only three of them achieved the fifth technology readiness level.

The first project, headed by Bychkov [8-12], aimed to develop a six cylinder MWT. The MWT working principle is based on lift force, posed by six cylindrical blades [8]. Overall dimensions wery 3.6 x 3.6 m, and aerodynamic simulation showed the yffectiveness of such a wind turbine at 4.55 m/s wind. His team also provided research in construction limitations, and, in [9], a MWT simulation with various types of cylindrical blades diameter/, from 0.05 So 0.099 ro, was presented. DC motors rotated the cylindrical blades with speeds of up to 88000 rpm. Other research in conslructive simulstion, speed, and power density simulations is featured in [10]. This research demonstraied a comparison between a traditional bladed wicd turbine anl a cylindrical MWT. In [11], this comparison research is expanded and

(a)

(b)

(c)

shows the benefits of MWT in wind speed of less than 8 m/s. The research conclusion takes into account constructive calculations, and power coefficients were given in [12].

The second project was under the lead of Sedaghat [13,14]. Based on experimental results in [12], he has led an algorithm for the development of MWT and has resulted in cylindrical blades' lift and drag forces experimental measurements being within the recommendation for calculations of power coefficients, with a discussion of power density. Another paper [13] focused on aerodynamic characteristics of MTW and design calculations. This paper discussed the possibility of maximum power-tracking algorithms for better control of the MWT based on power coefficient. The MWT prototype from Sedaghat's team is similar to Bychkov's team's MWT.

The third scientific group managed by Maro Jinbo focused its efforts on developing control systems of MWT [15,16]. They concluded that the most efficient approach is the maximum power point tracking (MPPT), based on the hill-climbing control algorithm (HIL) [15]. Based on the background, the authors designed an algorithm and method to realize the fixed-step HIL algorithm for MWT. The expanded description of the HIL algorithm was described in [16]. That paper focused on algorithm practical realization. The construction of their MWT was similar to the previous one, including a boost converter and DC motors for cylindrical blades.

The last team, led by Atsushi Shimizu [17], reported a practical realization of MWT with vertical axis blades. Nowadays, they have undertaken an active promotional campaign at energy markets in various countries. The main publication in their research is [18], where the principal scheme was shown, and the MPPT algorithm was described.

All these projects have tried to find the optimum solution for the control system and design the modern MWT. The main idea in all teams is to find the maximum power point for MWT. All commands try to find such a point using experimental results. Another study in synthesizing the MPPT algorithm for MWT is given in [19]. Here was given an algorithm to control four cylindrical blades of an MWT using the hill-climbing algorithm with divided steps. Simulation results confirmed the correctness of this method.

In [20], momentum theory and symbolic regression found the optimum number of cylinders for MWT. It was shown that the optimum number of cylindrical blades is two. This two-cylinder blade configuration allows for the generation of maximum energy. Increasing the number of cylindrical blades decreases the MWT output power. In this research, the construction of the two-cylinder blades was considered.

As seen from the literature review, there is little attention paid to MWT development nowadays. However, promising MPPT algorithms may have potential application in distribution grids, where energy balancing is critical. This paper presents a comparative analysis of MWT simulation models, and the development of a numerical model proposed to be used in the MPPT algorithm. From a sustainability point of view, this work provides a systematic analysis method that may help optimize the overall efficiency of novel renewable energy sources. Furthermore, the presented model (s) may be used to improve life cycle assessment and power management. Full experimental and methodical details are provided so that the research work results can be reproduced using the code published together with an article (see Data Availability Statement). The structure of the research paper is as follows: Section 2 presents the analysis of three available models and power coefficient calculation for studied MWT; Section 3 presents model adaptation for the MPPT algorithm; the implications of main findings and future research directions are discussed in Section 4.

2. Magnus Effect-Based Wind Turbine Simulation Methods

The wind turbine model can be separated into different subsystems, including a model of the wind, aerodynamic model, mechanical model of the turbine and drive train, electromechanical model of the generator, the electrical model of the power converter, and the model of the control system. The main goal of this work was to perform an aerodynamic simulation of a wind turbine based on the Magnus effect.

The aerodynamic simulation of the wind turbine is based on Blade Element Momentum theory and can be presented as a function of mechanical power extracted from the wind by the wind wheel, as shown in (1) [21].

1 3

P = _. A-p-V3 -Cp (1)

.Pis the pow er extracted from the flow, A is the rotor swept area, p is the air density, V is the wind sp eed, and Cp is the power coefficient of a wind energy conversion system, showing the ratio of extractable mechanical power P to the power contained in the air

stream P^ The maximal possible Cp value is known as the Betz limit and hits tin estimated vtlue of 0.593.

Based on [S2], the power coefficient Cp depends on the wind speed ratio and is calculated using the strip theory [23]. The book [2t] held extensive discussions on side dependence between pnwer coeff!cient Cp, the rotor power curnes, and the torque curves in wind turbines with various types of blades, including the one vertical rotation blade wind turbine;. The main parameters of wind eurbine dominating the Cp, under [22], are thee number of rator blades, length af blades, blades aerodynamic profile, and gondola profile. According to (1) the power P captured from the wind is linear ie the Cp. This electric output power vdrsus the tip- speed ratio provides us with the power curve depending on wind turbrne technical cCaracteristics, wind profile, and turbine design. Due to the linear relationship between output power and power coefficient, further discussion will focus on Cp.

Building the Cp function. for the se lected wicd turbine is required So cre ate en acc urrte aerodynamic model of the wind wheel. The power coefficient can be presenled as a function of a velncity ratio of the flow before and sffer inter acting with the wind wheel. The wind wheel is prcsente d i n Figure 2. Extraction o f mechaoic al energy from the free flow leads So a Occ rease in wind velocity V aner passing through the rotrr erea; by the flaw conservation law, this leads to an increase in area A.

V2

Figure 2. Flow conditions due to the extraction of mechanical energy from a free-stream airflow, according to the elementary momentum theory.

