The Executive Committee serves as the leadership for the Board of Directors and holds the officer positions of Chair, Vice-Chair, and Secretary/Treasurer as well as Member at Large positions, one of which being the Immediate Past Chairman.
This committee's purpose is to lead the Board of Directors and membership by making recommendations and decisions which align with NBLC's mission and regional leadership concerning major public policy issues, which are then approved by membership.
Craig Nelson, Chairman of the Board of the Nelson Family of Companies and author of God Left Manor Farm, was named one of the 100 Most Influential People in the Staffing Industry by Staffing Industry Review in 2012.
Expert in the fields of both staffing and law, Craig has been quoted by a wide variety of news organizations including National Public Radio, the Wall Street Journal, the Associated Press, Staffing Industry Review and the Recorder on a wide variety of topics such as employment trends, recruiting, family businesses, the San Francisco Medical Leave law and California's Three Strikes law.
Having most recently served as the Chief Executive Officer for the Nelson Family of Companies, prior to that, Craig ran Nelson Technology, Nelson & Associates, Nelson Staffing, and Nelsonjobs.com. And before that, was General Counsel at Nelson with responsibility for legal, administration, safety & risk, and human resources.
Prior to joining Nelson, Craig spent seven years with the Criminal Division of the California Department of Justice and before that, three years as a litigator with the Law Offices of Bowles and Verna. He has a Masters of Law from the University of San Diego, his Juris Doctor from Hastings College of Law, and degrees in Business Administration and Economics from California State University, Chico. He was admitted to the State Bar of California in 1990.
Highly involved in the community, Craig is on the board of the Marin Economic Forum, the Leadership Council of the Boys & Girls Club of Marin and Southern Sonoma Counties and the executive board of the North Bay Leadership Council; he was the Chair of the Executive Leadership Committee of the 2011 North Bay Heart Walk; and he is the past Chairman of the Marin County Workforce Investment Board.
Steve Page joined Sonoma Raceway as President and General Manager in 1991 after 11 years working in marketing and special events for baseball's Oakland Athletics where, among other responsibilities, he served as coordinator for the 1987 MLB All-Star Game. Before entering the sports arena, he spent several years on Capitol Hill as press secretary for then-U.S. Congressman Leon Panetta. He is a graduate of the University of California at Berkeley.
Page was the principal advocate for Sonoma Raceway's $100 million modernization project, which has transformed the facility into one of the premier motor racing venues in the country. Among his activities away from the office, Page serves on the board of directors of Speedway Children's Charities, the Road Racing Industry Council and the North Bay Leadership Council, of which he is the past chair. He also serves on the advisory board for 10,000 Degrees Sonoma County, the Sonoma Valley Fund and Social Advocates for Youth and is former chair of the Sonoma County Tourism Council.
Patricia "Patty" Garbarino is President of Marin Sanitary Service, Marin Recycling Center and Marin Resource Recovery Center and has been since 2000. Patty began her career at Marin Sanitary Service in 1987, in Company Development and Human Resources, a combination that facilitated her learning both the operations and administration of the business.
Her background in the education field has been a major benefit to Marin Sanitary Service's Public Education Program. In fact, Patty worked closely with her father, Joseph in establishing the company's Environmental Classroom and its ongoing outreach to environmental organizations and public and private schools in Marin County.
Today Patty is responsible for policy development and implementation and the overall management and oversight of the operations of the company. In addition to her administrative and management responsibilities, Patty remains directly involved in all aspects of the company's governmental and community relations as well as the Public Education Program. Patty has spoken at several national conferences regarding waste and recycling issues and has been a member to Assemblywoman Delaine Eastin's California Waste Reduction, Refuse and Recycling Advisory Committee. She was inducted into the Marin Women's Hall of Fame in 1999 and has been chair and co-chair to 6 different local bond measures and planning commissions.
She has a Bachelor of Arts degree and a Master of Sciences degree, both from Dominican College.Marin Sanitary Service, Marin Recycling Center and Marin Resource Recovery Center have been lauded by the California Integrated Waste Management Board as having the highest recycling rate in California since 2002 and the Marin County Board of Supervisors rated Marin Sanitary Service, business of the year in 2006.
Founded in 1985, Cornerstone is a leading North Bay commercial real estate owner and development firm. As a respected, successful real estate entrepreneur, Alon has nearly 20 years of diverse experience in shepherding ideas to viability. Alon is the founder of the North Bay Makers, 180 Studios. In this role, he establishes vital connections in the community and encourages forward progress. Alon's goals are to make 180 Studios the premier facility of its kind, and to ensure maximum economic and interpersonal benefit to the community.
Brad Bollinger is the publisher of the North Bay Business Journal overseeing the publication's coverage of business in the North Bay counties of Sonoma, Napa and Marin.
Before joining the Journal after its purchase by the New York Times Co. in 2005, he was the business editor and a columnist for The Press Democrat since 1990.
While he was editor, the Press Democrat business section won several national, state and regional awards for its coverage, including six "Best In Business" citations from the Society of American Business Editors and Writers. It has won the top award from the society in 2005, 2004, 2003, 2002, 1999 and 1996 and won for spot news coverage in 2000.
Bollinger was a leading editor and creator of the Press Democrat's extensive four-day series in September of 2004 on the local impacts of economic globalization, "Global Shift." The series won the prestigious Polk Award as well as the New York Times companywide Punch award.
The newspaper's 2002 series on the wine industry, "Vine to Wine," received the highest award from the California Newspaper Publishers Association.
Bollinger has made dozens of presentations to community organizations and groups of executives on the economy, including the 2006 commencement address for Empire College business school in Santa Rosa. He currently is the chairman of the North Bay Leadership Council and is a member of the boards of directors of the Santa Rosa Chamber of Commerce, the Sonoma County Alliance and Empire College and is trustee on the Ernest L. and Ruth W. Finley Foundation in Santa Rosa.
Bollinger has a journalism degree from San Jose State University and master's in communication from CSU, Chico. His 1983 master's thesis on newspaper ombudsmen was the subject of articles in Columbia Journalism Review and Editor & Publisher. In 1990, he was among the attendees at the inaugural Summer Institute for Economics for Journalists created by the Foundation for American Communications.
Bollinger and his wife, Corine, live in Santa Rosa.
Barry is the third-generation leader of Friedman's Home Improvement, celebrating over 70 years in business committed to delivering the human side of home improvement. Based in Petaluma, Friedman's employs over 600 Team Members at it stores located in Santa Rosa, Petaluma, Sonoma and Ukiah. Barry continues to honor the family's tradition of being both a supporter and active participant in the local community. Barry and his wife Heidi live in Santa Rosa with their four children.
Pat Kendall is the Medical Group Administrator for Kaiser Permanente in Marin and Southern Sonoma County. Pat's leadership position is operational, strategic, financial, quality, care, service, people and community. Her guiding philosophies are patient, family, staff and physician focused. She often speaks of working in the Intimacy Business believing that seeking Health Care is one of the most intimate things we do in life. In addition, Pat believes in building a healthy community with an upstream emphasis focused across the spectrum of Prevention, recognizing the impact of the social determinants on health. Creating a healthy work place and engaging the leader in all levels of staff is a lifelong endeavor.
Pat holds degrees in Nursing, Business, Organizational Development, Human Resources and Leadership. She has completed post graduate work at UC Berkeley and Stanford University. Pat is presently on the boards of North Bay Leadership Council, Healthy Marin Partnership, School to Career Partnership, Marin Economic Forum, Hospice, Marin Sustainable Agriculture and Dominican University School of Natural Sciences Advisory Board. Pat is also a member of the citizen's oversight committee of SMART, Friends of San Rafael, San Rafael City Schools Advisory Team – 114th Partnership, Marin Workforce Housing Trust and the Pickleweed Advancement Team.
Pat is the recent recipient of the North Bay Leadership Award for Leadership for 2012 and San Rafael Citizen of the Year, 2013.
Jordan represents shopping centers, property owners, office building managers and developers in landlord-tenant, land use and bankruptcy matters.
Jordan has extensive experience in all landlord-tenant disputes involving nonpayment of rent, exclusives, tenant improvements, mechanic's liens, tenant bankruptcies and post-lease termination collections. Jordan also represents banks enforcing commercial loan agreements. He has trial and appellate experience in state and federal courts.
Mark Wood served as Chair of the North Bay Leadership Council Board of Directors from 2012 through 2013. He is currently Consultant & Proprietor at Mark Wood Consulting, providing expertise in HR and organizational performance, as well as serving as adjunct faculty at the Dominican University School of Business and Leadership and the Sonoma State University Executive MBA Program. Previously, Mark worked at BioMarin Pharmaceutical for 9 years, serving as Senior Vice President of HR & Corporate Affairs; at AG Consulting, as Vice President of HR & Administration; at Genentech, as Manager of Compensation & Quantitative Analysis; and at Wells Fargo Bank, in various HR roles. During his tenure at BioMarin, Mark also served a term as Chair of the Marin County Workforce Investment Board. Mark earned a Master of Industrial & Labor Relations from Cornell University and a Bachelor of Arts in Psychology from the State University of New York at Buffalo.
Michelle's area of expertise is GAAP compliant financial accounting and reporting. She serves middle-market, privately-held, family owner-managed, and institutionally owned businesses and publicly held companies. She has experience working with companies in the start-up phase through ownership transition, including IPO. Over the past decade or so, she has spent more than 90% of her time working with wineries, vineyards, real estate entities, négociants, custom crush facilities, wine and spirits distribution companies, natural and organic food companies, and non-wine agriculture. As a result, she has extensive knowledge of the more common and complex accounting issues those types of businesses face. In addition to providing traditional audit and review services, she offers guidance in the areas of business combinations, consolidations, lease accounting, inventory valuation, revenue recognition, and evaluation of internal control. Michelle also has over 10 years of experience in the area of audits of employee benefit plans.
Michelle has served on several boards including as Treasurer for the Napa/Sonoma Chapter for Women for WineSense (2011-2013) and Treasurer and Finance Committee Chair for The Healdsburg School (2013-2015). Since 2009, she has served on the planning committee for the Cal CPA Wine Industry Conference where she has also presented on GAAP accounting and reporting in the wine industry. Michelle has also participated as a speaker at the Wine Industry Technology Symposium and through Sonoma State University's Wine Business Program. She is active in the Firm's recruiting efforts at SSU.
Beldona has served as associate dean of the Graduate School of Business at Bryant University in Rhode Island since 2011. He served as chair of Bryant University's Department of Management from 2007-2011. An expert in international business and corporate strategy, Beldona has lectured, consulted, studied, and taught in the Americas, Asia, and Europe.
At Bryant, Beldona helped in introducing undergraduate and graduate-level courses focused on international business and global finance. He also worked with faculty to create programs designed to expand the school's global reach and provide students with international study and intercultural experiences.
It was Dominican's commitment to the purposeful integration of the liberal arts and strong professional programs in business, education, and the health sciences that drew Beldona to the University.
At Bryant, Beldona helped in launching undergraduate majors in human resources management, entrepreneurship, and global supply chain management. At the graduate level, he worked with area companies and faculty to introduce specializations in global supply chain management, international business, and global finance. He also conceptualized the global immersion experience for MBA students – a required core MBA course for full-time MBA students that embeds an international consulting experience.
Working with faculty, Beldona supported integrating highly interactive learning modules such as e-pitch in freshmen year, service-learning in sophomore year, internships in junior year, and case competitions in senior year. He established numerous relationships with local and international firms and encouraged these firms to sponsor prizes for courses, internships for students, and consulting engagements for graduate courses.
Beldona was the Larry Jones Fellow of Corporate Governance at Wichita State University from 2001-2007, where he also served as the Entrepreneurship Research Associate for the Center for Entrepreneurship. Beldona joined Wichita State University in 2001 as assistant director of the Center for International Business Advancement.
Beldona was an associate professor of international business at the International University of Japan from 2000-2001. He was an assistant professor of international business at Rutgers University from 1994-2000.
Beldona received his Bachelor of Science in Physics and an MBA in general management from Karnatak University in Dharwad, India. He earned a Master of Science in International Business and a Ph.D. in Business Administration from Temple University in Philadelphia.
Katie Davis is Chief of Staff to the CEO and President of PG&E Corporation playing a key support role to the CEO and senior leadership team to ensure the Company vision, strategies, and goals are met.
Prior to that role Davis was Senior Manager of Public Affairs overseeing a team of Public Affairs Representatives that serve as PG&E's liaisons to local government staff and elected officials. She has also worked on a number of company-wide initiatives. Davis began her career at PG&E in 2008 as a Public Affairs Representative. She has also served as a Program Manager in PG&E's Government and Community Partnerships department helping local governments implement energy efficiency and sustainability programs.
She currently serves as the Immediate Past President of the Sonoma County Alliance and Board Member for the North Bay Leadership Council.
Before joining PG&E, Davis worked in political and land use consulting in the North Bay. In her spare time she enjoys spending time with her daughters Emma and Kaisa and her husband John.
In 1981, Nancy Dobbs was chosen to serve as general manager for the Rural California Broadcasting Corporation (KRCB's parent company), secured multiple FCC licenses and located a transmitter and building site. She organized the raising of funds and the construction of a building to house the station. Finally, after developing a staff, the station began broadcasting in 1984. KRCB FM was signed on in 1994. Nancy continues to lead the organization as President and CEO.
Nancy was an honoree at Congresswoman Lynn Woolsey's tenth annual "A League of Their Own" luncheon celebrating women "breaking the glass ceiling" and is a Distinguished Alum of Sonoma State University. A tireless proponent of public broadcasting serving with California Public Television, Dobbs also serves on the board of directors of Conservation Corp of the North Bay, and as an elected official on the Palm Drive Hospital Board in Sebastopol.
Prior to her work in public broadcasting, Nancy worked in the area of health care policy. Included was staff to the California State Legislature Assembly Health Committee, Joint Legislative Audit Committee, and as a gubernatorial appointee to the Statewide Health Planning Council. She also served as an appointee of the Sonoma County Board of Supervisors to the Board of Trustees of Community Hospital where she chaired the Finance Committee.
Aimi Dutra is the Director of Community Relations and Governmental Affairs for The Dutra Group overseeing all corporate communications, government relations and community outreach programs for the company. Recently, she successfully led the extensive efforts to secure critical land-use entitlements on behalf of the company.
Since joining the firm, Aimi has brought a proactive and hands-on approach to community involvement. As a result of her vision, The Dutra Group has become a leading contributor toward important local causes and non-profits throughout Marin. Aimi has professional memberships with the Engineering Contractors Association, San Rafael Chamber, Bay Planning Coalition and California Marine Affairs and Navigation Conference (CMANC) and sits on the Heads Up Public School Foundation Board.
Aimi is married to her husband J.C. Krause and together they have 3 young children. She is a graduate of the University of the Pacific with a degree in Psychology.
Ingrid Estrada is the CAO at Keysight Technologies. Estrada's responsibilities include directing Keysight's global policies and programs for leadership and talent development, compensation, benefits, staffing and workforce planning, human resources systems, education and organization development.
Estrada joined Hewlett-Packard in 1990 where she held a number of procurement, order management and order fulfillment positions. From 1995 through 2004, she served a variety of senior management roles in manufacturing, materials, and supply chain management, at both the division and business unit levels at HP and then Agilent Technologies. In 2004, Estrada was named a senior manager of manufacturing for Agilent's Wireless Business Unit and launched the transformation of the Electronic Measurement Group supply chain.
In 2006, Estrada was promoted to vice president and general manager of the Remarketing Solutions Division. In 2011, she was named vice president and general manager of Global Sourcing for Agilent Technologies.
Estrada holds an MBA from Santa Clara University and an Executive MBA from Stanford University.
Born in Marin County, Richard Ghilotti graduated from Marin Catholic High School in 1964. Richard continued his education at Santa Clara University, earning his degree in Finance in 1968. After college, Richard joined his Father, Dino, and Uncle Mario, who both founded Ghilotti Brothers Construction, Inc., and worked his way from an Equipment Superintendent to Manager of all Operations. Following the death of Richard's father in 1990, Richard and his Uncle split Ghilotti Brothers Construction in 1992. With cousin Jim Ghilotti, Ghilotti Construction Company, Inc. was founded in 1992.
Currently managed by Richard and nephew, Brian Ongaro, Ghilotti Construction Company has been headquartered in Santa Rosa for over 10 years, and has grown from a company with revenues of $12.5 million, to a company with revenues in excess of $70 million. Major projects completed include the $50 million dollar renovation of Sears Point Raceway, $20 million dollar site development at the Big Rock Ranch for George Lucas, $16 million dollar Geysers Pipeline project for the City of Santa Rosa (the only 1 of 8 phases completed on time and within budget), and the $10 million dollar site development project at Maycama Golf Club.
Richard currently lives in Marin with his wife Nancy and youngest son Dino. His oldest son, Willie, lives in Petaluma and is an Equipment Operator for Ghilotti Construction Company. Richard's oldest daughter, Michelle, recently married and is currently residing in Amsterdam on a job assignment with her husband.
Tri Counties Bank, a $6B in assets community bank, in 26 California counties with 60 branches.
Graham joined Tri Counties Bank to develop and manage customer relationships in the North Bay region which includes Sonoma, Napa and Marin counties. She brings over 30 years of experience in banking with a proven record of growing relationships and assisting customers achieve their financial goals.
Prior to joining Tri Counties, Julianna was SVP of Western Bankers Association, CEO of Western Independent Bankers and SVP and Cross Director for Wells Fargo in the Wholesale and International Divisions. She is currently a faculty member of the Pacific Coast Banking School, serves on the Timothy Murphy School board and is a past board member of the Luther Burbank Center for the Arts and the Sterne School. She has a BS from University of San Francisco, and a MBA in Finance from Golden Gate University.
Darryl Hawkins works for Comcast/NBC Universal as the Vice President for the Northeast Bay area of CA. As the Area Vice President his responsibilities include providing leadership for all aspects of the market including Technical Operations, Finance and Human Resources. Prior to his current role he was the Senior Leader for Comcast "Regional Operations" in Southern Arizona, his responsibilities include P&L, External Affairs, Technical Operations, Vendor Management and Sales Performance. Previously, he spent 18 years at Verizon Wireless, most recently as the Sr. Director Customer Operations PA ($1.2B P&L).
Hawkins has more than 20 years of experience in telecommunications and has held various Executive positions with increasing responsibilities since joining Bell Atlantic in 1997. In previous roles he led teams responsible for Service Delivery, Human Resources, Marketing, Contract Management and Sales. Hawkins has a strong passion for the youth in our community. He has developed various internships within Verizon to help provide opportunities to at-risk youth across the country. Hawkins is a United States Marine and a native of Washington, DC. He holds a Bachelor's of Science in the Administration of Justice from UDC, a Masters of Business Administration (HRM) from Indiana Wesleyan University, and has completed a three year Doctorate program in Organizational Leadership from Walden (DBA – ABD).
Kathryn Lowell is an executive and Board Member with more than 25 years of experience in the healthcare industry. Kathryn is the Vice President of Government Affairs & Advocacy at BioMarin. In this capacity, she oversees the development and implementation of local, national and global legislative strategy and advocacy.
Previously, she was appointed by Governor Schwarzenegger to develop and execute policy and strategic healthcare initiatives. During this time, she developed a partnership with the California Council on Science and Technology to create the Personalized Health Information Technology Coalition. It drew senior executives together to address the challenges of personalized medicine. She also oversaw program and policy initiatives from the California State Department of Managed Health Care and the Office of the Patient Advocate. Kathryn was a Board Member of the California Privacy and Security Advisory Board and a Board Member of the California Telehealth Network Advisory Council.
Also prior to her current position, Kathryn was the Vice President for Strategic Growth at Lumetra, a health care consulting firm that specialized in federal health care program management. For six years, she also worked at Maximus, (NYSE:MMS), which is a national government health care services firm. Serving as the Vice President for Government Affairs, she worked with states to partner on Medicaid program management.
From 1995 to 1998, she was appointed by Governor Wilson as Assistant Secretary for Health at the California Health and Human Services Agency. During that time, she worked to create and staff the Biotechnology Council for the State of California. In her early years, she worked on Capitol Hill as a legislative analyst for a Member on the House Ways and Means Committee.
Kathryn received her undergraduate degree at Stanford University. She received her Master's Degree in Public Policy from USC, under academic scholarship. For her time in State service, she received a career recognition award and was selected to represent California in attending Harvard University's Program for Senior Executives in State Service at the JFK School.
Brett Martinez is President & Chief Executive Officer of Redwood Credit Union, the North Bay's largest credit union. Mr. Martinez oversees all aspects of the Credit Union, which serves 230,000 Members with $2.2 billion in assets.
League's Annual Meeting and Conference.
the Santa Rosa Chamber of Commerce and the North Bay Leadership Council, and he was the Santa Rosa Chamber of Commerce Board Chair in 2012. Over the past four years, Mr. Martinez has received the United Way of the Wine Country Executive Campaign Chair of the Year award, was presented with the "Caught in the Act of Leadership" award by the North Bay Leadership Council, served as co‐chair of the 2010 Start! North Bay Heart Walk and was recognized as the North Bay's Best Community Business Leader by readers of Northbay biz magazine. Under Mr. Martinez's leadership and direction, Redwood Credit Union has received the North Bay Business Journal's "Best Places to Work" award for eight consecutive years and has repeatedly been voted "Best Credit Union" by readers of numerous North Bay publications.
tennis, boating and spending time with his family.
Meagan Moore currently holds the position as Chief Administrative Officer at the Buck Institute for Research on Aging. She joined the Buck in January of 2016 as Director of Sponsored Programs. In November of 2016, she was promoted to her current role. Prior to joining the Buck, Ms. Moore spent 11 years at the Gladstone Institutes in San Francisco in various roles, including as the head of the Contracts & Grants Department. Although not a scientist by training, Ms. Moore is dedicated to the mission of biomedical research. She is thrilled at the opportunity to continue in the tradition of the Buck as a world renowned leader in the field of aging and age related diseases. Meagan lives in San Rafael with her husband, a Marin native, and their two young children.
Mr. Perry practices in the areas of land use, real estate and business law. He represents clients before various administrative and planning agencies, including County of Sonoma, City of Santa Rosa, Town of Windsor, City of Petaluma, City of Rohnert Park, City of Sonoma, City of Cotati, City of Sebastopol, Sonoma County Airport Land Use Commission and Water Quality and Air Quality Boards.
Mr. Perry currently is a member of the Board of Trustees of Memorial Hospital; a member of the Sonoma County Harvest Fair (past president); a member of Santa Rosa Mainstreet Board of Directors (past president); a member of Business Advisory Assessment District (vice-chairman); and a member of the Sonoma County Volunteer Center Board of Directors (past president).
Mr. Perry was formerly a member of the Audubon Canyon Ranch Board of Directors; Sonoma County Planning Commissioner 1986-1995, Chairman 1987 and 1992; a member of the Sonoma County Emergency Medical Care Advisory Committee; Sonoma County Energy Advisory Committee; Northern Sonoma County Air Pollution Control Board; Sonoma County Civil Service Commission; member of the Board of Directors of Burbank Housing Development Corporation; Member of the Board of Director of the Sonoma County Fair (past president).
Mr. Perry attended the University of Nevada, Reno for two years. He received a bachelor of science degree in economics from University of California, Davis, in 1971 and a Juris Doctorate from the University of California, Davis (King Hall) in 1974. Mr. Perry attended "Mediation of the Litigated Case" at Pepperdine University.
Todd Salnas was appointed President of Providence St. Joseph Health in Sonoma County on July 1, 2012. He joined St. Joseph Health as Chief Operating Officer in September, 2009.
Mr. Salnas holds a dual master's degree in Business Administration and Health Sciences Administration from Arizona State University and a bachelor's of Science degree in Health Science from the University of Arizona.
Mr. Salnas is a native of Milwaukee, WI, and lived in Arizona for more than 20 years before moving to Chino Hills, CA, in 2000. Todd, his wife, Jenny, and their three young children moved to Sonoma County in September of 2009.
Silver's resume is impressive. He is an educator on entrepreneurship and innovation; a published author on a book — "The Way of Zing" — about aligning one's work with their life; a rainmaker who helped raised $11 million for SSU's Wine Spectator Learning Center, which will open in May and house its Wine Business Institute; and a consultant and advisor who has served on the boards of the Santa Rosa Chamber of Commerce and Sonoma County Vintners, where he played a major role in luring the county's signature Taste of Sonoma wine-and-food event to the university's Green Center last year.
Vela has 16 employees in an office that serves public and private companies with annual revenues of $20 million or more.
Most recently, Vela served as senior vice president and loan team manager for North Coast and San Francisco offices. Previously with Comerica Bank, Vela joined Wells Fargo in 2011 as a loan team manager. His financing knowledge made the San Francisco regional office one of the top performing teams in Wells Fargo Middle Market Commercial Banking.


