Data-driven Coordination of Distributed Energy Resources Alejandro - - PowerPoint PPT Presentation

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data driven coordination of distributed energy resources
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Data-driven Coordination of Distributed Energy Resources Alejandro - - PowerPoint PPT Presentation

Nov 28, 2023 288 likes •736 views

Data-driven Coordination of Distributed Energy Resources Alejandro D. Dom nguez-Garc a Coordinated Science Laboratory Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign PSERC Webinar Series

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SLIDE 1

Data-driven Coordination of Distributed Energy Resources

Alejandro D. Dom´ ınguez-Garc´ ıa

Coordinated Science Laboratory Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign PSERC Webinar Series November 5, 2019

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SLIDE 2

Outline

1 Introduction 2 DER Coordination for Active Power Provision 3 LTC Coordination for Voltage Regulation 4 Concluding Remarks

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SLIDE 3

Impacts of Renewable-Based Power Generation Resources

◮ Deep penetration of renewable-based generation imposes additional requirements on ancillary services including:

  • Frequency regulation (in bulk power systems)
  • Reactive power support (in distribution systems)

◮ Frequency regulation in bulk power systems is typically achieved by controlling large synchronous generators

  • Resources in distribution systems are not utilized for this task

◮ Reactive power support in distribution systems is provided by devices such as load tap changers (LTCs) and fixed/switched capacitors

  • These devices are not designed to manage high variability in voltage

fluctuations induced by renewable-based generation

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 1 / 32

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SLIDE 4

The Solution

◮ An increasing number of DERs are being integrated into distribution systems ◮ DERs could potentially be utilized to provide ancillary services if properly coordinated by, e.g., an aggregator

PV systems Electric Vehicles Fuel Cells Residential Storage

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 2 / 32

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SLIDE 5

Need for Data-Driven Coordination

G G

bulk power system power distribution system

tie line

aggregator

◮ DER aggregators needs to develop appropriate coordination schemes so DERs can collectively provide services that meet certain requirements ◮ Model-based schemes may be infeasible due to the lack of accurate models ◮ Data-driven schemes that only rely on measurements provide a promising alternative for developing efficient coordination schemes

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 3 / 32

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SLIDE 6

Presentation Overview

Objective

To develop data-driven coordination frameworks for assets (DERs, LTCs) in distribution systems ◮ Part I. Active power provision problem

  • total active power exchanged between the distribution and bulk systems

needs to equal to some amount requested by the bulk system operator

◮ Part II. Voltage regulation problem

  • the voltage magnitude at each bus needs be maintained to stay close

to some reference value

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 4 / 32

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SLIDE 7

Outline

1 Introduction 2 DER Coordination for Active Power Provision 3 LTC Coordination for Voltage Regulation 4 Concluding Remarks

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SLIDE 8

Optimal DER Coordination Problem (ODCP)

Synchronous generator Inverter-interfaced source Load Synchronous generator Inverter-interfaced source Load Bus

Legend

Synchronous generator Inverter-interfaced source Load

Legend tie line

distribution system

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bulk grid

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1

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2

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3

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4

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5

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5

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pd

3

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pd

2

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pd

4

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pd

1

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Determine the DER active power injection vector, pg, that minimizes total cost of operation while satisfying:

  • C1. The power exchanged with the bulk system, y, tracks some

pre-specified value, y⋆

  • C2. The active power injection from each DER does not exceed its

corresponding capacity limits, i.e., pg ≤ pg ≤ pg

  • C3. The power flow on each line does not exceed its maximum capacity,

i.e., −f ≤ f ≤ f

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 5 / 32

slide-9
SLIDE 9

Input-Output System Model

◮ y can be written as a function of pg, pd, qd as follows: y = h(pg, pd, qd) ◮ h captures the impacts from both the physical laws as well as the effect of any reactive power control scheme