The flow conservation law is shown in (2), where A is the rotor swept area, V' is the wind speed at the windwheel, A1 and A2 are cross-sectional upstream and downstream, where Vi and V2 are upstream and downstream wind speed, respectively.

Ai-Vi = A-V = A2-V2 The power coefficient can thus be specified as (3).

(2)

C = P = 1-Cp = Po = 2

1 -«V

1 - V2

. V1

(3)

Determining the function of the power coefficient is an essential step in modeling wind energy conversion systems. According to Vaimann et al. in [2], different methods of representing Cp can be used, including exponential models, sinusoidal models, and polynomial models. As it can be seen from the above-mentioned research work, one of the most commonly used models is an exponential model.

Cp (A, p) = Co + (Ci7 - C2 p - C3 pC - C5 )e-C6Y + C7 A (4)

Here, A is the tip-speed ratio, determined by (5), p is the pitch angle, 7 is determined by (6), R is the rotor radius, &> is the rotational speed of the rotor, C0-7 and d0-2 are coefficients of the exponential model, which depend on mechanical and aerodynamic characteristics of the rotor and can be obtained via experiment or simulation.

A = "V (5)

Y = _1___^ (6)

r A + do p + di 1 + p3 (6)

Previous equations show that wind turbine Cp can be varied by tuning the rotor speed and blade pitch. However, determining constructional coefficients requires prior knowledge of the blade shape parameters of the rotor. Thus, obtaining these coefficients in the initial stage of modeling presents a challenging task. Moreover, in horizontal-axis MWT, controlled via the cylinder rotational speed, should be used instead of turning the blade pitch. Therefore, modeling the wind turbine based on the Magnus effect requires using a Cp function different from the one used for conventional bladed turbines.

During this work, three methods of building a Cp function for horizontal-axis MWT were studied. The first approach, presented in [24], is based on the blade element momentum theory. This model is further referred to as the analytical model. This work aimed to estimate the wind power coefficient function for the MWT via an analytical solution. The authors present the power coefficient as the function of three parameters: tip rotor solidity , tip speed ratio (TSR) A1, and relative speed of the cylinder's rotation (or cylinder speed ratio, CSR) A2. These parameters are obtained in (7)-(9) and depend on constructional features of each wind turbine, such as the radius of the rotor swept area and the cylinder. The analytical solution itself is based on calculating the lift force of each cylinder blade element. The application of blade element momentum theory allows deriving the Cp function, which is shown in (10). The presented function was modified following data obtained by numerical Boundary Element Method (BEM) solutions. Here, W1 is the rotational speed of the rotor, W2 is the rotational speed of the cylinder, R is the rotor radius, r is the cylinder radius, and N is the number of cylinders.

The derived function shows that in the case of MWT, the MPPT can be achieved by varying the speed of the rotor and rotational speed of the cylinder. The maximal achievable value of Cp, which can be obtained using this function, is 0.593, which is the same value as the Betz limit. The presented method of estimating Cp does not take the drag of the cylinder into account, however, it can be used at a preliminary stage of the development of horizontal-axis MWT.

A1 = ^ (7)

A 2 = ^ (8)

2Nr

^ = 2nR (9)

Cp = ^^ (1 + V1 - ^rA1A2) (10)

Another method is proposed in [20]. At the first stage of work, a mathematical model of the wind turbine was developed using data obtained by Bychkov. For the sake of clarity, this model is referred to as the regression model. The authors used BEM and symbolic regression to create simplified models for different horizontal-axis MWTs with different cylinder numbers. Then, a direct method was applied to the model. Combinations of dimensionless parameters, such as the earlier-mentioned tip speed ratio, the relative speed of the cylinder's rotation, the aspect ratio, and Reynolds number, were generated using symbolical regression. As a result, the optimum function with the most significant correlation coefficient was selected for each configuration (number) of the cylinder. Experimental horizontal-axis MWT uses only two cylinders, thus, a function for a two-cylinder model is presented in (11). This function uses TSR and CSR as inputs, similar to the previous one, and introduces cylinder aspect ratio y, which can be calculated as shown in (12), where 2r is the diameter of the cylinder. Researchers conclude that a two-cylinder configuration is optimal for the MWT, and it allows the generator to reach the maximum power coefficient, which is equal to 0.2 in that particular case [19].

Cp =--r (11)

F 18.2 + 2.82y +(3.01 + Ai)À2

y=R (12)

Another method proposed (referred to as correlation model in text) by the same research group is presented in [25]. Similar to the previously mentioned studies, a mathematical model of the horizontal-axis MWT is derived, however, instead of the direct method, high correlation models are used to achieve optimal Cp functions for different configurations of wind turbines. The proposed mathematical model can be adapted to different cases according to the number of cylinder blades. In the case of a two-bladed cylinder, the function of Cp is presented in (13). Research confirms that a two-cylinder configuration is preferable, however, the maximum value of Cp reached during the simulation was not higher than 0.11. A decrease in drag coefficient, however, can lead to a higher Cp of up to 0.24. In both correlation and regressive models, the validity of obtained results is limited to a Reynolds number of 105 < Re < 106.

C =_À1 - 1.11À1 cos(0.876À2)_

p 8 31+A2 2.54 ^ '

Y + A1 (0.788-A2) + e(—+Y ) - eA1 cos(0.876A2)

Three models of the power coefficient were built using the presented algorithms. Parameters of the experimental turbines are presented in Table 1.

Table 1. Parameters of the experimental MWT.

Parameter Value

Radius of the Cylinder, r 6.5 cm

Rotor radius, R 0.6 m

Number of cylinders, N 2

Torsional damping, B 1.4 N-m-s/rad

Rotational inertia, J 2 kg-m2

Concerning these parameters, Cp against tip speed ratio A1 and cylinder speed ratio A2 was plotted for three formulas. Results for the analytical model are presented in Figure 3a. The plot shows steep dependency for both TSR and CSR. The maximum value of Cp reached during this simulation is 0.569, with multiple values of A1 and A2 corresponding to a maximum value of wind power coefficient. In the case of the regression model, as

shown in Figure 3b the maximum value Cp obtained by this model is 0.247 at Ai = 5 and

Xi = 1.7. Finally, a high correlation model is presented in Figure 3c. Here the maximum value reached is 0.047 at X1 = 1 and X2 = 1.7.