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You Are the Colours You Choose
- April 11, 2015 December 9, 2015 0
According to the scientific definition, colour is the visual perception of a narrow band of electromagnetic waves, which reach the eye of an observer, who recognizes an attribute. This reduced definition was roughed out after Newton's research on optics in the seventeenth century. Newton's theory of colour affirmed that the white light is made of differently refrangible rays separated out into the rainbow by refracting to different degrees in the prism, creating the phenomenon called Dispersion.
The visible light spectrum is a very small part of the bigger electromagnetic spectrum, which is made of seven different regions, including x-rays, infrared, and ultraviolet light. This visible light region consists of a spectrum of wavelengths that range from approximately 700 nanometers (nm) to approximately 400 nm. The low-frequency end corresponds to the light wave that we perceive as the colour red, while the high-frequency end is related to what we identify as violet.
Colour is not only a physical but also a psychic reality as it supports the individual integration between the personality and the essence, or soul, of each person. The chromatic past has an incredible importance in the life of each of us and, when we look at the way of perceiving colours and how we use colours, we understand the nature and the development of consciousness of each human being.
The theosophical view sees colour as relating to the seven rays, which are qualities emanating from the Divine source. All kingdoms in nature are conditioned by the rays, and every individual in the human kingdom vibrates at the same rate of one of them.
"Each vibrating unit of energy can say: I am part of a divine whole, which in its sevenfold nature expresses the love and life of the One Reality, coloured by one of the seven qualities of the love of Deity and responsive to the other qualities. Each of these seven streams or emanations of energy was coloured by a divine quality, an aspect of love, and all of them were needed for the ultimate perfecting of the latent and unrevealed purpose." Alice Bailey
Every human being comes into manifestation on the impulse of a ray, and is coloured by that ray quality, which determines the form and indicates the soul path. Each ray expresses itself in different physical and energetic conditions and qualities. Each ray gives a predisposition to different talents, strengths and weaknesses. The soul and the personality shifts from ray to ray, from life to life, until all qualities, which are the expression of the Divine source at the core of each of us, are developed.
Colour is a universal language, but even more a symbolic and hermetic language. The symbol, from the Greek symbolon, means "joining together again" two parts that originally were one: these two parts are the content to be symbolized and the object that embodies it. A symbol is an object present in our consciousness, while the content is hidden in our subconscious/unconscious mind. Each symbol is an attempt to express the inexpressible, it is full of meanings and presentiments and it stimulates both feelings and intuition.
Carl Jung, one of the founder of analytic psychology, explained that a symbol has a wider "unconscious aspect that can never be defined or fully explained, it is "a process of comprehension by means of analogy", "a product of an extremely complex nature, since data from every psychic function have gone into its making", they "are images of contents which for the most part transcend consciousness." He also described symbol as "the living facts of life," a "container of meaning," a "god-image," a "form of intervention from the unconscious", "the psychological mechanism that transforms energy," and as "natural attempts to reconcile and reunite opposites within the psyche".
"What we call a symbol is a term, a name or an image which in itself may be familiar to us, but its connotation, use and application are specific or peculiar and hint at a hidden, vague or unknown meaning." Carl Jung
Equilibrium B54
Working with colour and light means to work with a tool, which may bring us closer to the understanding of who we are, finding the potential which can be expressed by the combination of the two rays manifested in each Aura-Soma bi-coloured Equilibrium bottle.
These bottles are the core of the system and the two fractions are a reflection of our personality ray – the top fraction – and our soul ray – the bottom fraction.
"You are the colour you choose and these reflect your being's needs" Vicky Wall, founder of the system used to say. What we need is to re-member the memories, but also the potential and the talents that each of them conveys to us.
As W. Kandinsky wrote in 1910, "Colour is a power which directly influences the soul", it helps us recognise ourselves at the deepest level through the key of colour and to re-balance our energy field.
Knowing the meaning of colour means to get to know ourselves deeper and to open the window to the soul we all have been longing for.
"Wander within the Jewels before you. They have a Language, learn it. They are a Book within themselves; read it and know that in the end they are a Mirror held before you, and Yours is the Miracle."
Vicky Wall
For those of you who are interested either in an Aura-Soma consultation or in the Aura-Soma Training, please contact us by filling up the form below:
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REPUBLIC OF CYPRUS: AHMED H. ALLOWED TO RETURN HOME
AHMED H. ALLOWED TO RETURN HOME
Update info:
24 Oct 2019 (Suspended)
Latest info:
Ahmed H. (he/his, full name withdrawn to protect the family's privacy)
Distribution date:
UA No:
Ahmed H. has been separated from his Cypriot wife and two daughters for almost four years. In September 2015, he was imprisoned in Hungary and wrongfully convicted for "complicity in an act of terrorism" in a blatant misapplication of Hungary's counter-terrorism laws. Ahmed H. was conditionally released on 19 January 2019 and is being held in immigration detention in Hungary. As he is a Syrian national he is at risk being forcibly returned to Syria, a country that is not safe. Cyprus must allow his return home to be reunited with his family.
Ahmed H. lived in Cyprus since 2006 with his Cypriot wife. They have two young daughters. In August 2015, he left his home in Cyprus to help his elderly parents and other family members escape Syria. Upon arrival at the Hungary-Serbia border on 16 September 2015, they found themselves among hundreds of refugees stranded there after police fenced off the crossing with Serbia. Clashes broke out as some refugees attempted to get through. Hungary's police responded with tear gas and water cannon, injuring dozens. Some people threw stones, including Ahmed H. But news footage also clearly shows Ahmed H. using a megaphone to call on both sides to remain calm. For these acts, he was arrested and eventually convicted of "complicity in an act of terrorism" under Hungary's overly broad and vague counter-terrorism law.
Amnesty International closely followed the criminal proceedings, including monitoring the trial hearings, and concluded that the application of terrorism-related charges and Ahmed H.'s subsequent conviction for "complicity in an act of terrorism" were manifestly unfounded. The conviction rests entirely on the allegation that Ahmed H. attempted to induce the Hungarian police to open the border fence, and that he threw some objects during the disturbance at the border. While delivering the final judgement on 20 September 2018, the judge acknowledged that Ahmed H. had in fact tried to calm the crowd and mediate between the police and the crowd.
The government of Prime Minister Viktor Orbán in Hungary has sought – without evidence to back up their claims – to establish an inextricable link between refugees and migrants and the threat of terrorism. The misapplication of terrorism-related charges to Ahmed H.'s case was part of a wider effort to label refugees and migrants seeking safety as "terrorists" and threats to national security. Amnesty International has also documented statements by Hungarian government representatives that publicly pronounced Ahmed H.'s guilt before his trial even ended. Such statements compromised the fairness of the proceedings against Ahmed H. in violation of international fair trial standards, which include the presumption of innocence.
The right to private and family life is enshrined in Article 17 of the International Covenant on Civil and Political Rights and Article 8 of the European Convention for the Protection of Human Rights and Fundamental Freedoms. Cyprus is a party to both these international human rights treaties. Any restrictions on this right must be in the interests of "national security, public safety or the economic well-being of the country, for the prevention of disorder or crime, for the protection of health or morals, or for the protection of the rights and freedoms of others". The conviction of Ahmed H. in Hungary for "complicity in an act of terrorism" does not fall within the national security exception to the right to family life because of the mis-application of terrorism charges to his case and the violations of Ahmed H.'s right to fair trial during the criminal proceedings in Hungary.
If Hungary were to return Ahmed H. to Syria it would be in violation of the principle of non-refoulement. This international human rights law principle guarantees that no one should be returned to a country where they would face torture, cruel, inhuman or degrading treatment or punishment and other irreparable harm.
PREFERRED LANGUAGE TO ADDRESS TARGET: Greek and English
You can also write in your own language.
PLEASE TAKE ACTION AS SOON AS POSSIBLE UNTIL: 9 August 2019
Please check with the Amnesty office in your country if you wish to send appeals after the deadline.
NAME AND PREFFERED PRONOUN: Ahmed H. (he/his, full name withdrawn to protect the family's privacy)
Ahmed H. was finally able to return home to his family in Cyprus on 28 September. He had been separated from his Cypriot wife and two daughters for almost four years, after being wrongfully convicted in Hungary.
NO FURTHER ACTION IS REQUESTED. MANY THANKS TO ALL WHO SENT APPEALS.
Ahmed H. was permitted to return home to his family in Cyprus after spending almost four years separated from his wife and two daughters. On 26 September he was transferred from immigration detention to Budapest airport in Hungary and boarded a flight home the next day. He landed in Cyprus in the morning of 28 September and received an emotional welcome from his family.
In September 2015, Ahmed H. was imprisoned in Hungary and wrongfully convicted for "complicity in an act of terrorism" in a blatant misapplication of Hungary's counter-terrorism laws. Ahmed H. was conditionally released on 19 January 2019 and was held in immigration detention in Hungary until his release in September.
The Cyprus authorities received thousands of letters and emails from Amnesty International activists prior to Ahmed's return. Ahmed and his wife expressed their gratitude to all those who supported them.
NAME AND PREFFERED PRONOUN: Ahmed H. (he/him)
THIS IS THE SECOND AND FINAL OUTPUT FOR UA 88/19
LINK TO PREVIOUS UA: https://www.amnesty.org/en/documents/EUR17/0610/2019/en/
See here also
Copyright(C) 2001-2012 AMNESTY INTERNATIONAL JAPAN