Assumption 1

  • H1. The rate of change in y w.r.t. pg is bounded for bounded changes in

the DER active power injections

  • H2. The total active power provided to the bulk power system will increase

when more active power is injected in the power distribution system

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 6 / 32

slide-10
SLIDE 10

ODCP Formulation

◮ The ODCP can be formulated as: minimize

pg∈[pg,pg] c(pg)

subject to h(pg, pd, qd) = y⋆, −f ≤ M −1(Cpg − pd) ≤ f

pg DER active power injections pg, pg DER upper and lower capacity limits pd, qd Load active, reactive power demands y⋆ Requested power to be exchanged with bulk grid f Line flow limits M Reduced node-to-edge incidence matrix C Matrix mapping DER indices to buses

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 7 / 32

slide-11
SLIDE 11

Data-Driven DER Coordination Framework

◮ Data defining the ODCP problem:

  • Cost function, c(·)
  • DER capacity limits, pg, pg
  • Network topology and DER location, M, C
  • Load active and reactive power demand, pd, qd
  • Line flow limits, f
  • Input-output model, h(·, ·, ·) ← Assumed unknown

◮ Real-time measurements available:

  • DER active power injections, pg[k], k = 1, 2, . . .
  • Active power exchanged with the bulk system, y[k], k = 1, 2, . . .

Framework Building Blocks

◮ An input-output (IO) model estimator that uses available real-time measurement data ◮ A controller that uses uses the identified IO model to solve the ODCP

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 8 / 32

slide-12
SLIDE 12

Two-Timescale Coordination Framework

time controller solves ODCP time iterations in estimation process estimator updates sensitivities

Estimation Process

pd and qd remain approximately constant between two time instants; therefore, changes in y[k] that occur across time steps in the estimation process depend only on changes in pg[k]; thus, y[k] = h(pg[k], pd, qd), k = 0, 1, . . .

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 9 / 32

slide-13
SLIDE 13

Input-Output Model as a Linear Time-Varying System

◮ For notational simplicity, define u[k] = pg[k], u = pg, u = pg, and π = [(pd)⊤, (qd)⊤]⊤; then, the IO model can be written as: y[k] = h(u[k], π), k = 0, 1, . . . , ◮ For k > 1, the above equation can be transformed into the following equivalent linear time-varying model: y[k] = y[k − 1] + φ[k]⊤(u[k] − u[k − 1]) where φ[k]⊤ = [φi[k]] = ∂h ∂u

  • ˜

u[k]

, with ˜ u[k] = aku[k] + (1 − ak)u[k] ◮ φ[k] is referred to as the sensitivity vector at time step k ◮ The entries of φ[k] are bounded for all k by Assumption 1

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 10 / 32

slide-14
SLIDE 14

Estimator – Estimation Step

◮ At time step k, the objective of the estimator is to obtain an estimate

  • f φ[k], denoted by ˆ

φ[k], using available measurements of u and y ◮ Problem P1: ˆ φ[k] = arg min

ˆ φ∈Q=[b1,b1]n

Je( ˆ φ) = 1 2(y[k − 1] − ˆ y[k − 1])2 subject to ˆ y[k − 1] = y[k − 2] + ˆ φ⊤(u[k − 1] − u[k − 2])

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 11 / 32

slide-15
SLIDE 15

Estimator – Control Step

◮ The objective of the controller during the estimation process is to ensure that the output tracks the target [Different from the ODCP] ◮ Problem P2: u[k] = arg min

u∈U=[u,u]

Jc(u) = 1 2(y⋆ − ˆ y[k])2 subject to ˆ y[k] = y[k − 1] + ˆ φ[k]⊤(u − u[k − 1]) ◮ Note that ˆ φ[k] is used to predict the value of y[k] for a given u

