Copyright by Joannah Igbe Otashu 2020 the Dissertation Committee for Joannah Igbe Otashu Certiﬁes That This Is the Approved Version of the Following Dissertation

Copyright by Joannah Igbe Otashu 2020 The Dissertation Committee for Joannah Igbe Otashu certiﬁes that this is the approved version of the following dissertation:

Employing Chemical Processes as Grid-level Energy Storage Devices: demand response and frequency regulation

Committee:

Michael Baldea, Supervisor

Efstathios Bakolas

Thomas F. Edgar

Thomas M. Truskett Employing Chemical Processes as Grid-level Energy Storage Devices: demand response and frequency regulation

Joannah Igbe Otashu

DISSERTATION Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulﬁllment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN May 2020 Dedicated to my mum and dad, Esther Aboje Otashu and Abel Otashu. Acknowledgments

Many thanks to my graduate supervisor, Professor Michael Baldea for the immense support provided throughout my Ph.D. journey. Thank you for actively contributing to my professional growth including providing me with opportunities for sponsored research trips and a variety of collaborations.

Special thanks to Professors Thomas Edgar, Thomas Truskett and Efstathios Bakolas for your counsel and agreeing to serve on my dissertation committee.

Thank you Professor Ross Baldick for helping me understand better the operation of electricity markets and for always willing to be a sounding board for my research ideas.

Many thanks to my undergraduate professors; Professor Saidu Waziri – for helping me write my ﬁrst research grant proposal, Professor Nehemiah Maina – for encouraging me to get a doctorate abroad and supporting me throughout, Professors Mohammed Dabo, B.O. Aderemi, A.O. Ameh, M.T. Isa – for your support over the years.

Thank you Dalhatu Hamza for your patience and support during my application to graduate school.

I wish to thank the members of Baldea and Edgar research groups for your contributions and helpful discussions regarding my research.

Special thanks to my family; dad, mum, Victoria, Patience, Ene, Opyotu, and David – for your sacriﬁces, encouragement and support. I love you all.

v Thanks to the friends I met during grad school and my long-standing friends from years back – your friendship helped me maintain my sanity throughout graduate school.

Finally, thanks to the Schlumberger Foundation Faculty for the Future Program for ﬁnancial support.

vi Employing Chemical Processes as Grid-level Energy Storage Devices: demand response and frequency regulation

Joannah Igbe Otashu, Ph.D. The University of Texas at Austin, 2020

Supervisor: Michael Baldea

Reliable operation of the electric grid which comprises of instantaneously matching electricity demand and supply is a tier one security concern and has received increased attention in recent years. This is a challenging task as both electricity demand and supply are characterized by variability and uncertainty; the volatility on the supply end originating from increased integration of renewable energy in power generation. Electricity demand typically peaks in the late afternoons whereas wind and solar generation peak in the early mornings and mid-day respectively. To reconcile this electricity supply and demand mismatch, energy storage is employed for storing excess power generated during oﬀ-peak demand periods. Cor- respondingly, stored power is discharged during peak electricity demand hours. Electricity storage is however presently expensive and current grid storage devices are insuﬃcient to ensure reliable electric grid operation. Standby generators with fast ramp rates which are generally more expensive to run are also employed when needed. A solution to this problem involves motivating end-users of electricity to modify their demand patterns in manner that enhances grid stability. This activity, referred to as demand response (DR) can help relieve

vii the strain on the grid by tailoring end-user power consumption to match the time varying supply of electricity.

Chemical industries that are power-intensive can provide valuable DR as they represent localized grid loads with end products that are generally easier and cheaper to store compared to electricity. These industries can ramp production during oﬀ-peak power demand hours storing excess product and reduce production rates when power demand peaks while depleting inventory to meet any required product demand. However, the operation of chemical industries are subject to strict constraints including; economic (e.g., reliable product delivery), feasibility (e.g., limits on process variables) or safety concerns. Additionally, the dynamics of chemical processes are typically described by high-order non-linear models and ramp rates and capacity limits alone, which are typically employed on the grid side to manage demand response activities, are not suﬃcient to characterize the transient nature of chemical processes.