(a) (b) (c)

Figure 3. Simulation of Cp for different functions. (a) analytical model, (lb) regression model, (c) correlation model.

The direct comparison shows significant discrepancies between different approaches to simulate the power coefficient. The primary explanation for ruch discrepancy is a different approach to the deguction of Cp function. In the case of regresgion and correlrtion models, experimental data obtai ned by previous researchers was used to find depe nd encie s batween Cp and different parameters of the windwheel. In the case of the analytical model, hrwever, the formula for Cp is derived from the Blade Element Momentum )heory. Therefore, the first two models depend on the mechanical characCeristics of the generator, which were used to obtain experimental data. Some of these characterirtics can be exclusive to the wind turbine and were not taken )n)o account when deriving; Cp. Meanwhile, the analytical model is more universal and does not depend on the unique mechanicnl parameters of the wind rurbine. Thus, the difference between different models can bet explained by tire unique structural properties of the experimental generator. Another xxplanntion is thaX correlation and gegression models function in a spocified Reynolds number area, and the Re number of simuloted turbine exceeds this rangea To determina which model ir mora feasible, a full mechanic and aerodynamic simulation xf the wind turbine was performed and dascribed in the next chapter

3. Simulation Results

Three models were usrd for further processing: analytical, correlation, and regression. All simulation modelf in MATLAB/Simulink are availablx via the link available in the Data Availability Stafemsnt AO mechanical cnh axrodynamic model of the wing generator is presented in Figure 4. The simulation includes the input block used to tune control parameters (rotational speed of cylinders) and wind speed. The aerodynamic model is based on a BEM function (1). The mechanical model simulates the inertia of the rotor without a drivetrain or a generator (configuration during the experiment), and mechanical load is presented as a constant. In addition, a mechanical model is used to obtain the rotational speed of the windwheel. Thus, the torque on the rotor can be controlled with one parameter (cylinder speed), which was a primary requirement for the developed model.

Results of the experiment with experimental MWT are presented in Figure 5. Data points show experimental data obtained during the initial tests of the turbine without the generator and the drive train. The data points are interpolated using a cubic polynomial. The plot shows three distinct areas: the initial area, where the dependence between angular speed and torque is insignificant; the quasilinear area; the saturation area, where the increase in angular speed does not lead to an increase in torque. The quasilinear area is where the generator is to be operated, and thus, determining its beginning and end point is important for the further development of the control system. The plot clearly shows that rotor torque increases with an increase in wind speed, and the end point shifts towards the

higher end of the angular speed axis. Maximum torque points are highlighted with circular markers. To ensure effective control of the wind generator, it has to operate at these points.

(a)

(b)

Figure 4. Model of the experimental wind generator,, (a) shows a general over-view of the model with aerodynamic and mechanical blocks, (b) shows the structure of an aerodynamical model.

Results of the simulation for the analytical model are presented in Figure 6. Similarly, to the experiment, the angular speed of the cylinder was increased from 0 to 600 rad/s, and the mechanical torque was measured. The plot for each wind speed shows two distinct areas: saturation and quasilinear, without any distinct initial area. Otherwise, the presented model follows the same patterns as experimental data and shows the maximum rotor torque of 23.8 Nm.

Figure 5. Experimental data on rotor torque for the prototype wind generator.

Figure 6. Mechanical torque against the rotational frequency of the cylinder for the analytical model.

A similar simulation was performed for the regression model, and results are presented in Figure 7. The presented model shows the main dependencies between mechanical power and rotational frequency and, unlike the analytical model, presents a distinct initial area. However, since the maximum value of Cp this mod el ccn obtain is relatively low, the; maximum torque is sigrnficantly lower than in the previoue model, reachmg the value of (5.29 Nm.

7 r

~ 3 0

0 100 200 300 400 500 600 Rotational speed of the cylinder, rad/s

Figure 7. Mechanical Oorque against the rotaeional frequency of the cylindee for tire regreesion model.

The regre ssion and correlation models showe d po or accuracy when compared to experimental results. Figures 7 and 8 arc provided with smaller sixes limits for illustrative purposes.

50 100 150 200 250 300 350 Rotational speed of the cylinder, rad/s

Figure 8. Mechanical torque against the rotational frequency of the cylinder for the correlation model.

The last simulation was performed using the correlation model. The results are shown in Figure 8. This simulation doesnot show a distinct initial area but clearly showa quasilinear and saturation areas. In addition, the model shows a signiSicanrly lower rotational speed cequircd to reach the saturatidn area. However, the maximum torque on the rotor shaft is limited to 2.69 Nm, which is lower than previous models show.

As the next step of the experiment, data of maximum torque for each different wind speed was gathered and presented in Table 2 and visualized in Figure 9. Experimental data was used to assess the accuracy of each model. The data shows that an analytical model obtained the most accurate result. However, the correlation and regression models have shown less accuracy since the maximum value of Cp reached using this model is significantly lower. This discrepancy can be explained by the fact that the simulated turbine does not operate in the specified Re area.

Table 2. Results of the simulation.

Wind Speed,

Torque, Nm

m/s Experimental

4 5

6 12

8 19.2

10 27.2

Analytical Model

Correlation Model

Regression Model

6.02 11.1 17.1 23.8

0.33 0.82 1.6 2.69

1.09 2.38 4.12 6.29

To summarize, it can be said that the developed models follow the same dependencies as the experimental MWT. The analytical model showed the most accurate results. Other Cp models have shown to be highly dependent on the mechanical characteristics of the wind generator used. This can be explained by the fact that the correlation and the regression models were built for a very specific range of Reynolds numbers between Re 105 and 106. This Reynolds number is true for wind turbines with a radius of 2-5 m, however, the radius of the experimental wind turbine is 0.6 m. Meanwhile, the significant difference between the maximum torque of the correlation and the regression models can be caused by different approaches in both methods; the regression model uses the experimental data as an input, while the correlation model was based on dependencies between variables in equations describing the Magnus effect.