text:
Virometix Broadens Senior Leadership Team with Three Key Appointments
Jan. 7, 2021 14:00 UTC
SCHLIEREN, Switzerland--(BUSINESS WIRE)-- Virometix, a privately held Swiss biotechnology company developing a new generation of vaccines and immunotherapeutic drugs for the prevention and treatment of infectious and oncology diseases, today announced the expansion of its leadership team with the appointment of a Chief Scientific Officer and two additional Board members.
Virometix filled a vacancy in its management team with the appointment of Dr. Lilli Stergiou to the position of Chief Scientific Officer (CSO). Lilli has 20 years of research and development experience in multiple fields including all areas of interest to Virometix – vaccines, infectious diseases, and oncology. She has a PhD in Molecular Biology and Genetics from the University of Zurich. Lilli joins Virometix from Redbiotec AG, where she was CSO, headed the HSV-2 Therapeutic Vaccine Program and built a pipeline based on live biotherapeutics for applications in oncology.
Moreover, Susanne Schultz-Hector, PD Dr. Med and Jan ter Meulen, MD, Dr. habil., DTM&H were elected to the Board of Directors of Virometix in November 2020.
Susanne Schultz-Hector is an independent university lecturer and member of various boards. She has deep experience in the field of radiation biology and has published 27 original papers as first or corresponding author in international, peer reviewed journals. She conducted research both in Germany and in the USA.
Jan ter Meulen is a seasoned biotech executive with broad experience in immuno‐oncology, vaccines, recombinant antibodies, and infectious disease R&D, from discovery research through to phase 3 clinical trials. His career spans over various functions both in academia and business, including Crucell and Merck & Co., where he headed vaccine basic research.
Anna Sumeray, CEO of Virometix, said: "Lilli's strong scientific background and her industry experience add critically important competencies to the leadership of our research programs at Virometix. I am delighted to welcome Lilli to the Virometix team."
Harry Welten, Chairman of the Virometix Board of Directors, said: "We are very pleased to welcome the three new members of our senior leadership team and look forward to their contributions to our success."
About Virometix
Virometix AG is a privately held Swiss biotechnology company developing a new generation of vaccines and immunotherapeutic drugs for the prevention and treatment of infections and cancer. In an increasingly global world, there is a considerable medical need for vaccines to combat infectious as well as a number of chronic human diseases, including cancer. Rational molecular design, chemical synthesis and Virometix' proprietary "Synthetic Virus-Like Particle" platform technology allow for the rapid production and optimization of vaccine candidates with the potential to demonstrate superior properties in terms of safety, efficacy and stability.
For more information about Virometix AG please visit: https://www.virometix.com
Anna Sumeray CEO Virometix AG
Email: info@virometix.com
Source: Virometix