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 12 / 32

slide-16
SLIDE 16

Estimation Process Workflow

· · · u[k − 1] → y[k − 1] → ˆ φ

  • estimation step

control step

  • [k] → u[k] → y[k] → ˆ

φ[k + 1] · · · ◮ At the beginning of iteration k, y[k − 1] is used in Problem P1 to update the sensitivity vector estimate, ˆ φ[k] ◮ The updated sensitivity vector estimate, ˆ φ[k], is then used in Problem P2 to determine the control, u[k] ◮ Then, the DERs are instructed to change their active power injection set-points based on u[k] ◮ Problems P1 and P2 are not solved to completion for each k ◮ Instead, we iterate the projected gradient descent algorithm that would solve them for one step at each iteration k

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 13 / 32

slide-17
SLIDE 17

Simulation Setup

1 2 3 4 5 6 7 8 9 12 14 10 11 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 84 85 83 86 87 88 89 90 91 92 93 94 95 96 97 98 99 101 100 102 103 104 105 106 107 108 109 110 111 112 113 114 117 115 118 119 116 121 122 120 DER

Figure 1: The IEEE 123-bus distribution test feeder.

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 14 / 32

slide-18
SLIDE 18

Tracking Performance During Estimation

0.0 2.0 4.0 6.0 8.0 10.0

time (s)

−600 −400 −200

tracking error (kW)

  • 3000 kW
  • 2900 kW
  • 2800 kW
  • 2700 kW
  • 2600 kW
  • 2500 kW

Figure 2: Tracking error for βk = 0.02 under various tracking targets.

0.0 2.0 4.0 6.0 8.0 10.0

time (s)

−100 −75 −50 −25

tracking error (kW)

0.005 0.01 0.02 0.03 0.04 0.05

Figure 3: Tracking error for y⋆ = −3000 kW and various constant control step sizes.

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 15 / 32

slide-19
SLIDE 19

Estimation Accuracy

◮ Mean absolute error (MAE) of estimation errors: MAE = n

i=1

  • ˆ

φi[k] − φi[k]

  • n

, where n is the number of DERs

0.0 2.0 4.0 6.0 8.0 10.0

time (s)

0.00 0.02 0.04 0.06

MAE (p.u./p.u.)

0.005 0.01 0.02 0.03 0.04 0.05

Figure 4: Estimation error under various control step sizes.

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 16 / 32

slide-20
SLIDE 20

Outline

1 Introduction 2 DER Coordination for Active Power Provision 3 LTC Coordination for Voltage Regulation 4 Concluding Remarks

slide-21
SLIDE 21

Background

◮ Voltage regulation transformers—also referred to as Load Tap Changers (LTCs)—are widely utilized in power distribution systems to regulate voltage magnitudes along a feeder ◮ Model for a load tap changer on a line connecting buses i and j:

+ − + − i j Vi Vj rℓ + i xℓ tℓ : 1 + − Vj′ = Vi t2

ℓ

Vi Primary side voltage magnitude Vj0 Secondary side voltage magnitude Vj Line receiving end voltage magnitude t` Tap ratio r` Transformer + line resistance x` Transformer + line reactance

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◮ The tap ratio, tl, typically takes 33 discrete values ranging from 0.9 to 1.1, by an increment of 5/8%, i.e., tl ∈ T = {0.9, 0.90625, · · · , 1.09375, 1.1}

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 17 / 32

slide-22
SLIDE 22

Problem Motivation

◮ Current LTC control schemes are myopic:

  • Based on local voltage measurements
  • Do not account for future uncertainty effects on current control actions

◮ These schemes are no longer suitable because of increased variability and uncertainty in uncontrolled power injections arising from:

  • Residential PV installations
  • Electric vehicles

◮ In addition, the use of controlled DERs for providing frequency regulation to the bulk grid has an impact on voltage regulation

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 18 / 32

slide-23
SLIDE 23

Volt/VAR Control Architecture

t0 t1 t2 Slow Time-Scale Control time Fast Time-Scale Control

◮ Slow Time-Scale: LTCs are periodically dispatched so as to reduce mechanical wear ◮ Fast Time-Scale Control: Power-electronic-interfaced DERs with reactive power provision capability