Figure 9. Comparison of maximum torque points for different simulation techniques.

4. Discussion

The goal of MPPT techniques is to achieve maximal power production at the given wind speed. An MPPT can be achieved with different control methods, including variable blade pitch, yaw control, torque control, and aerodynamic braket. Variable blade pitch allows changing Cp by adjusting one of the function's parameters. This method allows for simple control; however, it also requires more complex blades with pitoh control drives. Yaw control aims to reducethe rotor swept area by rotating the nacelle againsO the flow direction. However, this control meqhod increases fatigue of roton blades due to variation in load, and therefore it is rarely used for large-scale wind turbines. Torque control is one of the most straightforward and affordablf techniques. It is based on varying torque on the generator, and it does not require additional mqchanical components inside the rotor or nacelle to conCrol yaw or piich. MeanwhUe, aerodynamic brakes are used to siall the; turbine. This is a relatively compiqx control technique that requires the mechanization of rotor blades.

In the case of MWT, the most feasible conventional control technique is torque control. However,, due to the fact thai lift in the rotor of MWT is wenerated lay cylinders and depends on their rotational speed, it becomes possible to control the wind turbine by adjusting the angular speed of Magnus cylinders. This approach har a potential aolvantage of over-torque aontrol becauoe it qllows for the running cylinders at optimal rotational speed to limit energy consumprion.

The mqdel developed in this research wor k was usei to stu dy tOe p ower charactefistics of the experimenial geneeator. For that reason, the mechanisql output power was cbtained at wind speeds from 2 to 10 m/s, and results are presented in Figure 10. Thus, the power curve shown in the figure represents the maximum mechanical power extracted from the wind at a given wind speed.

In the first region (wind speed 1-4 m/s), the wind turbine operation is not feasible due to low power output. In the second region (4-10 m/s), MPPT techniques should be used to achieve optimal mechanical output. With the increase in wind speed, the MWT reaches the third region (10+ m/s), and the rated power production should be achieved. The turbine has to be kept at a lower rotational speed in this region to ensure the rated power generation and avoid mechanical damage.

Wind speed, m/s

1 2 3 4 56789 10

185 267 332 384 436 474 515 549 583 614

Rotational speed of the cylinder, rad/s

Figure 10. Principles of MPPT for wind generators. 5. Conclusions

The technologiral level of todayfs industry and everyday life requires sustainable energy preservation and regenecation methods. Therefore, new and renewable energy sources should providr high efficiency in order to achieve than goal. MWT is one of the potential technologies that can contribute fowarda advanced control techniques (e.g., MPPT) being used. In the current research work, multiple goals required for developing an MPPT control for the MWT were achieved.

Primarily, the literature analysis shows that, when compared to conventional wind turbinen, MWT has ineerior coverage feom the scientific research groups. There were only three possible aimul ation models discussed in the literature, s o all approaches eo simulate oi tUe power coefficient sf MWT were studied. As a result, mechanical and aerodynamical mod els of the MWT were created.

The conducted study showed the possibility of designing three simulation models for describing MWT power charaeteristhcs. The literature review showed the linear relationship between the generated power and power coefficient of a wind energy conversion system. Three analytical models—analytical, regression, and correlation, using the power coefficient—were created. The simulation results for the presented mathematical models were later validated by comparing them to the experimental data on mechanical torque of the rotor of the experimental wind generator. The analytical model in the comparison was the most accurate with experimental results. Other models have been shown to be highly dependent on the mechanical characteristics of the wind generator used. The final comparison has shown that the analytical model presents the most accurate results, and it was selected to be a base model for further research in the field of MPPT for MWT.

All discussed in the paper models may be repeated by using the link given in the Data Availability Statement section.

Moreover, a power curve was obtained using the chosen analytical model. This curve is to be used to determine the optimal approach to MPPT. Further work in the field of MPPT for MWT requires the implementation of electrical, electromechanical blocks to the developed model, and simulating different MPPT techniques.

Author Contributions: Conceptualization, G.D. and A.R.; methodology, A.L.; software, A.L.; validation, D.L., T.V. and A.A.; writing—original draft preparation, A.L., G.D. and A.R.; writing—review and editing, T.V.; supervision, D.L. and A.A.; funding acquisition, A.R. All authors have read and agreed to the published version of the manuscript.

Funding: The research has been supported by the Estonian Research Council under grant PSG453 "Digital twin for propulsion drive of autonomous electric vehicle".

Data Availability Statement: Magnus effect-based wind turbine (MWT) simulation models presented in the paper are available via the link https://github.com/aelukin/MWT_model (accessed on 4 January 2022).

Conflicts of Interest: The authors declare no conflict of interest. Nomenclature

BEM Boundary Element Method

cp power coefficient of a wind energy conversion system

CSR Cylinder speed Ratio

DC Direct current

HIL hill-climbing control algorithm

MPPT Maximum Power Point Tracking

MWT Magnus effect-based Wind Turbine

TSR Tip Speed Ratio

References

1. Global Wind Report. 2021. Available online: https://gwec.net/global-wind-report-2021/ (accessed on 29 June 2021).

2. Vaimann, T.; Rassölkin, A.; Kallaste, A.; Märss, M. Feasibility study of a local power supply system for sparsely populated areas in Estonia. Agron. Res. 2016,14,1720-1729.