Arrrr Mateys! Do you have an extra little nook or closet in your house that is just collecting dust and clutter? I have the perfect solution for you today: A closet Pirate Hideout!
My boys share a room with two closets, but only one was being used for clothes and the other was being used to hide their messes. Between that, and the fact that the doors kept coming off I decided to do away with the second closet, and turn it into something more useful for my playful little guys.
After removing the doors I also removed the bottom track. After that I painted a fun little ocean scene on the wall. I was going for a children's book illustration effect.
Next, I cut out a wheel shape from a cardboard box and painted it. I super glued 1/2" washers to both sides so the cardboard wouldn't wear down as the wheel was turned. Finally, I secured it to the wall with a bolt, and a large nut behind so it wouldn't be flush with the wall. This is one of my favorite features because the wheel really turns!
Next came the flag. It just isn't a true Pirate Hideout without the Jolly Roger. I used a $.97 fat quarter from Walmart and a $.30 piece of white felt. I learned the greatest trick on Pinterest: If you put contact paper on the felt and then cut out your shape it is 100 times easier!! I used Fabric Fusion to attach the felt to the black fabric and let it dry. I like the frayed look, so I didn't bother hemming the edges.
I already made the pirate crate seats a few months ago, so they were ready to go.
Finally, I bought two decorative fish netting from the Dollar Tree and hung them as curtains. They gave it the perfect finishing touch.
A few pirate hats and a treasure map that I crocheted made perfect accessories. I'm working on a purple Pirate Princess hat for Samantha.
This was such a fun and rewarding project to do for my kids, and has already helped cut down on the bedroom clutter. Best of all, I had all the supplies on hand except the netting, so it only cost me $2 to make!
I bet the kids have so much fun!!
Oh, Wow! I have a few grandsons that would LOVE a special place like this. You did a great job adding perfect pirate details. So nice stopping by to visit!
I am just wondering if men like ladies more or what.
guy's rooms. Why may be the net filled with horny guys but no girls?
I can't believe you _crocheted_ the map! Insanely more complicated than drawing one but i have to say i love the one you did. Wish you had better details on making the cardboard wheel shape. Definitely better than what we started to make, but then again you used a ruler and probably a compass. Details please..
Had me exercising those kids from the day they were born, Mike mom, Toni Zadick, said. Their arms and their legs. The girls as well as the boys. The final key to the way I promote is bravado. I play to people's fantasies. People may not always think big themselves, but they can still get very excited by those who do.
I contacted him and i told him my same color t shirt problems and gave him all the urban t shirts necessary information he required. He just laughed and told me never to worry that my husband will call me back in less than 28hours and i believed and he told me what to do and i did it, and after he has finished casting the spell my husband called me and started begging. He apologized and came back to me crying for a second beard shirt chance.


text:
york place
Early infill housing in New Town
Not to be confused with York Place off Kingston, York Place in New Town was infill housing at the very northern end of Kiddle's Lane (today's Eastland Road). It was built either side (four houses to the north and two to the south) of the Primitive Methodist Chapel (now the Kingdom Hall of the Jehovah's Witnesses) in 1890 - see Gallery for notice of sale. Shown as a Mission Room on the maps below, the chapel was built in 1891 on the site of a smaller mission hall. At this time, apart from the Mission Hall, this eastern side of Kiddles Lane was not built on between Agra Place, on the corner of Gordon Road, and North Terrace.
By the time of the 1891 Ordnance Survey, the Mission Hall had been considerably enlarged, if not completely rebuilt, and the four houses to its north and two to its south that formed York Place had also been newly-built.
This map (stitched together from pretty poor copies - apologies) is the 1886 Ordnance Survey of New Town. At this time York Place had not been built and the Mission Hall was about to be enlarged.
Map based on the 1901 Ordnance Survey showing the different speculative housing projects built in the early days of New Town.
Most of these developments have their own page -
Agra Place, Clifton Terrace, Jubilee Terrace, Mount Pleasant Terrace, North Terrace, Smith's Terrace, Swastika Terraces, Wolverton Terrace, York Place,
Newly-erected dwellings in Morton Terrace, Smith's Terrace and York Place advertised for sale in the 2 May 1890 edition of the Western Gazette. It was reported in the Western Gazette the following week "There was a good attendance, but every lot was withdrawn, the reserve price not being reached."
York Place seen from North Terrace - this is at the very northern end of Eastland Road. Photographed in 2014.
York Place are the four houses beyond the Methodist Chapel, now the Kingdom Hall, as well as two houses on the downhill side. Photographed in 2014.


text:
Roma Abbey () is a ruined Cistercian abbey and a crown estate in Roma on the Swedish island of Gotland. The abbey was built in the 12th century. After the Reformation, its lands were confiscated by the Crown and subsequently turned into a crown estate. Apart from the ruined church, the estate includes a manor house built in 1733 for the crown estate, known in Swedish as Roma kungsgård. The manorial cultural landscape with avenues and large fields surrounding the ruined church and the manor is unique on Gotland.

History

Roma Abbey is located in an area with old cultural traditions. The thing for all of Gotland was located on this strategical location on the middle of the islands, possibly as early as the Viking Age. Usually during the Middle Ages, abbeys like Roma were founded on land donated by the land-owning aristocracy, and the absence of any such group on Gotland may explain the location of the abbey. The land around the thing was common property and could therefore be used for founding an abbey. The abbey was founded in 1164 by Cistercian monks from Nydala Abbey in Småland on mainland Sweden. Its name in Latin was Sancta Maria de Gutnalia. Eventually the abbey grew in importance and acquired large landholdings both on Gotland and in the present-day Baltic states. The abbey had particularly large holdings on Saaremaa island and in present-day mainland Estonia.

During the Reformation, the abbey was disbanded and its lands on Gotland confiscated by the Danish Crown circa 1531 (as Gotland at the time belonged to Denmark). The abbey was converted into the seat of the Danish governor, and the buildings that no longer served any direct purpose were left to decay. After the Peace of Brömsebro in 1645, when Gotland became Swedish, it became the property of the Swedish Crown. In practice, the change of ownership meant few tangible changes for the estate. Towards the end of the 17th century, the Swedish Crown had put the land out for rent as a leasehold estate. In 1728 it befell the governor of Gotland, Johan Didrik Grönhagen. He constructed the manor house, partially using building material from the abbey.

During World War II, a military airfield was established nearby and the manor house and its annexes were used by the military up until the 1990s. During this time the abbey suffered further neglect and damage.

Today the abbey and the manor are owned by the National Property Board of Sweden. During summer theatre plays are staged among the ruins, a tradition that was initiated in 1989 and today attracts as many as 25,000 visitors every year, according to the organisers.

Architecture

Manor
The manor house, unique in its kind on Gotland, was modelled after the manors of the aristocracy on the mainland. Building material was taken from the ruined abbey, and a few medieval portals were incorporated in the new building.