In this dissertation, we employ modeling techniques and production scheduling to evaluate the provision of demand response by chemical industries in highly volatile electricity markets. We develop and illustrate the use of suitable dynamic models to represent the chemical plant operation while providing fast-paced DR. As fast DR contribute signiﬁcantly to the solution pool available for handling power grid contingencies, this study will aid the smoother deployment of industrial chemical loads as grid support devices while ensuring safe and feasible operation of the chemical plant.

viii Table of Contents

Acknowledgments v

Abstract vii

List of Tables xiii

List of Figures xiv

Chapter 1. Introduction 1 1.1 Modeling industrial processes for fast demand response ...... 1 1.2 Guide to chapters ...... 3

Chapter 2. Grid-level “battery” operation of chemical processes and demand- side participation in short-term electricity markets 5 2.1 Introduction ...... 5 2.2 Problem formulation ...... 9 2.2.1 Problem Statement ...... 9 2.2.2 Process model ...... 11 2.2.3 Top tier problem: Engagement in LTM (Base case) ...... 13 2.2.4 Lower level problem: Engagement in STM ...... 14 2.2.5 Economics ...... 18 2.3 Case study ...... 20 2.3.1 Model parameters and scheduling implementation ...... 20 2.3.2 Results and discussion ...... 24 2.3.2.1 Optimal production proﬁles ...... 24 2.3.2.2 Economics ...... 28 2.3.3 Impact of process agility on performance ...... 29 2.4 Conclusions ...... 32

ix Chapter 3. Demand response-oriented dynamic modeling and operational optimization of membrane-based chlor-alkali plants 33 3.1 Introduction ...... 33 3.2 Background and Literature Review ...... 36 3.2.1 The Chlor-Alkali Industry ...... 36 3.2.2 Existing DR Eﬀorts ...... 38 3.3 A DR-relevant model of the membrane-based chlor-alkali process ...... 39 3.3.1 Mathematical model of the membrane cell ...... 41 3.3.1.1 Electrochemical model ...... 43 3.3.1.2 Material balance model ...... 44 3.3.1.3 Thermal model ...... 45 3.3.2 Liquefaction unit ...... 46 3.3.3 Storage unit ...... 48 3.4 Simulation case study ...... 49 3.4.1 Steady-state simulation ...... 49 3.4.2 Dynamic simulation ...... 50 3.5 Price-based demand response operation of the membrane chlor-alkali process 52 3.5.1 Production scheduling problem ...... 52 3.5.2 Optimization results ...... 54 3.6 Conclusion ...... 58

Chapter 4. Scheduling Chemical Processes for Frequency Regulation 61 4.1 Introduction ...... 61 4.2 Problem Formulation ...... 66 4.2.1 Time description ...... 66 4.2.2 Demand response scheduling and a correction based on grid frequency 67 4.2.3 Frequency-adjusted price ...... 69 4.2.4 Process responsiveness ...... 71

4.2.4.1 Case 1: Rf u 0 (minimum flexibility) ...... 74 4.2.4.2 Case 2: Rf u 1 (maximum flexibility with low proportionality) 75 4.2.4.3 Case 3: Rf u 0.5 (flexible and proportional response) . . . . . 76

4.2.4.4 Case 4: Negative Rf values (reverse and undesirable response) 76

x 4.2.5 Measure of desired process response ...... 78 4.3 Case study ...... 79 4.3.1 Chlor-alkali process description ...... 79 4.3.2 Dynamic model ...... 81 4.3.3 Implementation of production scheduling problems ...... 83 4.3.4 Frequency data ...... 83 4.4 Results and discussion ...... 84 4.4.1 Optimal production schedules ...... 84 4.4.2 Flexibility and proportionality of process response ...... 87 4.4.3 Energy consumption ...... 90 4.4.4 Choice of α ...... 92 4.5 Conclusions ...... 94

