2017

Towards Autonomous Power Management:

Extending the Holomorphic Embedding Loadflow Method

for NASA DC systems

EnergyTech Conference

November 28, 2016

Bradley C. Glenn, Ph.D.

Gridquant Technologies LLC

Antonio Trias, Ph.D.Jose Luis Marin, Ph.D.

Elequant Inc.

2017Outline

• The need for autonomous control of spacecraft power

• The role and relevance of powerflow in NASA’s

Intelligent Autonomous Control Architecture and

system engineering design needs

• HELMTM : quick overview

• SBIR Phase 2 Year 2 results:

• Completed model for the DC-DC converter (DCCU)

• Padé-Weierstrass method

• Developed HELMLAB DC prototype

• Vision for future applications

2017The need for Robust Load-Flow algorithms in Spacecraft and Turbo Electric Aircraft Power Systems

• Spacecraft and Turbo Electric Aircraft Power Systems are DC microgrids that must be extremely robust

• International Space Station (ISS) and manned space missions in near earth orbit have constant ground support from Houston Mission Control Center

• Deep Space Travel to MARS will require autonomous control due to communication latency

• Communication latency for MARS mission would be anywhere from 15 to 25 minutes depending on of proximity of MARS orbit in relationship to Earth

• Valuable High-Level system design tool for Turbo-Electric power system development process that utilizes models that are less computationally intensive than time domain models.

• HELM has the ability to work with Hybrid AC/DC systems.

2017The role of powerflow in DC & AC power systems

• Large Terrestrial AC Networks represent a dynamic system that constantly changes 24x7- Thousands of control actions taken daily to control voltage and frequency- Generator excitation control, load tap changers, phase shifters, reactive devices- AGC at a more macro level, and operator action in terms of schedule changes

• DC Microgrids on board spacecraft and on the ISS also represent very dynamic systems- SSU in PV array panels and voltage control by the DDCU- DC components such as PV arrays and batteries are inherently non-linear

• Terrestrial AC networks and DC microgrids have the luxury of operator intervention

• Power flows in terrestrial AC networks play a significant role in determining physical state:- AC power flows critical to determining whether in a stable state or near collapse- During times of extreme stress or large step changes, iterative methods have failed

• Autonomous control of spacecraft requires robust power flows that can solve at the limit

2017The role and relevance of powerflow in NASA’s

Intelligent Autonomous Control Architecture

From J. Soeder et al.,” Overview of Intelligent Power Controller Development for Human Deep Space Exploration”, IECEC 2014

• In utilities, powerflow is normally associated to analysis and planning tasks

• Here we have a very different type of application in mind: analytical tools for decision-support in network operations; and ultimately, for complete autonomous management.

Power System Model functions:1. Model of power generation2. Model energy storage3. Model power network (powerflow model of

the distribution system)4. Power System State Estimator

This module provides high fidelity models & simulation of the power system, which other control blocks need to make decisions regarding control actions.

2017HELM™ Overview

Current methods: lack of convergence, need initial seed solution

• Holomorphic Embedding Load Flow Method• Direct, constructive solution to powerflow equations• Non-iterative and deterministic, unlike traditional

methods• Uses a fundamentally new mathematical approach

• Based on Complex Analysis: Analytic Continuation, not numerical continuation or Homotopy

• New measures of distance to collapse (Sigma indicators)

• Similar to a Laplace transform that converts an ordinary differential equation into an algebraic problem, HELM converts a nonlinear algebraic problem into a sequence of linear systems that can solved with linear algebra (step prior to A.C.)

• As engineers we are not necessarily concerned with the derivation of a Laplace or integral transforms but how to execute their result.

2017HELM™ insights

• Not the same as the Series Load Flow Method• Holomorphic Embedding method is somewhat related to these ideas, but with one key

difference: the Series Load Flow uses real variables and HELM complex variables

• Not numerical Homotopy continuation • Homotopy methods compute the powerflow solution along a parameterized curve, but only

exploit continuity and single differentiability.• Therefore the path-following steps still use numerical iteration to track the solution (N-R is

typically used as the “corrector” in the predictor-corrector steps)

• The power series are not an approximation!• For Holomorphic functions (==complex analytic), the power series is the function• Many holomorphic functions are actually defined via their power series (e.g. ez)

• Padé approximants (in this case) are not an approximation!• The beauty of HELM is that voltages become an algebraic curve of the embedding parameter.

For these class of functions, Stahl’s theorem applies.• This means that the near-diagonal sequence of Padé approximants of the power series are

guaranteed to converge• Moreover the theorem states that they converge outside the radius of convergence of the

power series, in the maximal domain possible. Therefore they provide the maximal analytic continuation.

• This last bit provides completeness to the method: when the solution exists, the Padésequence will converge to it; when the solution does not exist, the sequence oscillates.

2017SBIR results: device modeling

• ISS models from PC Krause (Simulink, averaged)

• Derived their corresponding steady-state models for powerflow

V (volts)

I (a

mps)

50

60

Vengage

105 V

Von Vmin

Voff

Op. point

80 100 160

155 160 165

-40

-30

-20

-10

0

10

20

30

40

50

60

Input Voltage (V)

Inpu

t C

urr

en

t (A

)

Voltage Command

deadbanddischargeSlope

chargeSlope

imin

imax

Primary side(input)

Secondary side(output)

1 : M(D) vp vs

rs

ip is

Control input D

Power conversion efficiency h

v's

PV solarPanels(+SSU)

Batteries(BCDU)

DC-DC converters(DDCU)

Constant Power Loads (+converter)

2017

Control Limits

• Exhibited by both the SSU and DDCU

• Sequential Shunt Unit (SSU) (the SSU cannot have negative conductance, as this would mean that it can inject power into the network).