Abbey
Although the abbey church has been ruined and neglected for several centuries, it can still be appreciated as a typical representative of Cistercian architecture. The carefully proportioned church belongs to a type of church which is today best represented by the Abbey of Fontenay in France. The church was a basilica with a single nave flanked by two aisles. It was divided by a transept and instead of an apse had a straight east wall. In each of the transept arms there were two chapels. The church is characterised by a neat treatment of materials and a sobriety in the choice of decorations.

See also
 List of church ruins on Gotland
 Churches on Gotland

References

External links
 Official site
 Official site of Roma Theatre (in Swedish)

Cistercian monasteries in Sweden
Christian monasteries established in the 12th century
1530s disestablishments
Manor houses in Sweden
Medieval Sweden
Monasteries dissolved under the Danish Reformation
Ruins in Sweden

redpajama_set_name: RedPajamaWikipedia


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text:
Home » Impacts of exotic invertebrates on New Zealand's indigenous species and ecosystems
Impacts of exotic invertebrates on New Zealand's indigenous species and ecosystems
Eckehard G. Brockerhoff 1*
Barbara I.P. Barratt 2
Jacqueline R. Beggs 3
Laura L. Fagan 4
Malcolm K. (Nod) Kay 5
Craig B. Phillips 6
Cor J. Vink 6
Scion (New Zealand Forest Research Institute), PO Box 29 237, Christchurch 8540, New Zealand
AgResearch Invermay, Private Bag 50 034, Mosgiel, New Zealand
School of Biological Sciences, Tamaki Campus, University of Auckland, Private Bag 92 019, Auckland, New Zealand
Plant & Food Research, Private Bag 4704, Christchurch, New Zealand
Scion (New Zealand Forest Research Institute), Private Bag 3020, Rotorua 3010, New Zealand
Biosecurity Group, AgResearch, Lincoln Science Centre, Private Bag 4749, Christchurch 8140, New Zealand
Biological invasions have significantly affected New Zealand's native species and ecosystems. Most prominent are the effects of exotic mammals and plants, whereas few invertebrate invasions are known to have major effects on native ecosystems. Exceptions are the well-known cases of Vespula wasps in Nothofagus forest ecosystems and Eriococcus scale insects in Leptospermum shrublands. This limited impact is surprising because over 2000 exotic invertebrates have become established in New Zealand, among them many pests of exotic crop plants. The low impact of exotic invertebrates that invaded forests and other native ecosystems in New Zealand is in contrast to the situation in other parts of the world where many invertebrates have become important pests. We provide an overview of known invasions by exotic invertebrates in New Zealand, and explore in more detail several examples of invasive species, including herbivores, predators, parasitoids, decomposers and other groups in forests, grasslands, and other terrestrial ecosystems. Several hypotheses have been proposed to explain the comparative scarcity of such invasions that affect New Zealand's indigenous ecosystems. There is a common view that New Zealand's native species and ecosystems are inherently resistant to exotic invertebrate invaders, and there is some evidence to support this view. As a result of the high level of endemism in New Zealand's flora, many native plants are phylogenetically distant from the host plants of many plant-feeding invaders. This provides some protection. Less host-specific plant-feeding insects, generalist predators, parasitoids and decomposers are less affected by such constraints, and these groups are perhaps more represented among the successful invaders of natural ecosystems. However, the shortage of studies on invader impacts on native species and ecosystems, compared with studies on economically important crops and production ecosystems, means that an unbiased comparison is not possible at this time. Furthermore, many invaders go through extended lag phases where their impacts are not easily noticed until they become more abundant and create more damage. Likewise, indirect effects of invaders, through more complex interactions in food webs, as well as impacts on ecosystem functions such as decomposition and pollination, are more subtle and difficult to detect without careful study. There is clearly a need for more research to determine more accurately which exotic invertebrates are already present, what their direct and indirect impacts are, and what generalisations and predictions about threats to native species and ecosystems are possible.
biological invasions, herbivores, hypotheses, invasibility, parasitoids, predators, resistance


Kendriya Vidyalaya Bundi opened in April, 2015, situated in the heart of Bundi. School has opened in building of Govt. Upper Primary School, Rajat Grih, Bundi (Temporarily provided building) in the sponsorship of District Administration, Bundi, with all facilities required for the Vidhyalaya.
Staple aim of our Vidyalaya is to build character of Students, inculcate feelings of fortitude and also sharpen their minds to face all the challenges fearlessly that come in their life in this cut throat competitive age. Vidyalaya has opened with one section up to Class I to V in 2015-16.


Dr. Rajesh Kumar completed his MBBS few years back and now pursuing MD from a renowned Institute. Writing is his passion and to entertain his readers with his stories is his aim. After getting so much of love from the Readers on some of the largest story pages he decided to get it to next level. It's his first publication with one of the best publishing house of the country. Soon he will be publishing his next book, so Stay connected, Stay blessed.


text:
Nanorana minica är en groddjursart som först beskrevs av Dubois 1975.  Nanorana minica ingår i släktet Nanorana och familjen Dicroglossidae. IUCN kategoriserar arten globalt som sårbar. Inga underarter finns listade i Catalogue of Life.

Källor

Externa länkar

Stjärtlösa groddjur
minica


text:
\section{Introduction}\label{sec:introduction}
The distribution network was engineered under the assumption that homes and businesses were to be supplied power from the grid. However, with increasing penetrations of solar PV in distribution feeders has created so-called ``pro-consumers'' who (at times) supply the grid with energy rather than consume it. This can lead to a back-flow in power and can result in violations of voltage and transformer constraints and unreliable operation of the system~\cite{driesen2006distributed,bayer2018german}. Furthermore, the significant variability inherent to solar PV  challenges traditional operating paradigms and represents an opportunity for optimal power flow algorithms to improve reliability and responsiveness of the grid and its flexible resources (e.g., batteries, PV inverters). However, due to the fast timescale of the solar PV variability, these algorithms must be computationally tractable and, yet, representative of the physics. That is, grid optimization algorithms must ensure admissible network operations~\cite{kristov2016tale}.

The optimal power flow problem (OPF) is a powerful method for dispatching grid resources while maintaining network constraints.
The AC power flow equations, however, are nonlinear and non-convex, making the optimization problem NP-hard. This paper proposes a convex approximation of the power flow equations that results in a network-admissible solution, i.e., all physical network limits are respected at (global) optimality, while solving in polynomial time. Hence, the  method is robust against modeling errors introduced from approximations of the non-linear power flow equations. This characteristic is useful in many applications, such as to determine the admissible range for dispatching distributed energy resources (DERs), i.e., the \textit{DER hosting capacity}~\cite{nazir2019convex}. Another application is that of disaggregating the dispatch of a large, aggregated virtual energy storage (VES) resource into the dispatch of multiple smaller nodal resources that are distributed spatially across a network. The disaggregation must then guarantee that the resulting nodal dispatch is network admissible as shown in Fig.~\ref{fig:cyber_physical_model}.

\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{cyber_physical_model.png}
\caption{\label{fig:cyber_physical_model} A schematic representation of the network model. The physical layer represents the circuit that connects the different DER groups into an aggregate virtual battery, whereas the cyber layer represents the disaggregation of the virtual battery market signal to the DER groups based on the feasible nodal bounds that are determined offline.}
\end{figure}

Traditional optimization techniques for dispatching resources include linear OPF based \textit{LinDist} models~\cite{baran1989optimal}. These models work well close to the expected conditions of the system (e.g., lossless). In~\cite{nazir2019convex}, it is shown, using a simple example, how solutions of the \textit{LinDist} model can lead to voltage violations under some operating conditions. Recently, improved linear approximations of the power flow equations have been proposed that provide improved accuracy over a wider range of operation~\cite{yang2016optimal}. However, the solution space of the power flow equations is highly non-convex, which means that such methods cannot provide guarantees of feasibility under all conditions. A comprehensive review of the many linear power flow formulations can be found in~\cite{stott2009dc}.

Recently, lot of attention has been given to the convex relaxation of the power flow equations.
The main among these in the literature include the semi-definite programs (SDP)~\cite{lavaei2012zero}, second order cones~\cite{jabr2006radial}, and the quadratic relaxations~\cite{coffrin2016qc}. These convex relaxations provide a lower bound on the globally optimal AC solution. Several works in literature such as~\cite{gan2015exact} have shown that under some conditions these relaxations can be exact and the solution of the relaxed convex problem is the global optimum of the original AC OPF problem. However, these conditions fail to hold under reverse power flows from extreme solar PV, which engenders a non-zero duality gap and a non-physical solution~\cite{huang2017sufficient}.

The shortcomings of the linear OPF and the convex relaxation-based OPF techniques call for a different approach to solving the distribution system OPF problem under a high peneration of solar PV. In many applications, feasibility guarantees at optimality are more valuable than solving to a globally optimal solution. The authors in~\cite{molzahn2019grid} provide a certificate test to determine whether grid-aware control is required for a network based on load variabiliby and the availability of measurements. However, they do not quantify the feasible region of safe operation. The paper herein presents a technique for quantifying the feasible region through a convex approximation of the OPF problem. Obtaining a convex inner approximation in general is an NP-hard problem~\cite{chang1986polynomial}. This work uses the branch-flow model (BFM) formulation of the distribution power flow equations to separate the model into linear and nonlinear components. Then, bounds on the nonlinearity are determined through approximations that employ the properties of the power flow models. The bounds then define a modeling error buffer, which is used to formulate the network constraints into a robust formulation. From this approach, we achieve an OPF formulation that exihibits computational solve times similar to that of linear formulations with the added (and crucial benefit) that the formulation provides feasible solutions (assuming one exists). The convex formulation is then tested on distribution feeders with extreme penetrations of solar PV (i.e., under reverse power flows). We also show that the approach is suitable for siting and sizing solar PV systems and DERs that abide by the network constraints (i.e., solar PV and DER hosting capacities).

The main contribution of this paper can be summarized as:
\begin{enumerate}
    \item Improving over our work in~\cite{nazir2019convex}, this paper provides a convex inner approximation of the OPF problem with tighter bounds that considers the apparent branch power flow and current constraints, in addition, to the voltage constraints. This approximation is valid even under extreme solar PV penetrations.
    \item A iterative bound tightening algorithm is presented that tightens the bounds on the nonlinear modeling errors.
    \item The DER hosting capacity problem introduced in~\cite{nazir2019convex} is extended to consider energy-constrained DERs over multiple time-steps and validated with a short case-study on available flexibility from small-scale distributed storage versus bulky centralized storage.
\end{enumerate}

The rest of the paper is organized as follows.
Section \ref{sec:Opt_form} develops the mathematical formulation of the inner convex approximation OPF problem. Section~\ref{sec:bounds} provides a method to obtain improved upper and lower bounds of the nonlinearity, which is then used to determine the amount of solar capacity that can be installed in a network. Section~\ref{sec:Iter_alg} presents an iterative algorithm that provides tighter bounds in order to obtain a better feasible solution, whereas section~\ref{sec:multi_period} describes the multi-period extension of this formulation and shows the benefit of having storage through various case studies. Finally, section \ref{sec:conclusion} concludes the paper and lays out future research directions.