3. Windpark Damba. Available online: https://windpark.ru/projects-ru/vetropark-damba/ (accessed on 29 June 2021).

4. Kull, A. Estonian Wind Atlas; Ruhr-Universität Bochum: Bochum, Germany, 1995; p. 52.

5. Lukin, A.; Demidova, G.L.; Lukichev, D.V.; Rassölkin, A.; Kallaste, A.; Vaimann, T.; Belahcen, A. Experimental Prototype of High-Efficiency Wind Turbine Based on Magnus Effect. In Proceedings of the 2020 27th International Workshop on Electric Drives: MPEI Department of Electric Drives 90th Anniversary (IWED), Moscow, Russia, 27-30 January 2020; pp. 1-6. [CrossRef]

6. Lukin, A.; Rassölkin, A.; Demidova, G.L. Estimation of International Standards for Unconventional Wind Turbine Testing. In Proceedings of the 2020 55th International Universities Power Engineering Conference (UPEC), Turin, Italy, 1-4 September 2020; pp. 1-6. [CrossRef]

7. Prakht, V.; Dmitrievskii, V.; Kazakbaev, V.; Andriushchenko, E. Comparison of Flux-Switching and Interior Permanent Magnet Synchronous Generators for Direct-Driven Wind Applications Based on Nelder-Mead Optimal Designing. Mathematics 2021, 9, 732. [CrossRef]

8. Bychkov, N.M. Magnus wind turbine. 1. Results of model testing. Thermophys. AeroMech. 2004,11, 567-580.

9. Bychkov, N.M. Magnus wind turbine. 2. Characteristics of rotating cylinder. Thermophys. Aeromech. 2005,12,151-166.

10. Bychkov, N.M. Magnus wind turbine. 3. Calculated characteristics of the windwheel. Thermophys. Aeromech. 2008,15, 321-331. [CrossRef]

11. Bychkov, N.M.; Dovgal, A.V.; Kozlov, V.V. Magnus wind turbines as an alternative to the blade ones. J. Phys. Conf. Ser. 2007, 75, 012004. [CrossRef]

12. Bychkov, N.M.; Dovgal, A.V.; Sorokin, A.M. Parametric optimization of the magnus wind turbine. In Proceedings of the International Conference on Methods of Aerophysical Research, ICMAR, Novosibirsk, Russia, 30 June-6 July 2008; pp. 1-5.

13. Sedaghat, A. Magnus type wind turbines: Prospectus and challenges in design and modelling. Renew. Energy 2014, 62, 619-628. [CrossRef]

14. Sedaghat, A.; Mirhosseini, M.; Moghimi Zand, M. Aerodynamic design and economical evaluation of site specific horizontal axis wind turbine (HAWT). Energy Equip. Syst. 2014,2, 43-56. [CrossRef]

15. Jinbo, M.; Cardoso, G.; Farret, F.A.; Hoss, D.L.; Moreira, M.C. Fixed and adaptive step HCC algorithms for MPPT of the cylinders of Magnus wind turbines. In Proceedings of the 3rd Renewable Power Generation Conference (RPG 2014), Naples, Italy, 24-25 September 2014; pp. 1-6. [CrossRef]

16. Jinbo, M.; Farret, F.A.; Cardoso Junior, G.; Senter, D.; Franklin Lorensetti, M. MPPT of Magnus Wind System with DC Servo Drive for the Cylinders and Boost Converter. J. Wind Energy 2015, 2015,148680. [CrossRef]

17. Collaboration of the World's First Magnus Vertical Axis Wind Turbines and Satellite Communications for Electricity and Communications Services in Remote Island and Mountainous Areas All around the World. Available online: https://www. skyperfectjsat.space/en/news/files/pdf/news_sjc_en_20180130_01.pdf (accessed on 10 March 2021).

18. Challenergy Inc. The Model WLS Low-Torque Shaft Unit is Crucial in Making the Magnus Vertical-Axis Wind Turbine a Reality. Available online: https://www.thk.com/eng/csr/feature/pdf/2017/THK_CSR2017_en_f04.pdf (accessed on 10 March 2021).

19. Correa, L.C.; Lenz, J.M.; Ribeiro, C.G.; Trapp, J.G.; Farret, F.A. MPPT for Magnus wind turbines based on cylinders rotation speed. In Proceedings of the IECON 2013-39th Annual Conference of the IEEE Industrial Electronics Society, Gramado, Brazil, 10-13 November 2013; pp. 1718-1722. [CrossRef]

20. Richmond-Navarro, G.; Calderon-Munoz, W.R.; LeBoeuf, R.; Castillo, P. A Magnus wind turbine power model based on direct solutions using the Blade Element Momentum Theory and symbolic regression. IEEE Trans. Sustain. Energy 2016, 8, 425-430. [CrossRef]

21. Burton, T.; Jenkins, N.; Sharpe, D.; Bossanyi, E. Wind Energy Handbook; John Wiley & Sons: West Sussex, UK, 2011.

22. Hau, E.; von Renouard, H. Wind Turbines. Fundamentals, Technologies, Application, Economics; Springer: Berlin/Heidelberg, Germany, 2006; p. 887. [CrossRef]

23. Megson, T.H.G. Chapter 29—Wing problems, Butterworth-Heinemann. In Aircraft Structures for Engineering Students, 7th ed; Elsevier: Oxford, UK, 2022; pp. 887-909. [CrossRef]

24. Luo, D.; Huang, D.; Wu, G. Analytical solution on Magnus wind turbine power performance based on the blade element momentum theory. J. Renew. Sustain. Energy 2011, 3, 033104. [CrossRef]

25. Richmond-Navarro, G.; Urena-Sandi, N.; Rodriguez, G. High correlation models for small scale Magnus wind turbines. In Proceedings of the 2018 5th International Conference on Renewable Energy: Generation and Applications (ICREGA), Al Ain, United Arab Emirates, 25-28 February 2018; pp. 11-15. [CrossRef]

MPPT Algorithms for Wind Turbines: Review and

Comparison

Lukin Aleksandr, Galina Demidova, Omar Alassaf, Dmitry Lukichev Faculty of Control System and Robotics ITMO University Saint-Petersburg, Russia [e-mail], [e-mail]

Abstract— Wind turbines play an important role in energy generation using renewable sources. The wind is very changeable, and such turbines require techniques for finding and tracking maximum power density. The presented work includes an overview of maximum power point tracking techniques applicable to small-size wind turbines based on the Magnus effect. An overview of existing algorithms is presented. Overviewed algorithms are assessed using a simulation in MATLAB with a model of the experimental Magnus effect wind turbine.

