Methods and Algorithms for Economic MPC inPower Production Planning

Leo Emil Sokoler

Ph.D. DefenseKgs. Lyngby, Denmark. March 2016

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Presentation Outline

1 Background & Introduction

2 Economic MPC of Energy Systems

3 Optimization Algorithms

4 Integrated Planning and Control

5 Optimal Reserve Planning

6 Conclusions & Future Work

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Background & Introduction

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The Future Power GridI The penetration of wind, solar and hydro power is increasing

significantlyBE BG CZ D

K DE EE IE EL ES FR HR IT CY LV LT LU HU

MT NL

AT PL PT RO SI SK FI SE UK

0

20

40

Rene

wabl

eEn

ergy

[%] 2020 Target

2004 Progress2012 Progress

I New planning methodologies are required to accommodatethe intermittency of renewable energy resources

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Control Hierarchy

Information

Production PlanningI Hours-ahead unit commitment and economic

dispatch of the system generators

Balance ControlI Balancing of production and consumption in

near real-time

Frequency ControlI Real-time activation of reserved generation

capacity to maintain system stability

Com

puta

tion

Tim

e

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Case Study: The Faroe Islands

I Population of about 50,000people

I No interconnectors to othercountries (isolated power system)

I Some of the worlds bestconditions for wind power

I Target: 100% renewable energyby 2030

I Flexibility on both the productionand the consumption side ofenergy

Current challenges for the Faroe Islands are future challenges forlarger interconnected power systems

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Key Contributions

I Proof of concept for balance and frequency EMPC-basedcontrol schemes

I Mean-Variance EMPC accounts for the inherent uncertaintyand variability of renewable energy sources

I Integrated planning and control using a hierarchical EMPCalgorithm

I Computationally efficient algorithms overcome tractabilityissues of the proposed EMPC schemes

I An optimal reserve planning problem for unit commitment andeconomic dispatch in small isolated power systems

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Economic MPC of Energy Systems

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Economic MPC (EMPC)

Optimal Control Problem

min.u,x ,z

φ (u, x , z)

s.t. xk+1 = Axk + Buk , k ∈ N0

zk = Czxk , k ∈ N1

(u, x , z) ∈ X

I Prediction horizon Ni = {0 + i , 1 + i , . . . ,N − 1 + i}

I Input vector u = (uT0 , uT

1 , uT2 , . . . , uT

N−1)T ∈ RNnu

I State vector x = (xT1 , xT

2 , xT3 , . . . , xT

N )T ∈ RNnx

I Output vector z = (zT1 , zT

2 , zT3 , . . . , zT

N )T ∈ RNnz

Assumption: Cost function φ is a convex function and constraintset X is a convex set

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Two-Generator Case Study

Generator Specifications

#Generator Capacity Response Time Utilization Cost

1 Small Fast High2 Large Slow Low

Closed-Loop Simulation (Deterministic)

0 5 10 15 20 25 300

10

20

30

Time [min]

Powe

r[M

W]

zg1,k zg2,k zT ,k (zT ,k ,zT ,k )

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Uncertainty Management

Closed-Loop Simulation (Stochastic)

0 5 10 15 20 25 300

10

20

30

Powe

r[M

W]

zg1,k zg2,k zT ,k (zT ,k ,zT ,k )

Certainty-Equivalent EMPC does not perform well in the presenceof uncertainty

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Certainty-Equivalent EMPC (CE-EMPC)

I Linear stochastic system

xk+1 = Axk + Buk + wk , k ∈ N0

yk = Cy xk + vk , k ∈ N1

zk = Czxk , k ∈ N1

I Affine functions

x = Lx (u; x0,w)z = Lz(u; x0,w)

I Cost function

ψ(u; x0,w) = φ (u, Lx (u; x0,w), Lz(u; x0,w))

I Optimal control problemmin.u∈U

ΨCE = ψ(u; x0,E [w ])

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Mean-Variance EMPC (MV-EMPC)

I CE-EMPC does not minimize the expected costψ(u; x0,E [w ]) 6= E [ψ(u; x0,w)]

670 680 690 700 710 720 7300

200

400

600

800

ψ(u∗CE ; x0,w)

Coun

ts

ψ(u∗CE ; x0,E [w ])