Chapter 5. An Integrated Approach to Demand Response: Study on the Complete Integration of Chemical Processes in Grid-Level Op- erations 96 5.1 Introduction ...... 96 5.2 Background ...... 98 5.2.1 Power system model ...... 98 5.2.2 Transient load models for chemical processes ...... 101 5.2.2.1 Linear state space models ...... 102 5.2.2.2 Hammerstein-Wiener models ...... 103 5.3 Problem definition ...... 104 5.4 Case study ...... 106 5.4.1 Industrial process: chlor-alkali production ...... 106 5.4.2 Transient load model for chlor-alkali process ...... 107 5.4.3 Power grid topology ...... 109 5.4.4 Optimization implementation ...... 110 5.5 Discussion of results ...... 112 5.5.1 Flexible operation of concentrated end-user power demand ...... 112 5.5.2 Effect of flexible load location ...... 115 5.5.3 Increasing the contribution of chemical plant to overall load ...... 119 5.5.4 System-wide power demand ...... 119 5.6 Conclusions ...... 122

xi Chapter 6. Conclusions and Recommendations for Future Work 123 Dissemination of Research ...... 127

Appendices 129

Appendix A. Nomenclature and data – chapter 2 130 Nomenclature ...... 130 Electricity price data ...... 131

Appendix B. Nomenclature and data – chapter 3 133 Nomenclature and parameter values for model equations ...... 133 Calculation of lumped heat capacity ...... 135 Input variables for scheduling problem formulation ...... 136 Electricity price data ...... 136

Bibliography 139

Vita 159

xii List of Tables

1.1 Conventional and DR-relevant scheduling formulations ...... 2

2.1 Process model parameters ...... 22 2.2 Comparison of process economics for top and lower level scheduling formulations 28

3.1 Review of relevant models for electrolytic processes ...... 40

4.1 Interpretation of process response in terms of the responsiveness factor (decreasing desirability down the rows)...... 77

5.1 Structure of data driven SS and HW models for chlor-alkali plant ...... 108 5.2 Scenarios considered in solving P1 and P2 ...... 112 5.3 Results for plant load concentrated on bus 7 (data indicate percent change from nominal value) ...... 113 5.4 Results for plant load concentrated on bus 5 ...... 116 5.5 Results for distributed plant load (data indicate percent change from nominal value) ...... 118

A.1 Nomenclature and abbreviations ...... 130

B.1 List of symbols for electrolyzer model ...... 133 B.2 List of symbols for compressor model ...... 135 B.3 List of symbols for storage tank model ...... 135 B.4 Heat capacity at 298K [1] ...... 136 B.5 List of model input variables ...... 137

xiii List of Figures

2.1 Grid with DR participating chemical process and other loads (which we assume fixed) on the grid. The chemical process exploits the availability of storage and underutilized plant capacity to modulate production rate and engage in both LTM and STM...... 10 2.2 (a) Power consumption profile as predicted by the production schedule and (b) actual power consumption profile for a hypothetical chemical process in response to time-varying electricity price. The power consumption is minimized for a given time slot, i = ι of length ts. Energy cost is optimized for the rest of the horizon based on LTM and forecast STM prices. The difference between the long-term and short-term optimal power consumption profiles for slot ι represents extra DR capacity referred to as “DR headroom”, that can be deployed in STMs...... 17 2.3 Flowchart for process simultaneous engagement in long and short-term electricity markets. DR headroom is sold in the STM when predicted profitable. 21 2.4 Optimal power consumption profiles for top and lower level problems for sample day, 01/09/17. ‘SD DAM’ stands for same day DAM price and ‘DB FMM’ stands for day-before FMM price. Red stars indicate sale of DR headroom in the FMM. The black dashed line is the actual power consumption profile . . 25 2.5 Available DR headroom (top) and process bid prices as dictated by the DR headroom cost (bottom) for all three STM forecast scenarios for the first sample day, 01/09/17. Bid price is set to zero when DR headroom cost is negative ...... 26 2.6 Effect of process time constant on the agility of a process assuming perfect STM price forecast for 01/09/17. Higher time constants result in sluggish response to rapid set-point changes (actual power consumption, P, does not instantaneously track set points, P sp) leading to a transient (rather than predominantly steady-state) operation. Engagement in the STM (red stars) also diminishes with increasing time constant ...... 30 2.7 Comparison of (a) top: Energy cost and total revenue earned from the STM (b) bottom: Total profit to the process for top and lower level (perfect forecast, sample day 01/09/17) scheduling formulations, for increasing process time constant ...... 31

