Citation
Gan, Lingwen (2015) Distributed Load Control in Multiphase Radial Networks. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/Z9FQ9TJ0. http://resolver.caltech.edu/CaltechTHESIS:01272015214848277
Abstract
The current power grid is on the cusp of modernization due to the emergence of distributed generation and controllable loads, as well as renewable energy. On one hand, distributed and renewable generation is volatile and difficult to dispatch. On the other hand, controllable loads provide significant potential for compensating for the uncertainties. In a future grid where there are thousands or millions of controllable loads and a large portion of the generation comes from volatile sources like wind and solar, distributed control that shifts or reduces the power consumption of electric loads in a reliable and economic way would be highly valuable.
Load control needs to be conducted with network awareness. Otherwise, voltage violations and overloading of circuit devices are likely. To model these effects, network power flows and voltages have to be considered explicitly. However, the physical laws that determine power flows and voltages are nonlinear. Furthermore, while distributed generation and controllable loads are mostly located in distribution networks that are multiphase and radial, most of the power flow studies focus on singlephase networks.
This thesis focuses on distributed load control in multiphase radial distribution networks. In particular, we first study distributed load control without considering network constraints, and then consider networkaware distributed load control.
Distributed implementation of load control is the main challenge if network constraints can be ignored. In this case, we first ignore the uncertainties in renewable generation and load arrivals, and propose a distributed load control algorithm, Algorithm 1, that optimally schedules the deferrable loads to shape the net electricity demand. Deferrable loads refer to loads whose total energy consumption is fixed, but energy usage can be shifted over time in response to network conditions. Algorithm 1 is a distributed gradient decent algorithm, and empirically converges to optimal deferrable load schedules within 15 iterations.
We then extend Algorithm 1 to a realtime setup where deferrable loads arrive over time, and only imprecise predictions about future renewable generation and load are available at the time of decision making. The realtime algorithm Algorithm 2 is based on modelpredictive control: Algorithm 2 uses updated predictions on renewable generation as the true values, and computes a pseudo load to simulate future deferrable load. The pseudo load consumes 0 power at the current time step, and its total energy consumption equals the expectation of future deferrable load total energy request.
Network constraints, e.g., transformer loading constraints and voltage regulation constraints, bring significant challenge to the load control problem since power flows and voltages are governed by nonlinear physical laws. Remarkably, distribution networks are usually multiphase and radial. Two approaches are explored to overcome this challenge: one based on convex relaxation and the other that seeks a locally optimal load schedule.
To explore the convex relaxation approach, a novel but equivalent power flow model, the branch flow model, is developed, and a semidefinite programming relaxation, called BFMSDP, is obtained using the branch flow model. BFMSDP is mathematically equivalent to a standard convex relaxation proposed in the literature, but numerically is much more stable. Empirical studies show that BFMSDP is numerically exact for the IEEE 13, 34, 37, 123bus networks and a realworld 2065bus network, while the standard convex relaxation is numerically exact for only two of these networks.
Theoretical guarantees on the exactness of convex relaxations are provided for two types of networks: singlephase radial alternativecurrent (AC) networks, and singlephase mesh directcurrent (DC) networks. In particular, for singlephase radial AC networks, we prove that a secondorder cone program (SOCP) relaxation is exact if voltage upper bounds are not binding; we also modify the optimal load control problem so that its SOCP relaxation is always exact. For singlephase mesh DC networks, we prove that an SOCP relaxation is exact if 1) voltage upper bounds are not binding, or 2) voltage upper bounds are uniform and power injection lower bounds are strictly negative; we also modify the optimal load control problem so that its SOCP relaxation is always exact.
To seek a locally optimal load schedule, a distributed gradientdecent algorithm, Algorithm 9, is proposed. The suboptimality gap of the algorithm is rigorously characterized and close to 0 for practical networks. Furthermore, unlike the convex relaxation approach, Algorithm 9 ensures a feasible solution. The gradients used in Algorithm 9 are estimated based on a linear approximation of the power flow, which is derived with the following assumptions: 1) line losses are negligible; and 2) voltages are reasonably balanced. Both assumptions are satisfied in practical distribution networks. Empirical results show that Algorithm 9 obtains 70+ times speed up over the convex relaxation approach, at the cost of a suboptimality within numerical precision.
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  optimal power flow, distribution network, unbalanced network 
Degree Grantor:  California Institute of Technology 
Division:  Engineering and Applied Science 
Major Option:  Electrical Engineering 
Minor Option:  Applied And Computational Mathematics 
Awards:  DemetriadesTsafkaKokkalis Prize In Benign Renewable Energy Sources Or Related Fields, 2015 
Thesis Availability:  Public (worldwide access) 
Research Advisor(s): 

Thesis Committee: 

Defense Date:  28 August 2014 
NonCaltech Author Email:  ganlingwen (AT) gmail.com 
Projects:  ARPAE 
Record Number:  CaltechTHESIS:01272015214848277 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:01272015214848277 
DOI:  10.7907/Z9FQ9TJ0 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  8762 
Collection:  CaltechTHESIS 
Deposited By:  Lingwen Gan 
Deposited On:  04 Feb 2015 22:41 
Last Modified:  14 Apr 2016 22:07 
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