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 19 / 32

slide-24
SLIDE 24

Optimal LTC Dispatch Problem

Synchronous generator Inverter-interfaced source Load Synchronous generator Inverter-interfaced source Load Bus

Legend

Synchronous generator Inverter-interfaced source Load

Legend tie line

distribution system

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bulk grid

V0

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V1

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V2

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V3

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V4

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V5

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N = 5 L = 5 n = 4

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Load tap changer (LTC)

Objective

Find a policy for determining the LTC tap positions based on measurements of current ◮ tap ratios ◮ bus voltage magnitudes so as to minimize bus voltages deviations from some reference value as power injections change as time evolves

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 20 / 32

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SLIDE 25

Power Distribution System Model

◮ The relation between square voltage magnitudes and active/reactive power injections and LTC tap ratios at instant k can be written as V [k] = g(p[k], q[k], t[k])

V [k] Vector of voltage magnitudes at instant k p[k], q[k] Vector of active, reactive, power injections at instant k t[k] Vector of tap ratios at instant k

◮ Network topology and transmission line parameters are encapsulated into g

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 21 / 32

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SLIDE 26

Assumptions

  • T1. Changes in active and reactive power injections at instant k,

∆p[k] := p[k + 1] − p[k] and ∆q[k] := q[k + 1] − q[k], are random

  • T2. Active and reactive power injections k, p[k] and q[k], are not

measured and their joint probability distribution is unknown

  • T3. Network topology is known, i.e., M is known
  • T4. Line parameters are unknown, i.e., r and x are unknown

◮ Because of Assumption T1 is natural to formulate the problem as a Markov Decision Process (MDP) ◮ Because of Assumptions T2 – T4, we will have to resort to reinforcement learning (RL) techniques to solve this MDP

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 22 / 32

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SLIDE 27

LTC Coordination Problem as an MDP

◮ State space, S: the state, s, is composed of tap ratio and squared voltage magnitude vector, i.e., s = (t, v), t ∈ T Lt, v ∈ RN; thus, S ⊆ T Lt × RN ◮ Action space, A: the action, a, is the change in LTC tap ratio between two consecutive time instants, i.e., a = ∆t, ∆t ∈ ∆T Lt =: A, where ∆T = {0, ±0.00625, · · · , ±0.19375, ±0.2} is the set set of feasible tap ratio changes ◮ Reward function, R: the reward when the system transitions from state s = (t, v) into state s′ = (t′, v′) after taking action a = ∆t is R(s, a, s′) = − 1 N v′ − v⋆, i.e., we are penalizing voltage deviation from some reference value v⋆

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 23 / 32

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SLIDE 28

LTC Coordination Problem as an MDP

◮ State transitions: governed by random changes in active and reactive power injection vectors, ∆p := p′ − p and ∆q := q′ − q: s′ = h(s, a, ∆p, ∆q)

  • Network topology and transmission line paramters are encapsulated

into h

  • Probability of transitioning from s to s′ under action a, P(s′ | s, a),

could be computed if h and the joint pdf of ∆p and ∆q were know

Objective

Find a policy π : (t, v) → ∆t, t ∈ T Lt, v ∈ RN, ∆t ∈ ∆T Lt so that − 1 N

∞

  • k=0

γkE

  • v[k + 1] − v⋆
  • t[0] = t0, v[0] = v0
  • is maximized (equivalent to minimizing voltage deviations)

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 24 / 32

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SLIDE 29

Solving the LTC Coordination Problem

◮ Let ¯ R(s, a) denote the expected reward for the pair (s, a) ◮ The MDP is solved when we find Q∗(s, a), s ∈ S, a ∈ A, and π∗(s), s ∈ S, satisfying Q∗(s, a) = ¯ R(s, a) + γ

  • s′∈S

P(s′|s, a) max

a′∈A Q∗(s′, a′)

π∗(s) = arg max

a

Q∗(s, a) ◮ Two issues in our setting:

  • I1. We do not know P(· | ·, ·)
  • I2. Even if we knew P(· | ·, ·), we could not solve for Q∗(s, a) efficiently

because of the curse of dimensionality in the state and action spaces

◮ To circumvent Issue I1, we apply a model-free RL algorithm that utilizes transition samples obtained via a virtual transition generator ◮ To circumvent Issue I2, we use function approximation and a learning scheme for sequential estimation of the action-value function

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 25 / 32

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SLIDE 30

LTC Coordination Framework Building Blocks

Power Distribution System

Environment

Greedy Actor

Acting Agent action reward

Action-Value Function Estimator (Critic) History

Learning Agent

Virtual Transition Generator Exploratory Actor

state action action-value function estimate

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 26 / 32

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SLIDE 31

Action-Value Function Value Estimator

◮ Let ˆ Q(·, ·) denote an approximation of the optimal action-value function Q(·, ·) ◮ Methods for obtaining ˆ Q(·, ·) include:

  • Parametric functions
  • Neural networks

◮ Here we consider a linear parametrization of ˆ Q(·, ·): ˆ Q(s, a) = w⊤φ(s, a), where w ∈ Rf is the parameter vector and φ : S × A → Rf is some basis function ◮ Let D = {(s, a, r, s′) : s, s′ ∈ S, a ∈ A} denote a set (batch) of transition samples obtained via observation or simulation ◮ We use least-square policy iteration (LSPI) algorithm [Lagoudakis and Parr, 2003] to find w that best fits the transition samples in D

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 27 / 32

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SLIDE 32

Challenges

◮ The LSPI algorithm requires adequate transition samples that spread

  • ver S × A

◮ This is challenging in power systems since the system operational reliability might be jeopardized when exploring randomly

  • We address this by developing by generating samples via a virtual

transition generator that leverages historical system operational data

◮ The LSPI also suffers from the curse of dimensionality when the action space is large—the case in the LTC coordination problem

  • We address this by using a sequential scheme that breaks the learning

problem into smaller problems and uses the LSPI algorithm on those

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 28 / 32

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SLIDE 33

IEEE 123-bus Test Feeder

1 2 3 4 5 6 7 8 9 12 14 10 11 13 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 84 85 83 86 87 88 89 90 91 92 93 94 95 96 97 98 99 101 100 102 103 104 105 106 107 108 109 110 111 112 113 114 117 115 118 119 116 121 122 120 Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 29 / 32

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SLIDE 34

IEEE 123-bus Test Feeder Results

2 4 6 8 10 12 14 16 18 20 22 24 time (hour)

  • 8
  • 6
  • 4
  • 2

2 4 tap position

Batch RL

1 2 3 4

2 4 6 8 10 12 14 16 18 20 22 24 time (hour)

  • 8
  • 6
  • 4
  • 2

2 4 tap position

Exhaustive search

1 2 3 4

Figure 5: Tap positions.

2 4 6 8 10 12 14 16 18 20 22 24 time (hour) −10 −5 reward (×10−3)

batch RL exhaustive search conventional scheme

Figure 6: Immediate rewards.

1 2 3 4 5 6 7 8 time (day) −10 −5 reward (×10−3)

batch RL exhaustive search conventional scheme

Figure 7: Rewards over 7 days.

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 30 / 32

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SLIDE 35

Outline

1 Introduction 2 DER Coordination for Active Power Provision 3 LTC Coordination for Voltage Regulation 4 Concluding Remarks

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SLIDE 36

Conclusions

◮ We developed a data-driven coordination framework for coordinating assets in distribution systems to provide ancillary services: ◮ The proposed framework:

  • It assumes no prior information on the distribution system model,

except knowledge of network topology

  • It mainly relies on measurements
  • It is adaptive and robust to changes in operating conditions

◮ Refer to the following papers for more details:

  • 1. H. Xu, A. Dom´

ınguez-Garc´ ıa, and P. Sauer, “Data-driven Coordination

  • f Distributed Energy Resources for Active Power Provision,” IEEE

Transactions on Power System, 2019.