3.1 Process ﬂow diagram for chlor-alkali production through membrane electrolysis. Units in black dashed boxes are modeled in this chapter (as described later)...... 37

xiv 3.2 Schematic diagram of a single membrane cell electrolyzer...... 42 3.3 Chlorine gas liquefaction unit - Uhde system. Chlorine flows from first horizontal heat exchanger at −15oC into the the second liquefier, placed at an angle of 60o where it is further cooled to −55oC. Red dash lines show the refrigerant stream. [2] ...... 47 3.4 Plant response to step change in current density from 0.4A/cm2 to 0.6A/cm2 at time, t = 15 min...... 50 3.5 Temperature trends with current density changes for conditions specified in Table B.5. Left: steady state cell temperature for constant inlet feed conditions. Right: required inlet feed temperature for specified constant outlet cell conditions...... 51 3.6 Optimal schedule showing the actual power consumption (middle) and inventory (bottom) for the chlor-alkali plant over a 3 day horizon...... 55 3.7 Optimal production schedule for the chlor-alkali plant for a 24 h horizon . . 57 3.8 Evolution of cell variables for the optimal production schedule ...... 58 3.9 Optimal temperature profile for feed electrolyte (top) and resulting cell temperature for the optimal schedule (bottom)...... 59

4.1 Top: System frequency values over a three hour period Bottom: Correspond- ing frequency-adjusted prices compared with the nominal base prices for two diﬀerent values of α ...... 71

4.2 Range of process responsiveness factor, Rf (positive), values. Black curve indicates proportional power change to grid frequency deviations. The red star denotes the point where maximum flexibility is offered for maximum frequency deviation observed ...... 74 4.3 Chlorine production through brine membrane electrolysis. The liquefaction and storage units (in orange box) provide leverage for engagement in demand response activities ...... 80 4.4 Optimal base case and frequency-adjusted schedule for a sample set of frequency data; α =600...... 85 4.5 Evolution of stored product, liquid chlorine (left) and cell temperature (right) over time ...... 86 4.6 Process responsiveness factor values with scaled power and frequency deviations for the first (left) and third (right) hour for a scenario of frequency data ...... 88

4.7 Aggregate positive (left) and negative (right) process responsiveness, Rfs, for each hour ...... 89 4.8 Average distribution of process responsiveness according to classiﬁcation presented in Table 4.1. Average proportionality (Rfp) for each hour is about 2 - 3%. Flexible but disproportionate response (Rf value of 1) was not observed 89

xv 4.9 Percent change in total (left) and hourly (right) energy consumption rates versus increased product generation for the optimal schedules generated for fifty scenarios of frequency-adjusted prices ...... 91 4.10 Optimal production schedule and cell temperature evolution for different values of α ...... 93 4.11 Hourly (positive) flexibility for different values of α ...... 94

5.1 Monolithic and sequential formulations addressing the optimal power flow and demand response scheduling problem ...... 105 5.2 Reduced-order model versus validation data responses for the chlor-alkali plant109 5.3 Modified IEEE 24 bus reliability test system. Small red and large green circles represent loads and wind turbines, blue crosses represent generators and white rectangles represent batteries [3] ...... 110 5.4 Wind generation and power demand pattern for Houston hub, 09/13/18 - 09/14/18 (scaled relative to the maximum values observed). Data source: www.ercot.com ...... 111 5.5 Power demand profile and plant output variables for a single chlor-alkali plant under normal grid conditions, plant load on bus 7 ...... 113 5.6 Power demand profile and plant output variables for a single chlor-alkali plant under grid congestion, plant load on bus 7 ...... 115 5.7 Sample product inventory for single chlor-alkali plant (left) and state of charge of grid connected batteries (right) under normal grid conditions ...... 116 5.8 Power demand profile and plant output variables under grid congestion, plant load on bus 5 ...... 117 5.9 Costs for sequential formulation P2 compared to corresponding values for monolithic P1. Bubble size denotes total chemical plant capacity as a percent of the total system load ...... 120 5.10 Overall grid load under congestion, concentrated load on bus 7 ...... 121