I MV-EMPCmin.u∈U

ΨMV = αE [ψ(u; x0,w)] + (1− α)V [ψ(u; x0,w)]

with risk-aversion parameter α ∈ [0; 1]13 / 40

Monte-Carlo Approximation

I Uncertainty scenarios S = {1, 2, . . . ,S}

I Optimal control problem

min.u∈U ,{x s ,zs ,ψs}s∈S ,µ

αµ+ 1−αS−1

∑s∈S

(ψs − µ)2 ,

s.t. x sk+1 = Ax s

k + Buk + w sk , k ∈ N0, s ∈ S

zsk = Czx s

k , k ∈ N1, s ∈ Sψs = φ(u, x s , zs), s ∈ Sµ = 1

S∑s∈S

ψs

I Two-stage extension with non-anticipative constraints can beapplied for less conservative closed-loop performance

I Large-scale optimization problem even for small systems14 / 40

Performance of MV-EMPC

0 5 10 15 20 25 30 35

210

220

230

240

α = 0α = 1

s

µ

MV-EMPCCE-EMPC

Computationally Attractive Alternatives

I Safety margin using constraint back-off

I Augmented objective function, e.g. setpoint-based penaltyterms and/or regularization terms

MV-EMPC provides a baseline for performance evaluation15 / 40

Frequency Control via EMPC

I Objective 1: Avoid critical frequency fluctuations

0 5 10 15 20 25 30 35 40

48

49

50

Time [sec]

Freq

uenc

y[H

z]

I Objective 2: Minimize cost of operations

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Optimal Control Problem

Objective Functionφ(u, z) = βφeco(u, z) + (1− β)φsp(u, z)

with risk-aversion parameter β ∈ [0; 1]I φeco: Operate system at minimum cost

I φsp: Restore the frequency to the nominal frequency

Closed-Loop Simulation

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

−1

0

1

Time [min]

Freq

.D

evia

tion

[Hz]

β = 0 β = 0.5 β = 1

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Optimization Algorithms

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Computational Aspects of EMPC

Problem structure is utilized for real-time solution of the OCPs

s1 s2 s3

(a) Scenario Coupling

g1 g2 g3

(b) Generator Coupling

t1 t2 t3

(c) Temporal Coupling

I Case (a) and (b) are handled by decomposition methods

I Case (c) is handled using Riccati-based methods

I Nested structures occur (c)→(b)→(a)

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EMPC Decomposition Algorithms

Schematic Diagram

Aggregator (Master Problem)

Subproblem 1 Subproblem 2 Subproblem 3 Subproblem 4

Subproblems can be solved in parallel and warm-start is applicable

Methods

Method Problem Class Iterations Accuracy DimensionsDWD LPs Few High IncreasingADMM CPs Many Low Constant

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Example: Block-Angular LPs

I Problem formulation

mint

{∑j∈J

cTj tj | Gjtj ≤ gj , j ∈ J ,

∑j∈J

Hjtj ≤ h}

I DWD: Extreme point representation

I ADMM: Problem splitting using auxiliary variables

vj = Hjtj , j ∈ J

Formulation of modified problem and simplified recursion ischallenging

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Benchmark

CPU Time to Solve the OCP

101 102 103

10−1

10−2

100

101

102

No. of Generators

CPU

Tim

e[se

c]

DWempc ADMMempc GurobiCPLEX MOSEK

I Memory issue around M = 3000 for centralized solves

I The performance of ADMM is very problem dependent22 / 40

Further ADMM Results

MV-EMPC

Step Description

1 Solve a single OCP for each uncertainty scenario2 Minimize variance s.t. non-anticipative constraints

Input-Constrained EMPC

Step Description

1 Solve unconstrained OCP2 Solve input-constrained OCP with no dynamics

A speedup in computational speed of more than an order ofmagnitude is achieved for both cases

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Homogeneous and Self-Dual Interior-Point Method

I Solution of the OCP minx{gT x |Ax = b, Cx ≤ d} is obtained

from solution of (z , s, τ, κ) ≥ 0 and

AT y + CT z + gτ = 0, Ax − bτ = 0Cx − dτ + s = 0, −gT x − bT y − dT z + κ = 0

I Warm-start works well for homogeneous and self-dual IPMs

tk Time

Inpu

t

u∗(k − 1) u∗(k)

I Search direction is computed using a Riccati-iterationprocedure

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LP Solver Comparison

101 10210−2

10−1

100

101

102

103

104

No. of Generators

CPU

Tim

e[se

c]

LPempc

Gurobi

CPLEX

MOSEK

FORCES

101 10210−2

10−1

100

101

102

103

Horizon Length

Warm-start reduces the CPU time by further 40% on average25 / 40

Integrated Planning and Control

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Production Planning

I Binary decisions b = (bT0 , bT

1 , . . . , bTL )T

I Problem formulation (simplified)min.