  • 2. H. Xu, A. Dom´

ınguez-Garc´ ıa, and P. W. Sauer, “Optimal tap setting of voltage regulation transformers using batch reinforcement learning,” IEEE Transactions on Power System, 2019.

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 31 / 32

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SLIDE 37

Questions?

Alejandro D. Dom´ ınguez-Garc´ ıa e-mail: aledan@ILLINOIS.EDU url: https://aledan.ece.illinois.edu

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 32 / 32

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SLIDE 38

References I

[Lagoudakis and Parr, 2003] Lagoudakis, M. G. and Parr, R. (2003). Least-squares policy iteration. Journal of Machine Learning Research, 4:1107–1149.

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SLIDE 39

A Primer on Markov Decision Processes

◮ A Markov Decision Process (MDP) is defined as a 5-tuple (S, A, P, R, γ) where

  • S is a finite set of states
  • A is a finite set of actions
  • P is a Markovian transition model
  • R : S × A × S → R is a reward function
  • γ ∈ [0, 1) is a discount factor

◮ Let s[k] and a[k] denote random variables (r.v.’s) respectively describing the value the state and action take at time instant k ◮ Let r[k] denote a r.v. describing the reward received after taking action a[k] in state s[k] and transitioning to state s[k + 1]; then r[k] = R

  • s[k], a[k], s[k + 1]
  • ◮ A deterministic policy π is a mapping from S to A, i.e.,

a = π(s), s ∈ S, a ∈ A

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 33 / 32

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SLIDE 40

A Primer on Markov Decision Processes

Objective

Given some initial state, s0, we want to find a deterministic policy, π∗, that maximizes the expected value of the cumulative discounted reward, i.e., π∗ = arg max

π ∞

  • k=0

γkE

  • r[k]
  • s[0] = s0
  • Dom´

ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 34 / 32

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SLIDE 41

A Primer on Markov Decision Processes

◮ The action-value function under policy π is defined as Qπ(s, a) =

∞

  • k=0

γkE

  • r[k]
  • s[k] = s, a[k] = a; π
  • ,

s ∈ S, a ∈ A ◮ The optimal action-value function, Q∗(s, a), s ∈ S, a ∈ A is the maximum action-value function over all policies, i.e., Q∗(s, a) = max

π

Qπ(s, a) ◮ Q∗(s, a), s ∈ S, a ∈ A, satisfies the following Bellman equation: Q∗(s, a) = E [r | s, a] + γ

  • s′∈S

P(s′|s, a) max

a′∈A Q∗(s′, a′)

◮ The optimal policy, π∗(s), s ∈ S, is obtained as follows: π∗(s) = arg max

a

Q∗(s, a) ◮ The MDP is solved if we find Q∗(·, ·) and the corresponding π∗(·)

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 35 / 32

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SLIDE 42

LSPI Algorithm Iteration

◮ Let wi denote the estimate of w at the beginning of iteration i ◮ For each transition sample (s, a, r, s′) ∈ D, compute a′ = arg max

α∈A

w⊤

i φ(s′, α)

◮ Update the estimate of w as follows: wi+1 = B−1b where B =

  • (s,a,r,s′)∈D

φ(s, a)

  • φ(s, a) − γφ(s′, a′)

⊤

  • rank-1 matrix

b =

  • (s,a,r,s′)∈D

φ(s, a)r

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 36 / 32

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SLIDE 43

Timeline

t0 t1 t2 Slow Time-Scale Control time Fast Time-Scale Control

◮ Policy updated every K∆T units of time, e.g., 2 hours ◮ Updated policy used for tap setting for K time instants

Dom´ ınguez-Garc´ ıa (ECE ILLINOIS) Data-driven Coordination aledan@ILLINOIS.EDU 37 / 32