Keywords—wind energy, wind power generation, automatic control systems, power system simulation

I. Introduction

Magnus effect (or Magnus force) is a force that appears when a rotating cylindrical or spherical body is submerged into a moving gas or fluid with a relative motion between the body and the medium. It can often be observed in sports in the form of the curved trajectory of a thrown ball. This physical phenomenon has attracted more attention during the last two decades thanks to its possible application as a sail for marine vessels [1]. Another application of the Magnus force is in the field of renewable energy generation. Magnus Wind Turbines (MWTs) present a novel approach and provide energy generation at a wider range of wind speeds than conventional wind turbines [2].

Studied MWT is controlled by varying the rotational speed of the cylinders, which, in turn, generate lift and create torque on the generator shaft. Cylinders are powered by DC motors, making the whole system very sensitive to the balance between generated power and power consumed by the motors. This leads to a problem of finding and tracking the maximum power point at which the speed of a motor can be minimized, and power generation can be maximized. The goal of this paper was to perform research to assess the applicability of modern maximum power point tracking (MPPT) techniques to MWTs and provide a review of currently existing work in the field of MPPT for MWTs.

II. State of Art In MPPT For Wind Turbines

An extensive review of MPPT techniques is presented in [3]. Overviewed algorithms include Tip Speed Ratio (TSR), Optimal Torque (OT), Power Signal Feedback (PSF), conventional and modified Hill Climb Search (HCS), Incremental Conductance (INC), Optimal Relation Based (ORB), Hybrid, Fuzzy based (FLC), Neural Network (NN), adaptive and Multivariable Perturb and Observe. The authors separate algorithms into three distinct groups: Indirect Power Controllers (IPS) are focused on optimizing mechanical

Anton Rassolkin, Toomas Vaimann Department of Electrical Power Engineering and Mechatronics Tallinn University of Technology Tallinn, Estonia [e-mail]

power; Direct Power Controllers (DPS) directly maximize the output electrical power of the generator; novel advanced algorithms which don't fit into the first two groups. The overviewed algorithms were also classified according to their other parameters, such as complexity, convergence speed, computational memory requirement, wind sensor requirement, performance under varying wind conditions, and prior training or knowledge.

A different approach to the classification of MPPT techniques with relation to the type of generator is proposed by Thongam and Ouhrouche in [4]. This paper classified MPPT algorithms for Permanent Magnet Synchronous Generator, Squirrel Cage Induction Generator, and Doubly-Fed Electric Generator.

Another review is given by Kazmi et al. in [5]. In this paper, the authors analyze literature in the field of MPPT control and provide critical analysis of advantages and drawbacks. According to this paper, the most effective control algorithms up to date are self-tuning sensorless adaptive HCS and adaptive TSR. The main advantage of these types of MPPT algorithms is self-tuning capability. The conventional algorithms require building a generation lookup table and additional wind speed sensors, while HCS provides an easier-to-implement sensorless approach.

Smal-scale turbines:

• HI dmb Search

• Fuzzy logic

Fig. 1. MPPT techniques with relation to wind turbine size according to [6].

An analysis of existing wind turbine control algorithms and their applications is presented by Rathi and Sandhu in [6]. The authors used a simulation approach to determine which algorithm is more suitable for such an application according to the size and power rating of the wind turbine. Overviewed algorithms include TSR, non-adaptive HCS, and Fuzzy Logic (FL)-based MPPT, which were applied to models of three wind turbines: small-size Vestas (13.5 m, 225 kW), medium-size General Electric (50 m, 2 MW), and large Sea-Titan (95 m, 10 MW). MATLAB simulations were performed for each turbine, and the maximum amount of generated power was assessed. Authors conclude that TSR is mostly suitable for large-scale wind turbines, while HCS is suitable for all sizes. The scholars highlight the main disadvantage of non-adaptive fixed-step MPPT as slow response time. Meanwhile, the FL-based MPPT algorithm has shown higher convergence speed and overall better performance than all the previous ones. The results of this paper are summarized on a diagram in Fig. 1.

Another review of MPPT algorithms was presented by Mirecki et al. in [7]. In this work, four approaches to optimal control were studied: Torque Control, Speed control, Current controlled diode bridge, and FL MPPTs. The results were assessed by three different parameters. The analysis has shown the good performance of torque control and current control MPPT. FL control MPPT was shown to be less effective than other techniques, mostly due to high complexity and lower power extraction at certain parameters. However, the authors highlight that the ability to function without wind sensor data and wind turbine characteristics is a significant advantage of these algorithms.

More deep analysis of FL algorithmsshows significant advances in Fuzzy MPPT algorithms with multiple papers in the field published. Yaakoubi et al. in [8], a proposed FL

control algorithm is compared to conventional HCS MPPT. The results show that applying an advanced FL algorithm allows extraction of more power than the conventional techniques. Another application of FL-based control for MPPT is presented by Li et al. in [9]. In this paper, the effectiveness of the FL algorithm was estimated using four performance indicators: energy captured by the turbine, MPPT time during a slow change in wind speed, MPPT time during a fast change in wind speed, and power fluctuation magnitude during steady state. The authors have shown that adaptive algorithms in the form of FL control can significantly improve the performance of wind energy conversion systems.

The summary of reviewed papers is presented in Table I. The literature analysis shows that the most promising algorithms for MWT are HCS, FLC, and variations of adaptive and hybrid algorithms. Currently, the field of MPPT for MWT remains relatively unexplored, with only a few published works, due to the relative novelty of the technology and no commercial solutions available. Overview of algorithms for MWT control is presented in the next chapter.

III. MWT CONTROL ALGORITHMS

A. Fixed-step HCS

Research in the field of non-adaptive fixed-step HCS algorithms for MWT was carried out by Babaeva et al. [10].