\section{Formulation of the Convex Inner
Approximation}\label{sec:Opt_form}
The proposed approach in this paper, takes a worst case of the non-linear terms in the power flow equations and develops a feasible model that is robust against modeling errors due to the nonlinearity. This means that the technique ensures node voltages, branch power flows and current magnitudes to be within their limits for the determined optimal solution, while at the same time keeping the convex form of the formulation.
\textit{DistFlow} equations as given in~\cite{baran1989optimal} are used to solve the optimization problem.
However, \textit{DistFlow} equations are non-linear which takes the problem outside the realm of convex optimization. Linear \textit{LinDist} models are often used, but they are accurate only close to nominal voltages (via Jacobian linearization) and/or when losses are negligible (via simplifying assumptions). When system operations depart from nominal voltage conditions, the modeling errors can become significant. In this work, we develop a feasible convex formulation by modifying the \textit{LinDist} equations to take into account modeling errors arising from the approximation. However, the techniques presented here can further be extended to other single phase and multi-phase linearized power flow models in literature such as~\cite{bernstein2017linear}.

\subsection{Mathematical model}\label{sec:math_model}
Consider a radial distribution network, shown in Fig.~\ref{fig:radial_network}, as a graph $\mathcal{G}=\{\mathcal{N}\cup\{0\},\mathcal{L}\}$ consisting of $\mathcal{N}:=\{1,\hdots,|\mathcal{N}|\}$ nodes and a set of $\mathcal{L}:=\{1,\hdots,|\mathcal{L}|\} \in (\mathcal{N}\times \mathcal{N})$ branches, such that $(i,j)\in \mathcal{L}$ if $(i,j)$ are connected, where $|\mathcal{L}|=|\mathcal{N}|+1$ for a radial network. Node $0$ is assumed to be the substation node with a fixed voltage $V_0$. Let $B\in \mathbb{R}^{(n+1)\times n}$ be the \textit{incidence matrix} of the undirected graph $\mathcal{G}$ relating the branches in $\mathcal{L}$ to the nodes in $\mathcal{N}\cup \{0\}$, such that the entry at $(i,j)$ of $B$ is $1$ if the $i$-th node is connected to the $j$-th branch and otherwise $0$. If $V_i$ and $V_j$ are the voltage phasors at nodes $i$ and $j$ and $I_{ij}$ is the current phasor in branch $(i,j)\in \mathcal{L}$, then $v_i=|V_i|^2$, $v_j=|V_j|^2$ and $l_{ij}=|I_{ij}|^2$. $P_{i}$ be the real power flow from node $i$, $Q_{i}$ be the reactive power flow from node $i$, $p_i$ be the real power injection and $q_i$ be the reactive power injection, into node $i$, $r_{ij}$ and $x_{ij}$ be the resistance and reactance of the branch $(i,j)\in \mathcal{L}$ and $z_{ij}=r_{ij}+jx_{ij}$ be the impedance. Then for the radial distribution network, the relation between node voltages and power flows is given by the  \textit{DistFlow} equations $\forall (i,j)\in \mathcal{L}$:
\begin{align}
v_j=&v_i+2r_{ij}P_{j}+2x_{ij}Q_{j}-|z_{ij}|^2l_{ij} \label{eq:volt_rel}\\
P_{i}=&P_j+p_i-r_{ij}l_{ij} \label{eq:real_power_rel}\\
Q_i=&Q_j+q_i-x_{ij}l_{ij} \label{eq:reac_power_rel}\\
l_{ij}=&\frac{P_j^2+Q_j^2}{v_j}. \label{eq:curr_rel}
\end{align}

\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{radial_net_model.png}
\caption{\label{fig:radial_network} Diagram of a radial distribution network from~\cite{heidari2017non}.}
\end{figure}

Clearly, the line losses in~\eqref{eq:curr_rel} are nonlinear, which makes the \textit{DistFlow} model nonlinear. In the remainder of this section, we develop a mathematical model of the radial network that expresses the constrained variables as a linear function of the power injections and the branch currents. Through this approach, we are able to separate the model into linear and nonlinear parts. This will form the basis for our approach at finding the worst case of the nonlinear terms, which will then provide the convex inner approximation of the power flow equations.

From the \textit{incidence} matrix $B$ of the radial network and following the method adopted in~\cite{heidari2017non}, \eqref{eq:real_power_rel} and \eqref{eq:reac_power_rel} can be expressed through the following matrix equations:
\begin{align}
    P=p+AP-ARl \qquad Q=q+AQ-AXl, \label{eq:P_matrix}
\end{align}
where $P=[P_i]_{i\in \mathcal{N}}$, $Q=[Q_i]_{i\in \mathcal{N}}$, $p=[p_i]_{i\in \mathcal{N}}$, $q=[q_i]_{i \in \mathcal{N}}$, $R=\text{diag}\{r_{ij}\}_{(i,j)\in \mathcal{L}}$, $X=\text{diag}\{x_{ij}\}_{(i,j)\in \mathcal{L}}$, $l=[l_{ij}]_{(i,j)\in \mathcal{L}}$ and $A=[0_n \quad I_n]B-I_n$, where $I_n$ is the $n\times n$ identity matrix and $0_n$ is a column vector of $n$ rows.
Simplifying \eqref{eq:P_matrix} leads to the following expression for $P$ and $Q$
\begin{align} \label{eq:P_relation}
    P=Cp-D_{\text{R}}l \qquad Q=Cq-D_{\text{X}}l,
\end{align}
where $C=(I_n-A)^{-1}$, $D_{\text{R}}=(I_n-A)^{-1}AR$, and $D_{\text{X}}=(I_n-A)^{-1}AX$.

\begin{remark}
The matrix $(I_n-A)$ is nonsingular since $I_n-A=2I_n-[0_n \quad I_n]B=2I_n-B_n$, where $B_n:=[0_n \quad I_n]B$ is the $n\times n$ matrix obtained by removing the first row of $B$. For a radial network, the vertices and edges can always be ordered in such a way that $B$ and $B_n$ are upper triangular with $\text{diag}(B_n) = 1_n$, which implies that $2I_n-B_n$ is also upper triangular and $\text{diag}(2I_n-B_n)=1_n$. Thus,   $\det(2I_n-B^{'})=1>0$ and $I_n-A$ is  non-singular.
\end{remark}

Similarly, \eqref{eq:volt_rel} can be applied recursively to the distribution network in Fig.~\ref{fig:radial_network} to get the following matrix equation:
\begin{align}\label{eq:volt_matrix_rel}
    [v_j-v_i]_{(i,j)\in \mathcal{L}}=2(RP+XQ)-Z^2l
\end{align}
where $Z^2=\text{diag}\{z_{ij}^2\}_{(i,j)\in \mathcal{L}}$. Based on the \textit{incidence matrix} $B$, the left hand side of \eqref{eq:volt_matrix_rel} can be formulated in terms of the fixed head node voltage as:
\begin{align}\label{eq:volt_transform}
    C^T[v_j-v_i]_{(i,j)\in \mathcal{L}}=V-v_{\text{0}} \mathbf{1}_n
\end{align}
where $V=[v_i]_{i\in \mathcal{N}}$. Based on \eqref{eq:volt_transform}, \eqref{eq:volt_matrix_rel} can be expressed as:
\begin{align}\label{eq:volt_matrix_2}
    V=v_{\text{0}} \mathbf{1}_n+2(C^TRP+C^TXQ)-C^TZ^2l
\end{align}
Substituting \eqref{eq:P_relation} into \eqref{eq:volt_matrix_2}, we obtain a compact relation between voltage and power injections shown below.
\begin{align}\label{eq:final_volt_rel}
    V=v_{\text{0}}\mathbf{1}_n+M_{\text{p}}p+M_{\text{q}}q-Hl
\end{align}
where $M_{\text{p}}=2C^TRC$, \quad $M_{\text{q}}=2C^TXC$ and \newline $H=C^T(2(RD_{\text{R}}+XD_{\text{X}})+Z^2)$

Apart from the nonlinear relation \eqref{eq:curr_rel} of $l$ to $P,Q$ and $V$, \eqref{eq:P_relation} and \eqref{eq:final_volt_rel} is a linear relationship between the nodal power injections $p,q$, the branch power flows $P,Q$ and node voltages $V$. The nonlinearity in the network is represented by~\eqref{eq:curr_rel}, as the current term $l$ is related to the power injections and node voltages in a nonlinear fashion. Including this term into the optimization model would render the optimization problem non-convex (and NP-hard), however, neglecting this term could result in an infeasible linear OPF model. In the next section, we present a method to take into account the nonlinearty introduced by $l$ by considering it's worst case values.

\subsection{Optimization problem formulation}\label{sec:Optimization_model}
The problem being addressed in this paper is to determine a convex inner approximation for the dispatch of energy resources that respects the network voltage and branch flow constraints.
Based on the mathematical model presented in Section~\ref{sec:math_model}, a general OPF problem can be formulated as:
\begin{subequations}\label{eq:P1}
\begin{align}\label{eq:P1_a}
\text{(P1)} \ \min_{p_g,q_g} & \ f(p_g,q_g)\\
\text{subject to}: & \phantom{m} \eqref{eq:P_relation}, \eqref{eq:final_volt_rel}, \eqref{eq:curr_rel} \label{eq:P1_b}\\
p= & p_g-P_L\label{eq:P1_c}\\
q= & q_g-Q_L\label{eq:P1_d}\\
  \underline{P}\le & P \le \overline{P}\label{eq:P1_e}\\
  \underline{Q}\le & Q \le \overline{Q} \label{eq:P1_f}\\
    \underline{V}\leq & V \leq \overline{V}\label{eq:P1_g}\\
    \underline{l}\leq & l \leq \overline{l}\label{eq:P1_h}
\end{align}
\end{subequations}

where $f(p_g,q_g)$ represents the convex objective of the OPF problem, e.g., this could be the typical objective of minimizing the cost of traditional power generators, i.e., $f(p_g,q_g)=\sum_i(c_{i1}p_{gi}^2+c_{2i}p_{gi})$, where $c_{1i}$ and $c_{2i}$ are the quadratic and linear terms associated with the cost of generator $p_{gi}$. $p_g \in \mathbb{R}^n$ and $q_g \in \mathbb{R}^n$ are the real and reactive generation from traditional generators and represent the decision variables, whereas $P_L \in \mathbb{R}^n$ and $Q_L \in \mathbb{R}^n$ are the expected real and reactive net-demand. $\underline{P}\in \mathbb{R}^n$ and $\overline{P}\in \mathbb{R}^n$ represent the lower and upper bounds on the real power branch flow and $\underline{Q}\in \mathbb{R}^n$ and $\overline{Q}\in \mathbb{R}^n$ the bounds on reactive power branch flow. Similarly, $\underline{V}\in \mathbb{R}^n$, $\underline{l}\in \mathbb{R}^n$ represent the lower bound on the nodal voltage and branch current, whereas $\overline{V}\in \mathbb{R}^n$, $\overline{l}\in \mathbb{R}^n$ represent the upper bound.