A.1 Electricity prices for day ahead market and ﬁfteen minute market for ten consecutive days (2nd week and previous weekend) in the months of January - July, 2017. Data source: www.ercot.com ...... 132 A.2 Day ahead market and ﬁfteen minute market electricity prices for sample day, 01/09/17. Data source: www.ercot.com ...... 132

B.1 Electricity price for the ﬁfteen minute electricity market, January 06-08, 2017. Data source: www.ercot.com ...... 138

xvi Chapter 1

Introduction

This chapter is based on material published in paper “Otashu, J. I., & Baldea, M. (2018). Grid-level battery operation of chemical processes and demand-side participation in short-term electricity markets. Applied energy, 220, 562-575”[4]. Joannah Otashu was the primary contributor to this paper.

1.1 Modeling industrial processes for fast demand response

Conventional plant production scheduling approaches (Table 1.1) rely on steady state plant models [5, 6] to determine the production sequence of a given process over a period of time. Such steady state models are suitable for schedules with decision targets that change at a slower pace compared to the dominant process time constant – i.e., steady state is achieved before changes are made to the input variables. However, demand response scheduling of chemical processes often require frequently varying input signals in the timescales of several seconds – minutes as the production schedule must be adapted to reﬂect the dynamic market or operating conditions (e.g., electricity price signals). Such frequent schedule change may overlap the process time constant (which is typically in the order of several minutes – hours) leading to transient plant operation. As a result, transition time arrays or rate of change constraints which are usually used to represent dynamic information in scheduling calculations [7, 8, 9, 10] do not adequately predict the dynamic response of the plant [11, 12].

1 A modiﬁcation to conventional scheduling in which detailed transition proﬁles are computed

Table 1.1: Conventional and DR-relevant scheduling formulations Conventional scheduling [13] DR-relevant scheduling [11] Max/Min Economic objective Max/Min Economic objective Objective hourly - monthly several seconds - decision variables hourly decision variables Subject to Steady state process model Inventory/storage model Inventory/storage model Capacity and flow constraints Capacity and flow constraints Demand satisfaction constraints Demand satisfaction constraints Constraint on rate of change of pro- (Reduced order) dynamic model of duction targets (stand-in for process process and product quality con- dynamics) straints off-line from dynamic process models [14] is similarly unsuitable since this formulation still assumes that steady state is attained between production targets.

On the one hand, detailed, ﬁrst-principles process models can be used to express the dynamic response of the process. However, such models are typically of high order and highly nonlinear, requiring impractical computational time e.g., yielding results in time periods longer than is required to exploit opportunities in electricity markets. A solution to this problem involves developing computationally tractable representations of the chemical process using reduced order models. The low-order models could be data-driven [15] or physics-based e.g., scale bridging models (SBMs) [16].

SBMs are a set of low-order representations of the closed-loop dynamic behavior of a chemical process, that is relevant to scheduling calculations (including, e.g., the evolution of production rate and product quality following changes in production targets, and hence, electricity consumption). Unlike conventional models utilized for scheduling, SBMs are adapted for use in scheduling calculations geared towards DR participation. By incorporating process

2 dynamics in the scheduling framework, SBMs guarantee feasibility of process operations even during transient operation of a plant. The associated scheduling models exhibit good computational performance. For example, SBMs developed for cryogenic air separation process have been shown to reduce computation time by two orders of magnitude compared to using a full-order, first-principles model of the plant [11, 17]. SBMs that capture the evolution of process input (manipulated) variables that affect the economic objective function have also been identified and incorporated into scheduling calculations for sample processes [18].

We capitalize on the benefits of SBMs to 1) develop novel frameworks for the demand response activities of chemical processes in fast-paced electricity markets 2) build DR-oriented dynamic models for an electrochemical process and illustrate the DR operation of the given process 3) propose a novel unified cooperative approach to industrial DR; integrating power flow calculations and suitable chemical process models for mutual benefits of both stakeholders.

1.2 Guide to chapters

The rest of this dissertation is organized as follows. In chapter 2, we develop a novel scheduling framework for the provision of demand response by chemical processes in two electricity markets of different timescales with the goal of; i) maintaining the desired service/supply levels for chemical customers, ii) improving grid operations as described above and, iii) ensuring safe, dynamically feasible and profitable operation of the chemical process. We focus on capturing the benefits in the faster-paced markets since short-term (i.e., time scales of the order of 15 minutes or less) interactions can be the most critical for the power infrastructure as well as the most profitable for the chemical plant operators. The underly-

3 ing strategy for the framework focuses on deploying extra DR capacity (from slower-paced markets) for trade in the faster-paced market.