u,x ,z,bfR(u, x , z , b) + fZ(b)

s.t. xk+1 = Axk + Buk + Edk , k ∈ N0

zk = Czxk + Fzdk , k ∈ N1

cR(u, x , z , b) ≤ 0cZ(b) ≤ 0

I Two time scales

τ0 τ1 τ2 τL−1 τL

t0 t1 t2 t3 t4 tf−2 tf−1 tf

∆t

∆τ27 / 40

Hierarchical Algorithm

0 2 4 6 8 10 12 14 16 18 20

horizon of LL-OCP

Time [min]

Gene

rato

rSe

tpoi

nt[M

W]

UL-OCPLL-OCP

I The UL-OCP (MIQP/MILP) is closely related to the unitcommitment problem

I The UL-OCP may be solved with a low frequency

I Tailored algorithms can solve the LL-OCP (QP/LP) efficiently28 / 40

Three-Generator Example

0 20 40 60 80 100 120 140 160 1800

20

40

Time [min]

Powe

r[M

W]

r d d0 d900

I Direct solution of the full OCP is 15 minutes

I Solution times are 2s (UL-OCP) and 0.1s (LL-OCP)

I Single resolve of the UL-OCP is performed

I Cost increase is less than 1% for the hierarchical approach29 / 40

Optimal Reserve Planning

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Unit Commitment in Isolated Power Systems

The conventional unit commitment and economic dispatchproblem can be posed as an MILP

min.x ,y

f T x + gT y

s.t. Ax + By ≤ bx ∈ Rn

y ∈ {0, 1}m

I Constraints: Power balance, fixed reserves, production limits,ramping limits, etc.

I Variables: Production levels, reserve levels, on/off decisions,etc.

The solution of the MILP provides a ≈24-hours ahead productionplan with a ≈15-minute resolution

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Operational Reserves

t = 0

Post-contingentstate (stationary)

f ′(t) ≈ 1K ∆P(t) (ttr, f tr)

f nom

Pre-contingentstate

Post-contingentstate (transient)

Time [sec]

Freq

uenc

y[H

z]

I Primary reserves are critical to avoid power outages(blackouts) in the event of a contingency ∆P(t) 6= 0

I Primary reserves are activated in direct proportion to thefrequency deviation from the nominal frequency

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Minimum Frequency Constraint

It is critical that f (t) ≥ f for some cut-off frequency f

I Large interconnected systemsSystem inertia is large and approximately constant⇒ A fixed amount of primary reserve is sufficient

I Small isolated power systemsSystem inertia is small and varies considerably⇒ Minimum frequency constraints are required

The constraint f (t) ≥ f is intractable to handle usingmixed-integer linear programming

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Alternative Formulation

The minimum frequency constraint

f (t) ≥ f

may be expressed as

EPR(t) + ∆E rot ≥ P lostt

I EPR(t) =∫ t

0 PPR(τ)dτ is the energy contribution from theactivation of primary reserves

I ∆E rot is the energy contribution from the system inertia

I P lostt is the energy lost as a result of the contingency(generator trip)

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Sufficient Conditions

I Minimum frequency occurs no later than time tc

PPR(tc) ≥ P lost

I Satisfy f (t) ≥ f for t ≤ tc, i.e.EPR(t) + ∆E rot ≥ P lostt, t ≤ tc

t = 0 t = tc

ff tr

Time [sec]

Freq

uenc

y[H

z]

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Optimal Reserve Planning Problem (ORPP)

I Unit commitment and economic dispatch problem withminimum frequency constraints

I Compared to a conventional production and reserve planningproblem (BLUC)

I Simulations show that several potential blackouts are avoidedat a cost increase of 3%

I Tested in the Faroe Islands in 2015

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Conclusions & Future Work

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Conclusions

MethodsI MV-EMPC overcomes performance issues of CE-EMPC in

operation of uncertain systemsI MV-EMPC provides a baseline for approximate methods

AlgorithmsI Tailored decomposition schemes significantly reduces

computational requirements of the proposed EMPC methodsI Additional speedup is achieved using Riccati-based IPMs

ApplicationsI Simulations demonstrate that EMPC-based methods for

balance and frequency control reduce cost and riskI Unifying framework for balance control and unit commitmentI Frequency-constrained planning in isolated power systems

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Future Work

Feedback From ExperimentsI Use feedback from the Faroe Islands to improve the proposed

planning and control methods

Risk Measures in MV-EMPCI Employ other risk measures than the varianceI Increase sensitive to the tail shape of the cost distributionI Develop algorithms to solve the resulting OCPs efficiently

Algorithms for EMPCI Quadratic programming extensions of LP solversI Tuned and parallel implementationsI Scenario reduction in MV-EMPC

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Thanks! Questions and Comments?

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