A Parallel Solution to Stochastic Power System Operation with Renewable Energy

5th Southeast Symposium on Contemporary Engineering Topics (SSCET), 2014

Yong Fu, Ph.D. Associate Professor

Electrical and Computer EngineeringMississippi State University

New Orleans, LA September 19th, 2014

Parallel Computing

o With development of high performance computing technique, parallel computing technique can significantly improve computational efficiency of optimization problem with utilization of multi-processors and multi-threads.

o These improvements cannot be achieved by the architectures of the machines alone, it is equally important to develop suitable mathematical algorithms and proper decomposition & coordination technique in order to effectively utilize parallel architectures

A Typical Power System Operation Problem – Security Constrained Unit Commitment

Objective Function – Minimize

Generating Unit Constraints

System Operation Constraints

Generation capacity Minimum ON/OFF time limits Ramping UP/DOWN limits Must-on and area protection constraints Forbidden operating region of generating units

Power balance System reserve requirements Power flow equations Transmission flow and bus voltage limits Limits on control variables Limits on corrective controls for contingencies

Generation and startup/shutdown costs

100

150

200

250

300

1 4 7 10 13 16 19 22 Hours

Load (MW)

Unit 1

Unit 2

Unit 3

Large Scale, Non-Convex, Mixed Integer Nonlinear

Problem

Who Use SCUC and How?

GENCOs TRANSCOs

ISO

Security-Constrained Unit Commitment

DISTCOs

ISOs: PJM, MISO, ISO New England, California ISO, New York ISO and ERCOT

ISO (SCUC) and Market Participants

Day Ahead Market (DAM) determines the 24-hourly status of the generating units for the following day based on financial bidding information such as generation offers and demand bids.

Day Ahead UC for Reliability (RUC), which focuses on physical system security based on forecasted system load, is implemented daily to ensure sufficient hourly generation capacity at the proper locations.

Look-Ahead UC (LAUC), as a bridge between day-ahead and real-time scheduling, constantly adjusts the hourly status of fast start generating units to be ready to meet the system changes usually within the coming 3-6 hours.

Real-Time Market (RTM) further recommits the very fast start generating units based on actual system operating conditions usually within the coming two hours in 15-minute intervals.

Stochastic SCUC

In stochastic programming, the decision on certain variables has to be made before the stochastic solution is disclosed, whereas others could be made after.

The set of decisions is then divided into two groups: A number of decisions are made before performing experiments. Such decisions are called

first-stage decisions and the period when these decisions are made is called the first stage. A number of second-stage decisions are made after the experiments in the second stage.

Stochastic models containing above two groups of variables, first-stage and second-stage decision variables, are called two-stage stochastic programming.

KkeFyHx

xbAxts

ydpMinxcMin

kk

K

kk

Tk

T

yyx k

,,1,

,binaryis,..1,,, 1

Stochastic SCUC --- Example

G3 16 $/MWh10MW~40MW

Load

G1 13 $/MWh40MW~80MW

G2 42 $/MWh15MW~ 40MW

System

L1 75MW

L2 75MW

1 2

20 MW

80 MW

50 MW

50 MW

52.5 MW

52.5 MW

75 MW0 MW 0 MW0 MW

100 MW 105 MW 95 MW

Base Case Scenario 1 Scenario 2

W

0 MW

15 MW

75 MW

15 MW

23 MW

? MW

? MW

G3 can adjust dispatches by 5 MW G2 is quick-start unit with 30 MW QSC

20 MW

65 MW

42.5 MW

42.5 MW

52.5 MW

52.5 MW

75 MW15 MW 0 MW20 MW

100 MW 105 MW 95 MW

Base Case Scenario 1 Scenario 2

0 MW

15 MW

60 MW

30 MW

23 MW

52 MW

0 MW

Solution 1

Solution 2

CasesEquipmen

tOutage

Wind(WM)

Load(MW)

Base case - 20 100

Scenario 1 G3 15 105

Scenario 2 L2 23 95

Current Work

o Amdahl’s law: an upper bound on the relative speedup achieved on a system with multi-processors is decided by the execution time of the application operating sequentially.

Optimal Power Flow

Calculation

Optimal Power Flow

Calculation

Optimal Power Flow

Calculation

Optimal Power Flow

Calculation

Time=1:NG

End

Start

Unit Commitment

calculation

Unit=1:NG

Unit Commitment

calculation

Unit Commitment

calculation

Unit Commitment

calculation

Proposed Approach

o Structure of Algorithm: Scenario-based stochastic model is adopted to analyze the uncertainties of load and wind energy in this paper. Instead of master-and-slave structure, UC and OPF subproblems are solved simultaneously in the proposed parallel calculation method.

o Convergence performance: In an iterative solution process, the number of iterations affects the overall computational time. Several convergence acceleration options, including initialization and update of penalty multipliers, truncated auxiliary problem principle and trust region technique, are used to improve the convergence performance and efficiency in a scenario-based study.

Optimal Power Flow

Calculation

Optimal Power Flow

Calculation

Optimal Power Flow

Calculation

Optimal Power Flow

Calculation

Time=1:NG

End

Unit Commitment

calculation

Unit=1:NG

Unit Commitment

calculation

Unit Commitment

calculation

Unit Commitment

calculation

Start

Decomposition Strategy

Mathematically, the stochastic SCUC can be formulated as a mixed integer programming (MIP) problem as shown in

Variable Duplication Technique

Augmented Lagrangian Method

hEydbyAxts

yxFMin

ss

ss

NS

ss

,..