Following this, we introduce an industrial-scale example processes – chlor-alkali production – in chapter 3. A novel first-principle model that depicts the transient properties of the process under highly dynamic operation while remaining computationally efficient is developed. Subsequently, the provision of fast price-based demand response by the facility is demonstrated through extensive simulation and optimization case studies. In chapter 4, we shift discussions towards incentive-based demand response demonstrating the role chemical processes can play in the short-term power balancing endeavor of the electric grid. We discuss a new approach to frequency regulation (an incentive-based demand response activity) that is appropriate for chemical process industries in terms of guaranteeing dynamically feasible production schedules. New metrics for describing the flexibility of the process for frequency regulation are provided.

In the final sections (chapter 5), we consider an integrated approach towards industrial demand response evaluating the impacts from the perspective of both the grid and process operators. In the developed unified approach, the chemical process models are explicitly incorporated into grid-level calculations to determine the optimal power grid operation. First, the chemical plant models are simplified to appropriate forms for smooth assimilation into power flow calculations (e.g., to linear dynamic models). We conclude in chapter 6 with some future work directions and remarks.

4 Chapter 2

Grid-level “battery” operation of chemical processes and demand-side participation in short-term electricity markets

This chapter is based on material published in papers “Otashu, J. I., & Baldea, M. (2018). Grid-level battery operation of chemical processes and demand-side participation in short-term electricity markets. Applied energy, 220, 562-575”[4] and “Otashu, J. I., & Baldea, M. (2018, June). A Two-tiered Shrinking Horizon Framework for Participation of Chemical Processes in Short-term Electricity Markets. In 2018 Annual American Control Conference (ACC) (pp. 5902-5907). IEEE” [19]. Joannah Otashu was the primary contributor to this paper.

2.1 Introduction

Electricity generation from renewable resources has increased signiﬁcantly over the past decade [20]. The introduction of intermittent sources like wind and solar generation into the power generation portfolio has complicated the already challenging task of balancing power demand and supply. Time-sensitive electricity pricing can encourage demand response (DR), which entails modifying power demand to match the supply of electricity, from electricity end-users. In this context, electricity is typically sold at higher rates during peak hours (when grid demand is high) and at lower rates during oﬀ-peak hours (when demand

5 declines). Users can voluntarily modify their consumption patterns to incur less cost. In incentive-based DR programs, participants provide ancillary services to the grid, and are typically rewarded on a per occurrence basis for helping improve grid reliability by oﬀering system operators handles with which real-time power generation and load can be balanced [21, 22, 23, 24].

DR programs achieve different levels of engagement from the target participants. For industrial users, the extent of participation is highly dependent on the operational flexibility of the industrial plant [25, 26, 27]. Engagement in incentive-based programs is particularly demanding in terms of flexibility and agility; since these programs are designed to sustain grid reliability in the event of short-term supply-demand imbalances [21, 28] and typically require fast and abrupt changes in the operating pattern of the production facility.

In the present study, we distinguish between long-term and short-term energy markets based on how far in advance electricity prices are known with certainty and the frequency at which prices change. We define these markets based on the operation of the Electricity Reliability Council of Texas (ERCOT), as described in [29]. Under this definition, we refer to the day-ahead market (whereby electricity prices are established once a day and known for 24h in the future with hourly granularity) as a long-term market (LTM). Conversely, in this work, the short-term market (STM) covers frequent, sub-hourly (specifically, fifteen minute) changes in electricity tariffs, which are known at most one hour in advance of the tariff becoming effective.

Electricity prices exhibit varying amounts of volatility (defined as the price rate of change, or the difference in price between two subsequent time intervals), as illustrated in Fig. A.1. The STM is typically more volatile. Also, there can be significant discrepancies

6 between STM and LTM prices at any given point in time. STM prices could be lower than the corresponding LTM price, spike to values that are more than one order of magnitude higher than the LTM counterpart or even decline to negative values. While participation in the STM carries risks, these variations can be exploited by users to save cost or earn revenue.