In this paper, the authors present a simple non-adaptive HCS MPPT algorithm with a fixed step. The implementation of this technique allowed to perform automatic tracking of the maximum power point. At the same time, the authors point out the significant power consumption by motors at the initial stages of the transitional process and highlight the

TABLE I. REVIEW OF EXISTING MPPT ALGORITHMS

Method Description Advantages Disadvantages

Tip Speed Ratio (TSR) Based on tracking of optimal relation between rotor speed and wind speed • Fast • Easy to implement • Requires wind speed sensor • Ineffective for small turbines

Direct Optimal Torque (OT) Based on regulating mechanical torque to reach optimal value • Fast • Easy to implement • Requires rotor speed sensor • Requires known opt. torque

Power Signal Feedback (PSF) Tracking generated power and maintaining its optimal value • Fast • Easy to implement • Power stability • Requires wind speed and rotor speed sensors • Requires known power curve

Hill Climb Search (HCS) Measuring and evaluating system's response to variation in control parameter • Easy to implement • Sensorless • Fixed-step HCS leads to torque instability

Incremental Conductance (INC) Based on generated electric power measurement • Easy to implement • Sensorless • Low convergence speed

I Optimal relation Based (ORB) Based on known functions of generated power and current • Easy to implement • Sensorless • Requires known power-current function

Hybrid A combination of different control methods • High convergence speed • Good tracking • Harder to implement than nonhybrid methods

Fuzzy Based (FLC) Based on Fuzzy Logic Controllers • High convergence speed • Good performance under varying wind conditions • Hard to implement • Requires developed rules • More complex controllers

Neural Networks (NN) Based on Neural Networks • High convergence speed • Good performance under varying wind conditions • Requires training • Requires complex NN controllers

M o Adaptive • Quick switching between previous algorithms • Adapting to environment • High convergence speed • Good performance under varying wind conditions • Requires training and knowledge • Requires complex NN controllers

Multivariable Perturb and Observe (MP&O) Using controlled perturbations and observing its reaction • No training required • Good performance under varying wind conditions • Low convergence speed • Large wind farms only

Fig. 4. Half-step HCS MPPT

Fig. 3. Fixed step HCS MPPT

importance of further investigations in net power generation. The proposed algorithm is presented in Fig.2.

B. Adaptive step HCS

A comparison between fixed and adaptive step HCS algorithms is presented by Jinbo et al. [11]. This model uses an analytical wind power coefficient function applied to a wind turbine model. The control is based on the varying rotational speed of the cylinders. Two HCS algorithms were studied. The first was based on a fixed-step HCS, which is relatively easy to implement. The presented algorithm uses a constant voltage reference value to perform the MPPT. The increase in the size of this fixed step leads to a higher response rate. However, as the authors highlight, it also leads to higher controller instability around the maximum power point. To combat these problems, the authors propose an adaptive HCS algorithm, shown in Fig.3., which measures the size of the power increment and adapts the step size for the MPPC controller of the cylinders accordingly. The paper shows that both algorithms are fully applicable to MWTs, with adaptive HCS showing fast convergence and good stability around the maximum power point.

The same team presents a more advanced version of this research [12]. In this paper, the authors modify the previously developed system by adding the model of PMSG and boost converter. These modifications allow adding an additional MPPT controller for the boost converter, thus creating a system where both load and rotational speed can be controlled via HCS algorithms. The simulation results show that this approach presents a feasible solution to the problem of Magnus MPPT. The system automatically activated when the wind speed is high enough to ensure net power positive, which is around 4.5 m/s (an effective operational area for MWTs).

C. Half-step HCS

Another team working in the field of MPPT for MWTs was led by Correa [13]. This team presented a different approach to HCS algorithms by utilizing an adaptive control based on the divided step method. The method is similar to the simple non-adaptive HCS. However, each time the maximum power point (or maximum generation point, MGP) is reached, the increment of change in cylinder speed

is halved, as shown in Fig.4. Therefore, this control gradually reduces oscillations around the MGP, significantly reducing fluctuations of generated energy. The authors note that even though the presented method is more straightforward than the previously mentioned adaptive control, it still has a high impact on the efficiency of the MWT. In case of presented work, the algorithm was modified to save the step value after each increment.

The literature review has shown four works in the field of MPPT for MWTs. The analysis of the presented work highlighted multiple trends in control of MWTs. The first trend is the widespread use of HCS techniques. This fact aligns with the research results that showed that HCS algorithms show good performance in small wind-turbine systems, such as a vast majority of experimental MWTs.

All presented works are based on either simple or more complex adaptive HCS algorithms. This fact can be explained by the fact that the main advantage of HCS is that it doesn't require prior knowledge of the wind turbine. This advantage is very important in the field of MWTs, where simulation and modeling present a challenging task due to the complex nature of the Magnus effect. Another trend is the use of cylinder rotation speed as the main control parameter. This goes against the general trend in small-scale wind turbine control, where the main control parameter is the

Fig. 2. Adaptive Step HCS MPPT

generator torque (since small-scale WTs with variable blade pitch are very rare due to mechanical complexity). Even though examples of torque-controlled MPPT for MWTs exist, they are coupled with speed-based controllers. This trend can be explained by the fact that in the case of MWTs, the cylinders' rotation speed plays a role similar to pitch control in the case of conventional horizontal axis wind turbines. But unlike pitch control, speed control is a fundamental feature of MWTs, and thus, it is reasonable to utilize this type of control to minimize the amount of energy consumed by the motor and increase the amount of energy generated by the turbine. The literature review has also shown that the FL-based control has demonstrated its effectiveness in MPPT for small wind turbines. Despite that, the FL algorithms were not yet studied with application to MWTs.