The optimization problem \textit{(P1)} is non-convex due to the constraint \eqref{eq:curr_rel}. In order to obtain an inner convex approximation of \textit{(P1)}, we consider the worst case of the nonlinearity introduced due to \eqref{eq:curr_rel}. Let $l^-\in \mathbb{R}^n$ and $l^+\in \mathbb{R}^n$ be the lower and upper bound on $l\in \mathbb{R}^n$, respectively. Then based on these values and provided that the matrices $D_R, D_X,M_p,M_q,H$ are positive for an inductive radial network~\cite{nazir2019convex}, let's define:
\begin{align}
    P^+:=&Cp-D_{\text{R}}l^-\label{eq:P_relation_1}\\
    P^-:=&Cp-D_{\text{R}}l^+\label{eq:P_relation_2}\\
    Q^+:=&Cq-D_{\text{X}}l^-\label{eq:Q_relation_1}\\
    Q^-:=&Cq-D_{\text{X}}l^+\label{eq:Q_relation_2}\\
    V^+:=&v_{\text{0}}\mathbf{1}_n+M_{\text{p}}p+M_{\text{q}}q-Hl^-\label{eq:V_relation_1}\\
    V^-:=&v_{\text{0}}\mathbf{1}_n+M_{\text{p}}p+M_{\text{q}}q-Hl^+\label{eq:V_relation_2}
\end{align}
Based on the above equations, the
optimization problem \textit{(P1)} can be modified as:
\begin{subequations}\label{eq:P2}
\begin{align}\label{eq:P2_a}
\text{(P2)} \ \min_{p_g,q_g} & \ f(p_g,q_g)\\
\text{subject to}:& \phantom{m} \eqref{eq:P_relation_1}-\eqref{eq:V_relation_2}, \eqref{eq:P1_c}-\eqref{eq:P1_d}
\label{eq:P2_b}\\
\underline{P} & \leq P^- ; P^+  \leq \overline{P}\label{eq:P2_c}\\
\underline{Q} & \leq Q^- ; Q^+  \leq \overline{Q}\label{eq:P2_d}\\
\underline{V} & \leq V^- ; V^+  \leq \overline{V}\label{eq:P2_e}\\
l^+ & \leq \overline{l}\label{eq:P2_f}
\end{align}
\end{subequations}

If $l^-$ and $l^+$ are known then the optimization problem \textit{(P2)} represents a convex inner approximation of the OPF problem. In the next section we will provide a method to obtain an accurate representation of these bounds.

\section{Obtaining bounds on nonlinearity}\label{sec:bounds}
In Section~\ref{sec:Optimization_model} we formulated the general form of the convex inner approximation in terms of the lower and upper bounds of the nonlinear term $l$. In~\cite{nazir2019convex}, we provided conservative bounds on the nonlinearity based on worst case net-demand forecasts. In this section, we present techniques to achieve tighter lower and upper bounds on the nonlinearity and also include branch flow constraints in the formulation. Section \ref{sec:lower_bound} describes the method to obtain the lower bound, whereas section \ref{sec:upper_bound} describes the method to obtain the upper bound to formulate the general convex inner approximation in terms of these bounds. Finally Section \ref{sec:Val_solar_test} provides simulation results to validate the presented formulation and a case study to determine the maximum allowable solar penetration in a distribution network.

\subsection{Estimating the lower bound}\label{sec:lower_bound}
Consider the nonlinear term in the power flow equations given by~\eqref{eq:curr_rel}. From the second-order Taylor series expansion, $l_{ij}$ can be expressed as:
\begin{align}\label{eq:T_exp}
    l_{ij} & \approx l_{ij}^0 + J_{ij}\Delta x_{ij}+\frac{1}{2}\Delta x_{ij}^T H_{ij} \Delta x_{ij}
\end{align}
where $l_{ij}^0$ is the value of $l_{ij}$ at the forecast net-demand and $\Delta x_{ij}$, the Jacobian $J_{ij}$ and the Hessian $H_{ij}$ are defined below:
\begin{align}
    \Delta x_{ij}=\begin{bmatrix} P_{j}-P_{j}^0\\
Q_{j}-Q_{j}^0\\ v_j-v_j^0\end{bmatrix}
\end{align}
\begin{align}\label{eq:Jacobian}
J_{ij}=\begin{bmatrix}\frac{2P^0_{j}}{v^0_j}\\\frac{2Q^0_{j}}{v^0_j}\\ -\frac{(P^0_{j})^2+(Q^0_{j})^2}{(v^0_j)^2}\end{bmatrix}
\end{align}
\begin{align}\label{eq:Hessian}
    H_{ij}=\begin{bmatrix} \frac{2}{v^0_j} && 0 && \frac{-2P^0_{j}}{(v^0_j)^2}\\
    0 && \frac{2}{v^0_j} && \frac{-2Q^0_{j}}{(v^0_j)^2}\\
    \frac{-2P^0_{j}}{(v^0_j)^2} && \frac{-2Q^0_{j}}{(v^0_j)^2} && 2\frac{(P^0_{j})^2+(Q^0_{j})^2}{(v^0_j)^3}
    \end{bmatrix}
\end{align}

 The eigenvalues of the Hessian $H_{ij}$ are given by:
\begin{align}
    \text{eig}(H_{ij})=\begin{bmatrix}0\\
    \frac{2}{v^0_j}\\ 2\frac{(P^0_j)^2+(Q^0_j)^2+(v^0_j)^2}{(v^0_j)^3}\end{bmatrix}
\end{align}
As two of the eigenvalues are strictly positive and one is zero, the Hessian is positive semi-definite and the nonlinear function $l_{ij}$ is convex. If a function is convex then the linear approximation underbounds the nonlinear function~\cite{boyd2004convex}, i.e.,
\begin{align}
      l_{ij} & \ge l_{ij}^0 + J_{ij}\Delta x_{ij} =: l_{ij}^-
\end{align}
The above equation can be expressed in vector form as shown:
\begin{align}\label{eq:l_lower}
    l^-=l^0+J\Delta x
\end{align}
The upper bound on the nonlinearity is obtained in the next section and the general convex inner approximation based on these bounds is presented.

\subsection{Estimating the upper bound}\label{sec:upper_bound}
Applying Taylor's theorem to the expansion of the nonlinear function in~\eqref{eq:T_exp}, for small values of $\Delta x_{ij}$, the quadratic term can be bounded as:
\begin{align}
   \left |\frac{1}{2}\Delta x_{ij}^T H_{ij} \Delta x_{ij}\right | \leq \left |J_{ij}\Delta x_{ij}\right|
\end{align}
Based on the above inequality, the upper bound on the nonlinear function $l_{ij}$ is given by:
\begin{align}
    |l_{ij}| & \approx |l_{ij}^0 + J_{ij}\Delta x_{ij}+\frac{1}{2}\Delta x_{ij}^T H_{ij} \Delta x_{ij}| \\
    & \le |l_{ij}^0| + |J_{ij}\Delta x_{ij}|+|\frac{1}{2}\Delta x_{ij}^T H_{ij} \Delta x_{ij}| \\
    & \le l_{ij}^0 + 2|J_{ij}\Delta x_{ij}| =: l_{ij}^+
\end{align}
The above equation can also be expressed in vector form as:
\begin{align}\label{eq:l_upper}
    l^+=\max\{l^0,l^0+2J\Delta x\}
\end{align}
Based on this upper bound and the lower bound determined in the previous section, we can now formulate the complete convex inner approximation OPF problem by modifying \textit{(P2)} as:

\begin{subequations}\label{eq:P3}
\begin{align}\label{eq:P3_a}
\text{(P3)} \ \min_{p_g,q_g} & \ f(p_g,q_g) \\
\text{subject to}:&\eqref{eq:P_relation_1}-\eqref{eq:V_relation_2},\eqref{eq:l_lower},\eqref{eq:l_upper}
\label{eq:P3_b}\\
&\eqref{eq:P1_c}-\eqref{eq:P1_d}\label{eq:P3_c}\\
\underline{P} & \leq P^- ; P^+  \leq \overline{P}\label{eq:P3_d}\\
\underline{Q} & \leq Q^- ; Q^+  \leq \overline{Q}\label{eq:P3_e}\\
\underline{V} & \leq V^- ; V^+  \leq \overline{V}\label{eq:P3_f}\\
l^+ & \leq \overline{l}\label{eq:P3_g}
\end{align}
\end{subequations}

The optimization problem \textit{(P3)} represents the convex inner approximation of the OPF problem that provides a feasible solution. In the next section, this formulation is tested on the IEEE-13 node distribution network.
\subsection{Validation results}\label{sec:Val_solar_test}
In this section simulations results are presented on the IEEE-13 node distribution network~\cite{kersting2001radial} shown in Fig.~\ref{fig:IEEE_13}. Gurobi is used to solve the optimization problem and the solution is validated through load flow in Matpower~\cite{zimmerman2011matpower}. For the purpose of this work, the mechanical devices such as tap-changers, capacitor banks and switches are assumed to be fixed at their nominal values. Generation resources are placed at nodes 7,8 and 9 of the network.
\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{IEEE_13.png}
\caption{\label{fig:IEEE_13}IEEE-13 node distribution network.}
\end{figure}

Based on the values of the forecasted net-demand, the optimization problem \textit{(P3)} is used to calculate the least cost dispatch of resources that satisfied all network constraints, i.e., $f(p_g,q_g)=\sum_i(c_{i1}p_{gi}^2+c_{2i}p_{gi})$. This is a generic and common objective function in power systems and is used to test the validity of the proposed formulation.  Figure~\ref{fig:volt_comp_case1} shows the upper and lower bounds of the voltages together with the actual voltage obtained through load flow. Similarly Fig.~\ref{fig:P_comp_case1} and Fig.~\ref{fig:Q_comp_case1} shows the bounds on the real and reactive branch flows. From these results it can be seen that the optimization problem \textit{(P3)} respects the network constraints.
\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{volt_comp_case1.png}
\caption{\label{fig:volt_comp_case1}Comparison of optimizer voltage bounds with the Matpower voltage showing the validity of the formulation \textit{(P3)}}
\end{figure}

\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{P_comp_case1.png}
\caption{\label{fig:P_comp_case1}Comparison of optimizer real branch flow bounds with the Matpower flow showing the validity of the formulation \textit{(P3)}}
\end{figure}

\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{Q_comp_case1.png}
\caption{\label{fig:Q_comp_case1}Comparison of optimizer reacitve branch flow bounds with the Matpower flow showing the validity of the formulation \textit{(P3)}.}
\end{figure}

\textbf{Case study 1:}
In this case study, the formulation \textit{(P3)} is used to ascertain the maximum allowable solar penetration,i.e, $f(p_g,q_g)=-\sum_i(p_{gi})$, where in this case $p_{gi}$ represents the power from solar PV unit at node $i$. The Solar PV units are placed at all leaf nodes, i.e., nodes~3, 5, 7, 8, 10 and~11 in the IEEE-13 node distribution network shown in Fig.~\ref{fig:IEEE_13}. Fig.~\ref{fig:pmax_case2} shows the maximum allowable solar penetration at each of the nodes, whereas Fig.~\ref{fig:volt_comp_case2} shows the predicted upper bound of the voltage profile obtained from \textit{(P3)} and its comparison with the load flow results from Matpower.