Being aware of these electricity price variations is paramount for industries with high power usage. Industrial chemical processes that incur significant operating expenses from electricity purchases (e.g. air separation, aluminum smelting, chlor-alkali) can lower their energy costs by scheduling production in response to variable electricity prices. Effec- tively, energy intensive industries offer load reductions earning income as “electricity storage” providers in the energy market. By overproducing during off-peak demand hours and storing excess product to supplement a reduced production rate during peak hours, a chemical plant can serve as a “grid-level storage battery”. This production pattern requires that both excess production capacity and product storage facilities be available. Recent studies indicate that industrial plants can improve their economic performance by 40% to 100% [30] and mitigate risks associated with uncertain electricity price [31] by engaging in multiple electricity markets simultaneously. Scheduling plant production to allow for participation in both long-term and short-term markets will likely lead to greater cost savings compared to sole engagement in LTMs.

This work aims to explore the benefits of the participation of energy intensive processes in STMs in addition to LTMs. To do this, we focus on continuous (rather than batch) processes and propose a two-tier scheduling formulation. In the top level problem, an optimal production schedule for the process is computed over a fixed time horizon based on LTM price. This optimal profile may not necessarily imply operation at the limits of the plant and

7 additional production and storage capacity may be available. Thus, in the lower tier problem, we compute the maximum load reduction available per sub-hourly interval that can be attained by the process with consideration of process dynamics, feasibility and safety. This extra load reduction is deﬁned as the plant “DR headroom”. Based on incentives presented in the STM, this DR headroom if available, will be deployed (i.e. production schedules from the lower tier formulation will be implemented). The lower level problem is repeatedly solved over a shrinking time horizon to accommodate changes in the STM and plant production proﬁle, e.g., ramping up production to make up for down times. Throughout this work, we neglect any constraints related to the power grid, and assume that the grid is able to fully utilize the DR capabilities of the industrial facility.

In the context of existing literature, we highlight the key contributions of our work:

• deﬁnition of a utility-relevant process variable called “DR headroom” for aiding “battery operation” of process industries

• development of a multi-level scheduling framework for the engagement of continuous energy intensive industries in multiple electricity markets including active engagement in STMs

• employing dynamic process models in the scheduling framework developed above and highlighting the impact of the dynamic agility of the process (represented by its time constant) on its ability to participate meaningfully in STMs

In the sections that follow, we present the problem formulation and demonstrate how a process can beneﬁt from participating in both LTM and STM via simulation studies based on a large set of electricity price data collected from ERCOT, the Texas grid operator.

8 2.2 Problem formulation 2.2.1 Problem Statement

Given the LTM price forecast (DAM price) which we assume is certain and the availability of product storage, the intention is to:

• Compute a plant’s optimal production schedule for a base case that reﬂects participation in LTM only. To this end, determine production targets for every time slot in the scheduling horizon. A time slot describes a decision block within the scheduling horizon. The length of a time slot is the period during which decision variables (e.g. production target) are held ﬁxed. The time slot for the LTM scheduling is chosen to be the same as that for the STM scheduling calculation and is set such that its granularity depicts the STM dynamics. Since the STM conditions vary every few minutes/seconds, the length of each time slot should be less than one hour.

• Evaluate any available excess DR capacity relative to previously computed optimum (which we refer to as ‘DR headroom’) for each time slot given previous decisions made while satisfying plant, process and product constraints in a shrinking horizon

• Compute the cost of dispatching such DR headroom per time slot in the STM. Hence, specify bidding price per slot. The bid prices are determined for each slot forward in time before the actual trading hour. This is done to submit bids before speciﬁed market gate closure (which we assume to be 75 minutes according to [32])

• Use this information to bid load reduction capability (which can be construed as using/releasing ‘stored energy’) in the STM.

9 We assume the process purchases power from the DAM only, but participates in the FMM as an ‘energy storage provider’ by selling DR headroom while not exceeding the dynamic abilities of the plant (Fig. 2.1). In bidding, the process oﬀers to consume less power than it had committed to previously (based on DAM price) and therefore earns revenue accordingly. We assume a ﬁnancial reward that is equivalent to being paid the FMM settlement point price per kWh reduction of energy consumed.