IV. Wind Turbine Description

The experimental MWT operates with the help of two rotating cylinders. Each cylinder is powered with the help of a DC motor, which allows them to rotate with a speed of up to 6000 rpm, which is enough to cover the wide variety of wind speeds. The cylinders are located on a beam connected to the shaft via the hub. The speed of the turbine and cylinders was measured using [15], while the control of the generator was implemented using MTPA algorithm considered in [16], which was adapted for operation in generation mode. A Permanent Magnet Synchronous Generator (PMSG) with a gearbox is used to generate power, however, in the case of this work, the generator is not simulated, and only the mechanical power of the shaft is studied. A more detailed description of the generator is given in [14].

The MWT model was built using MATLAB Simulink. It consists of three distinct parts: wind model, aerodynamic model, and mechanical model, as shown in Fig. 5. For the purposes of this simulation, the moment of inertia of the motor and the cylinders is ignored. The wind model used in this work is a constant wind speed of 3 m/s (the speed at which the MWT is presumably more effective than conventional turbines).

The aerodynamic model is based on the Blade Element Momentum (BEM) theory. The BEM shows the amount of kinetic energy extracted by the windwheel of determined radius from the wind at a given speed as mechanical torque on the shaft of the windwheel, as shown in (1)

1

P = -. A-p-y3 • C .

2 p

(1)

Here, P is the mechanical power, A is the area of the windwheel, p is the air density, V is the wind speed. These parameters show the maximum amount of kinetic energy contained in a moving mass of air. Meanwhile, Cp is a variable called the wind power coefficient. This coefficient shows the effectiveness of wind turbine and usually varies between 0 and 0.569 (so-called Betz limit, a theoretical limit on the amount of energy that can be extracted from the wind). Cp is a function that depends on multiple parameters, such as tip speed ratio, dimensions of the wind wheel, and constructive features of the wind turbine. Multiple models for Cp of Magnus wind turbines exist, and in the presented work, the analytical model was used [17]. The analytic function for Cp is presented in (2)

Mechanical model Fig. 5. Model of the wind turbine with MPPT

Fig. 6. Aerodynamic model of the wind turbine

C = 2

In this equation, Xi is the tip speed ratio (3) „ ■ R

A = ■

y

X2 is the relative speed of the cylinder's rotation (4)

2 y

. (2)

(3)

(4)

and or is a mechanical parameter - the rotor tip solidity (5)

2Nr

2-R

(5)

Here, mm is the rotational speed of the rotor, mc is the rotational speed of the cylinder, R is the rotor radius, r is the cylinder radius, and N is the number of cylinders. The aerodynamic model is presented in Fig. 6.

The equation shows that an important parameter for the power coefficient is the speed of the rotor. To estimate the rotor speed, simple one-mass inertia was used. It takes into account the inertia of the rotor and cylinders and the viscous friction coefficient in the ball bearings; however, it does not take into account tortional effects and the load torque on the rotor shaft. The generator is also not included in the model. This inertia model (6) allows to estimate the speed of the windwheel shaft and perform dynamic simulations

daCt)

T = J-—^- + B-w(t) ' dt w

V. Simulation Results

(6)

Three simulations were performed in the course of this work. The first simulation was running to study power generation of the MWT with cylinder running at maximum speed. Wind speed was set to 3 m/s since it's close to the theoretical value at which the MWT becomes effective in terms of net power generation. During the first simulation, the speed of the cylinder increased gradually from 0 to 614 rad/s, simulating acceleration from the standby mode.

The results of the simulation for mechanical power are presented in Fig. 7. The plot shows that studied wind speed, there is a point where cylinders rotate high enough to pass the maximum power point. This mode of operation is to be avoided since, in this mode, the cylinders consume extra power required to run the motor and at the same time generate less energy than physically possible at the given wind speed. In the case of the MWT, the maximum generated power at 3 m/s is 11.1 W at 337 rad/s. These values were used as a benchmark to further assess different MPPT techniques.

A. Fixed-step HCS

To combat the effect of passing maximum power point, MPPT algorithms must be utilized. The simulation results for generated mechanical power are presented in Fig.8. The cylinder speed variation is shown in Fig.9. Numerical results of the simulation are presented in Table II.

The first MPPT algorithm reviewed in the course of this work is a simple non-adaptive fixed-step HCS. For the presented algorithm, the step size increment was chosen to be 30 rad/s, and the MPPT controller is tuned to analyze the power coefficient with 1 second interval. The simulation shows that with a simple MPPT, the model was able to reach the MGP and tune the rotational speed of the cylinders so it would remain there, maximizing power generation. However, the plot also shows significant fluctuations of mechanical power and the maximum power point.

This effect can be explained by observing the MPPT signal for the cylinders. The plot shows the speed of the cylinders during the operation of the MWT. The chosen speed increment of 30 rad/s shows a good convergence time. However, it also leads to significant speed ripples at the maximum power point. This leads to fluctuations of mechanical torque and an overall decrease in the quality of produced energy. This ripple effect can be minimized by utilizing adaptive MPPT algorithms.

100 200 300 400 Cylinder speed, rad/s

Fig. 7. Power curve for MWT at wind speed of 3 m/s

B. Adaptive Step HCS

The mechanical power plot shows that introducing the adaptive step algorithm allowed minimizing torque ripples while simultaneously achieving high convergence time. However, the simulation has shown that the studied algorithm reaches a steady-state with some error, which leads to increased power consumption by the cylinder drives and decreases power production simultaneously.

C. Half-step HCS

Half-step HCS showed the same convergence time as the conventional algorithm. However, as the system reached its maximum power point, the size of the step started to decrease and rapidly reached the target value with no torque ripples observed. The cylinder speed plot shows a steady decrease in rotational velocity with each incremental step.

VI. Conclusions

In the course of the presented work, state of art in the field of MPPT for wind turbines was presented. Initial analysis has shown that HCs and Fuzzy control techniques present interest due to their high response speed, simplicity, and ability to function without sensors. In [3] the authors show that HCS, INC, and ORB algorithms are easier to implement and provide reliable control. However, TSR, OT, and PFS show better response speed.

TABLE II. RESULTS OF THE SIMULATION

HCS Adaptive HCS Half-Step HCS Benchmark

Average power, W 10.96 10.99 11.08 11.08