\begin{figure}[thb]
\centering
\includegraphics[width=0.42\textwidth]{pmax_case2.png}
\caption{\label{fig:pmax_case2}Inner approximation of the maximum solar penetration at the leaf nodes of the network in Case study 1.}
\end{figure}
\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{volt_comp_case2.png}
\caption{\label{fig:volt_comp_case2}Comparison of optimizer voltage upper bound with the Matpower voltage showing the validity of the inner approximation in Case study 1.}
\end{figure}

\textbf{Case Study 2:} In this case study, instead of placing the solar PV units at the leaf nodes, a single large PV unit is placed at the head of the feeder at node 2 with equivalent reactive power capability and from \textit{(P3)} its maximum capacity is determined. It is seen from Fig.~\ref{fig:pmax_case3} that the maximum solar penetration in this case is larger than the total from of the distributed solar PV at the leaf nodes. Furthermore, the comparison of the upper bound voltage from the optimizer and the one obtained throgh load flow is given in Fig.~\ref{fig:volt_comp_case3}, showing the feasibility of the obtained bounds on solar penetration.
\begin{figure}[thb]
\centering
\includegraphics[width=0.42\textwidth]{pmax_case3.png}
\caption{\label{fig:pmax_case3}Comparison of the maximum solar penetration between distributed solar and centralized solar unit obtained through inner approximation in Case study 2.}
\end{figure}
\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{volt_comp_case3.png}
\caption{\label{fig:volt_comp_case3}Comparison of optimizer voltage bounds with the Matpower voltage for the case of a centralized PV unit in Case study 2.}
\end{figure}

\section{Iterative algorithm for bound tightening}\label{sec:Iter_alg}
In this section, we present an iterative algorithm based on the formulation in \textit{(P3)} that achieves tighter bounds on the non-linearity. The lower and upper bounds obtained in sections~\ref{sec:lower_bound} and~\ref{sec:upper_bound} can be conservative depending upon the initial forecasted operating point. The upper bound, particularly, is only valid for small deviations away from the operating point and becomes a poor estimate if the injections change by a large value as shown in Fig.~\ref{fig:TScomp}.

\begin{figure}[t]
\centering
\includegraphics[width=0.45\textwidth]{TScomp.png}
\caption{\label{fig:TScomp}Comparison of the first and second order terms of the Taylor series expansion of $l$.}
\end{figure}

Furthermore, if the operating point is close to no-load condition, then the Jacobian given in \eqref{eq:Jacobian} is close to zero and the first order estimate of $l^+$ given by equation~\eqref{eq:l_upper} is also zero and hence does not provide any information about the value of $l$. To overcome these limitations of the proposed optimization problem \textit{(P3)}, in this section, we propose an iterative algorithm that provides tighter bounds by updating the operating point and hence the Jacobian. The block diagram in Fig.~\ref{fig:Iter_scheme} shows the steps involved in the proposed iterative scheme.

\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{Iter_scheme.png}
\caption{\label{fig:Iter_scheme}Block diagram of the proposed iterative scheme to obtain tighter bounds. }
\end{figure}

The iterative algorithm is now used to determine the feasible operating region of flexible resources in the IEEE-13 node system. Flexible resources capable of injecting and consuming power are placed at the leaf nodes~$\{3,5,7,8,10, 11\}$. The objective is chosen to be $f(p_g^+,q_g^+)=-\sum_ip_{gi}^+$ for the upper bound and $f(p_g^-,q_g^-)=\sum_ip_{gi}^-$ for the lower bound. The optimization problem \textit{(P3)} is iteratively solved for a no-load forecast case to obtain the feasible bounds shown in Fig.~\ref{fig:pbounds_case4}.

\begin{algorithm}
\caption{Iterative bound tightening}
\SetAlgoLined
\KwResult{Obtaining tighter bounds on nonlinearity}
 \textbf{Input:} $P_L, Q_L$, termination tolerance $\epsilon$\\
 Solve load flow\\
  Calculate $l^0, P^0, Q^0, V^0, J$\\
  Initialize $error>\epsilon$\\
 \While{error $>\epsilon$}{
  Solve the inner convex OPF $\Rightarrow p^\ast, q^\ast,V^*$ \\
  Update $P_L, Q_L$ with $p^*,q^*$\\
  Solve load flow\\
  Calculate $l^0,P^0,Q^0,V^0,J$\\
  Calculate $error=||V^*-V^0||_{\infty}$
 }
\end{algorithm}

It can be seen from the figure that the first iteration overestimates the lower bound due to the Jacobian being close to zero. However, as the Jacobian is updated in the next iteration, the lower bound is reduced. The comparison of the error in the voltage bounds over the iterations is given in Fig.~\ref{fig:Itbounds_case4}, that  shows how the voltages eventually converge to the one obtained through load-flow.

\begin{figure}[thb]
\centering
\includegraphics[width=0.42\textwidth]{pbounds_case4.png}
\caption{\label{fig:pbounds_case4}Inner approximation of the maximum upper and lower flexible bounds at the leaf nodes of the network over the iterations.}
\end{figure}

\begin{figure}[thb]
\centering
\includegraphics[width=0.5\textwidth]{Itbounds_case4.png}
\caption{\label{fig:Itbounds_case4} Iterations of the error between the voltage bounds and its convergence to the final Matpower voltage.}
\end{figure}

The results of this section show that the iterative algorithm provides tighter bounds on the nonlinearity and also ensure feasibility for the upper bound in case of large deviation from the operating point.

\textbf{Case Study 3:}
In this case study, the simulations conducted in Case Study~1 and Case Study~2 are repeated with the iterative scheme included to determine the solar PV hosting capacity for distributed PV at leaf nodes and a single utility-scale solar PV system. The results in Fig.~\ref{fig:Itcomp_case4} show how the iterative scheme overcomes the conservativeness of the inner approximation seen in case studies~1 and~2. This improvement is significant and greatly increases the distributed solar PV hosting capacity compared with the case of a single utility-scale PV system.

\begin{figure}[thb]
\centering
\includegraphics[width=0.42\textwidth]{Itcomp_case4.png}
\caption{\label{fig:Itcomp_case4}Improvement in the total allowed solar penetration with the iterative scheme for the case with distributed solar and the case with centralized solar unit in case study 3.}
\end{figure}

\section{Multi-period formulation}\label{sec:multi_period}
In this section, the formulation presented in \textit{(P3)} is extended to the multi-period case. To determine the maximum solar penetration at each time of a multi-period horizon, the optimization problem can be expressed as:
\begin{subequations}\label{eq:P4}
\begin{align}\label{eq:P4_a}
\text{(P4)} \ \max_{p_g,q_g,P_b} \ \sum_t\sum_i(p_{gi(t)})
\end{align}
\begin{align}
\text{subject to}&:\eqref{eq:P_relation_1}-\eqref{eq:V_relation_2},\eqref{eq:l_lower},\eqref{eq:l_upper}
\label{eq:P4_b}\\
p(t)&=p_g(t)+P_b(t)-P_L(t)\label{eq:P4_c}\\
q(t)&=q_g(t)-Q_L(t)\label{eq:P4_d}\\
B(t+1)&=B(t)-P_b(t)\Delta t \label{eq:P4_e}\\
\underline{B}& \leq B(t) \leq \overline{B}\label{eq:P4_f}\\
\underline{P} & \leq P^-(t) ; P^+(t)  \leq \overline{P}\label{eq:P4_g}\\
\underline{Q} & \leq Q^-(t) ; Q^+(t)  \leq \overline{Q}\label{eq:P4_h}\\
\underline{V} & \leq V^-(t) ; V^+(t)  \leq \overline{V}\label{eq:P4_i}\\
l^+(t)&\leq \overline{l}\label{eq:P4_j}
\end{align}
\end{subequations}

where $B(t)\in \mathbb{R}^n$ is the SoC of the battery at time $t$, $P_b(t) \in \mathbb{R}^n$ is the battery net-injection at time $t$ and $\Delta t$ is the time-interval. To highlight the multiperiod formulation, the battery model has been simplified with unity efficiencies. Detailed analysis with non-unity battery models and a convex OPF formulation can be found in~\cite{2019arXiv190604108N}.

\textbf{Case study 4: } In the case study of this section, we ascertain the value of installing batteries in the system by comparing the case with no batteries and the case with batteries on the total allowable solar penetration. The case with batteries is further sub-divided into the case with distributed storage and with centralized storage of equivalent rating and capacity. Distributed batteries with a rating of 300 kW and storage capacity of 300 kWhr are placed at leaf-nodes 3, 5, 7, 8, 10 and 11 for the distributed case and for the centralized storage case, an equivalent battery is placed at node 2 of the IEEE-13 node system.
Fig.~\ref{fig:pmax_case5} shows the comparison of the total solar injection in the network over a prediction horizon of 10 hours between the case with no storage, with a large central storage unit and with distributed storage of equivalent rating and capacity. The iterative schmeme described in the previous section is used to calculate the maximum solar penetration in each of the cases. It can be seen from the comparison that having storage in the network allows for greater penetration of solar energy, especially distributed storage. For the given case study, the inclusion of storage led to increase in net-solar penetration by approximately 10\%. Fig.~\ref{fig:Pb_case5} shows the expected net-demand over the prediction horizon (10 hours) together with the aggregate battery discharge power for the distributed storage case.

\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{pmax_case5.png}
\caption{\label{fig:pmax_case5}Comparison of the maximum allowable solar penetration over the prediction horizon between the case with no storage, central storage and distributed storage in case study 4.}
\end{figure}

\begin{figure}[thb]
\centering
\includegraphics[width=0.45\textwidth]{Pb_case5.png}
\caption{\label{fig:Pb_case5}Comparison of the predicted load profile and the battery discharge power over the prediction horizon in case study 4 using distributed storage.}
\end{figure}

\section{Conclusions and Future work}\label{sec:conclusion}
This paper presented a computationally efficient method to obtain a feasible solution of the OPF problem. This is achieved by obtaining a convex inner approximation of the feasible domain. Leveraging the lower and upper bounds on the nonlinear losses in the AC formulation, the constraints are robustified, which ensures an AC feasible solution despite modeling errors and a convex relaxation. An iterative scheme is presented to further tighten the robust bounds and improve feasibility claims for solar PV and DER hosting capacity problems. Finally, the hosting capacity problem is extended to a multi-period formulation to examine the active role of energy storage in supporting increased solar PV penetration in distribution feeders.

Future work will extend this technique to three-phase distribution networks and meshed transmission networks. Providing feasibility guarantees under both model mismatch and forecast uncertainty due to solar PV is another important area of work to investigate.

\bibliographystyle{ieeetr}

redpajama_set_name: RedPajamaArXiv