),(0

ssss

ss

NS

ss

yyhyEdbyAxts

yxFMin

ˆ,ˆ,..

),(0

hyEdbyAxts

yyc

yyyxFMin

ss

ssssT

ss

NS

ss

ˆ,..

ˆ2

)ˆ(),(2

0

Algorithms for Parallel Solutions

Auxiliary Problem Principle (APP) Method

Diagonal Quadratic Approximation (DQA) Method

Alternating Direction Method of Multipliers (ADMM)

Analytical Target Cascading (ATC) Method

Iterative Solution Procedure

Two separated auxiliary problem:

NT

t

NG

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kitsg

kitsgitsgitsitsgitsgitititsg

NS

ssuc PPPcPcSUDIPFMinL

1 1,

)1(,

)1(,,

2,,,

0]))ˆ(()(),([

NT

t

NG

iitsg

kitsg

kitsgitsgitsitsgitsg

NS

ssopf PPPcPcMinL

1 1,

)1(,

)1(,,

2,,

0]ˆ))ˆ(()ˆ([

Original Augmented Lagrangian Problem

UC Subproblem

Unit 1

UC Subproblem

Unit 2

UC Subproblem

Unit NG

Unit Commitment Subproblem

Optimal Power Flow Subproblem

OPF Subproblem Period 1

OPF Subproblem Period NT-1

OPF Subproblem Period NT

OPF Subproblem Period NT-1, Scenario 0

OPF Subproblem Period NT-1, Scenario 1

OPF Subproblem Period NT-1, Scenario NS

Decomposition structure:

Given values from the previous

iterationDecision variables for the current iteration

Case Study – IEEE 118-bus Testing System

o Case 1: Deterministic caseo Case 2: Stochastic case with 3 scenarios

Zone 1 Zone 2 7 2 13 33 43 44 54 55

1 117 45 56

15 34 53 3 12 14 46 57 36 52 6 11 17 18 35 47 37 42 58 4 16 39 51 59 19 41 48 5 40 49 50 60 38 8 20 9 30 31 113 73 66 62 10 29 32 21 69 67 61 65 64 28 114 71 81 26 22 75 118 76 77 115 68 80 63 25 27 23 72 74 116 24 98 99 70 78 79 97 87 86 85 88 96 90 89 84 83 82 95 112 91 93 94 107 106

92 106 109 111 100 105 103 104 102 101 108 110

Zone 3

54 thermal units 3 wind farms 118 buses 186 branches

Deterministic Case Study

The converged result is obtained after 39 iterations.

0 10 20 30 400

50

100

150

200

Iterations

Po

wer(

MW

)

0 10 20 30 400

50

100

150

Iterations

Po

wer(

MW

)

0 10 20 30 400

50

100

150

200

250

300

Iterations

Po

wer(

MW

)

0 10 20 30 400

50

100

150

200

250

300

Iterations

Po

wer(

MW

)

Popf,36,5

Puc,36,5

Popf,45,5

Puc,45,5

Popf,36,21

Puc,36,21

Popf,45,21

Puc,45,21

Unit 45 at Hour 5

Unit 45 at Hour 21

Unit 36 at Hour 5

Unit 36 at Hour 21

Deterministic Case Study

ItemsCentralized

SCUCParallel SCUC

Changes

Total Cost ($) 1,583,700 1,584,997 +0.08%

Time (Seconds) 19 8 -58%

Stochastic Case Study (3 scenarios)

Items 

Centralized SCUC

Parallel SCUC

Changes

Cost ($) 1,582,840 1,583,565 +0.046%

Time (Seconds) 1,083 20 -96%

Case Study – A 1168-bus Power System

o A practical 1168-bus power system with 169 thermal units, 10 wind farms, 1474 branches, and 568 demand sides.

o It could be nearly impossible to get a near-optimal stochastic SCUC solution for this system by applying a traditional centralized SCUC algorithm.

o However, the proposed parallel stochastic SCUC algorithm provides solutions.

Unit 8 at Hour 1

0 50 100 150 200 2500

200

400

600

800

1000

1200

Iterations

Po

we

r O

utp

ut (

MW

)

Popf,8,1,1

Puc,8,1,1

Case Study – A 1168-bus Power System

# of Scenarios # of Iteration Total Time (sec.)

0 315 109.551 330 146.062 299 139.313 327 163.474 277 142.285 278 140.286 248 139.527 243 130.948 242 133.259 237 148.33

10 231 131.95

Conclusionso The proposed stochastic SCUC approach minimizes the operation cost of

system by possibility expectation of each scenarios, which can adaptively and robustly adjust generation dispatch in response to constraints in different scenarios.

o In comparison with traditional stochastic SCUC, optimal power flow problem does not have to wait for unit commitment decision, both problems can be solved simultaneously, which is more computational efficient in both day-head and real-time power markets.

o The ideas can be applied to various power system applications: state estimation, economic dispatch, and planning.

fu@ece.msstate.edu

Thanks !