• The DC-DC converter Control Unit (DDCU). the converter’s regulation capability is limited by a maximum current output imax. The regulated voltage will be maintained until this output current limit is reached, at which point the regulation changes to constant-current mode, iA = -imax (and the voltage on the secondary bus is then allowed to increase in response to higher load).

9

2017The Padé-Weierstrass method

• Invented by Dr. Antonio Trias, as an extension of his HELMmethod

• Developed by Dr. Trias’s team at AIA/EleQuant, initially for AC utility grid systems

• Successfully adapted by AIA/EleQuant to DC systems in Phase II of this project

• WHAT IS IT? — A new analytic continuation procedure, exploiting the specific nature of the power flow equations

• BENEFITS:1. Improves the numerical accuracy of HELM in the presence of singularities (e.g.

proximity to voltage collapse)

2. Therefore allows treating the problem of control limits efficiently and reliably usage in autonomous control applications

2017Precision improvement in ill-conditioned cases

• Case right at the point of voltage collapse

• The singularity prevents obtaining more than 2 significant digits with Padé

• 8 PW steps achieve maximum possible precision (IEEE dp arithmetic)

0 5 10 15 20 25 30

power series order

-16

-14

-12

-10

-8

-6

-4

-2

0

2

log

10

(err

or)

Update errors vs order (case very close to collapse)

PW step 0PW step 1PW step 2PW step 3PW step 4PW step 5PW step 6PW step 7PW step 8

2017How P-W “pushes away” singularities

• P-W exploits the covariant nature of the power flow equations under certain changes of variables

• Each P-W transformation is a conformal map that “zooms in” around s=1

• Sequence of equivalent problems, each with the singularities farther away

0 1

-0.4

0.4

0 1

-0.4

0.4

0 1

-0.4

0.4

complexs-plane

s0=0.44

s'0=0.75

etc.

2017Control limits in power flow

Examples from dc PMAD systems in spacecraft:

• SSU: regulates output voltage of PV solar panels by shunting current➢ Upon reaching zero conductance, the voltage is no longer regulated

• DC-DC converter: regulates output voltage by changing its duty cycle➢ Upon reaching a maximum current, it regulates current (at Imax); voltage is allowed to drop

This results in inequality constraints, of complementary nature:

• If the control is within limits: the resource is a variable, and the setpoint condition is an equation (setpoint achieved)

• If the control hits a limit: the resource is no longer a variable, and the equation is removed (setpoint not necessarily achieved)

The traditional approach: “type-switching”: solve for each type of possible behavior, and keep the solutions that are consistent with the assumed constraints.

➢ combinatorial explosion in the number of possible type-switched states to explore

➢ there may be more than one valid combination. So which one should be selected?

➢ Type-switching heuristics help in some cases, but are not fully reliable

2017P-W applied to control limits

• HELM frames this problem in terms of optimization

• It completely sidesteps type-switching

• If there are several valid type-switched states, the solution selected by the method can be physically characterized as having the lowest amount of losses

• Uses barrier functions, like interior point methods optimization—but the method is based on a complex embedding, algebraic curves, and analytic continuation

• The singularity induced by the barriers is overcome by the Padé-Weierstrass method

2017

• 9,000 bus case with generator Mvar limits enforced

• 26 P-W steps yield the solution to extremely good accuracy

0 5 10 15 20 25 30 35 40 45

power series order

-14

-12

-10

-8

-6

-4

-2

0

2

log

10

(err

or)

Update error vs. order (case9241pegase)

PW step 2PW step 6PW step 10PW step 14PW step 18PW step 22PW step 26

P-W applied to control limits

2017

HELMLAB DC• prototype power-flow solver that has been developed for this SBIR Project implemented in MATLAB

• the input file format is loosely modeled after the one used in HELMLAB AC. In that case, the HELMLAB format was designed as a superset of the one used by PSERC’s MATPOWER.

• The whole input file is one MATLAB struct, and devices are grouped under struct fields. Each field is a 2D array, where each row represents one particular device. Here is a short description of these fields, and the devices that they represent:

• Global parameters:

• basekW: base value of power for per-unit magnitudes, in kW

• bus: buses

• line: transmission/distribution lines

• dcconv: DC-DC converter (“transformer”), including automatic regulation of the output voltage and limiting current

• pvarr: PV arrays (solar cells with optional shunt regulation)

• batt: batteries, including charge/discharge curves with limiting currents

• load: constant conductance loads and constant power loads (including optional constant-conductance and constant-current regimes, induced by their point-of-load converter)

• colors: not a type of device, but a list of buses with their specified “colors”, in order to request the calculation of other solution branches besides the white branch

page 16

2017Vision for future applications to NASA and terrestrial systems

• Our Phase II SBIR demonstrated that HELM solves reliably the powerflow in DC microgrids. It provides the stable operating points, in cases where several are possible.

• HELM, along with advanced intelligent applications (in the spirit of those used in AGORA for terrestrial grids) can be integrated in the future for autonomous control of aircraft/spacecraft

• HELM can be key for the power system management of future technologies such as Turbo Electric Propulsion, where the system undergoes abrupt state changes.

• Valuable High-Level system design tool in the power system development process that utilizes models that are less computationally intensive than time domain models.

• Emerging terrestrial microgrids will need HELM technology in the future

Figure 1. Schematic of microgrid

Source of Picture: J. Soeder et al., “Application of Autonomous Spacecraft Power Control Technology to Terrestrial Microgrids”, IECEC 2014