Transmission & STM price signal distribution LTM price signal

Utility management load

LTM price signal Modified load providing DR headroom Other loads on the grid

STM price load signal

Bulk generation Production rate modulation

Storage

Product delivery to customers Process

Figure 2.1: Grid with DR participating chemical process and other loads (which we assume ﬁxed) on the grid. The chemical process exploits the availability of storage and underutilized plant capacity to modulate production rate and engage in both LTM and STM.

10 2.2.2 Process model

In general, chemical processes are described using high order non-linear systems of differential algebraic equations (formulation (2.1)) [33]. The differential equations reflect the material and energy balances for the system while the algebraic equations typically stem from constitutive property relations. These models relate state, input and output vectors of the system (x, u, y respectively)

x˙ = f(x) + g(x)u

y = h(x) (2.1)

x ∈ Rn where n is large and represents the order of the system

Such high order ﬁrst-principles models can be reduced to lower order models that are de- scriptive of the process dynamic behavior for use in scheduling calculations as discussed in Section 1.1. In this study, we employ ﬁrst order dynamic models (formulation (2.2a) and (2.2b)) to describe the generic process [34].

τP˙ = P sp − P (2.2a)

S˙ = Sin − Sout (2.2b)

The dynamic models above relate the evolution of the power consumption of the process,P , to its target value P sp, illustrating the fact that the actual power consumption does not instantaneously track its target. Rather, the evolution of the process consists of a transient, characterized by the time constant τ.

We employ this canonical representation of the process with the intent of illustrating the concept of utilizing reduced order scheduling relevant models that solve quickly and

11 preserve process dynamics. It is worth noting that the reduced order dynamic models may not necessarily be of the ﬁrst order; the approach discussed can be easily extended to higher order reduced models so long as the models capture the scheduling-relevant process dynamics and te associated scheduling problems can be solved within time frames reasonable for short- term scheduling. Illustrative examples of using low-order models of the process dynamics for scheduling calculations are provided in recent publications [11, 35, 36].

The nomenclature for process variables and symbols used is introduced in Table A.1.

Stored material accumulation is computed based on product ﬂows into and out of the storage unit. The above diﬀerential equations (Eq. (2.2)) are discretized using a forward Euler scheme as

sp 60 × τ × (Pk,i,j − Pk,i,j−1) = Pk,i − Pk,i,j (2.3a)

in out 60 × (Sk,i,j − Sk,i,j−1) = Sk,i,j − Sk,i,j (2.3b) where set k describes hourly intervals within the scheduling horizon, set i, the number of time slots within an hour and k, every minute within a time slot. For example, consider a 24 h scheduling horizon and participation in the FMM. Every time slot will be of length 15 mins: k ∈ {1, 2 ... 24}, i ∈ {1, 2, 3, 4} and j ∈ {1, 2 ... 15}. The set ‘i’ deﬁnes how often production targets will be changed to accommodate changes in the STM. By sampling the process state variables every minute, we capture the dynamic nature of the process model as well. We employ the above discretized model in the scheduling problem discussed later.

The instantaneous product demand is satisﬁed by the amount of material produced less the amount of product committed to storage (Eq. (2.4a)). Power consumption is assumed to be proportional to the production rate (i.e., Pk,i,j = × production rate, where

12 is the conversion factor that reﬂects the amount of power consumed per unit of product

generated. We assume that product demand Dk,i,j is constant throughout the scheduling horizon. In addition, the power consumed, stored material and draw rate from storage are constrained by bounds that describe the size of the plant and storage facility (Eq. (2.4b)). Lastly, the amount of stored material at the end of the scheduling horizon must at least be equal to the amount of product available at the start of the horizon (Eq. (2.4c)).

1 D = P − Sin + Sout (2.4a) k,i,j k,i,j k,i,j k,i,j

in out 0 ≤ Sk,i,j,Sk,i,j ≤ Sdrawrate

Pmin ≤ Pk,i,j ≤ Pmax (2.4b)

0 ≤ Sk,i,j ≤ Smax

S1,1,1 ≥ Sκ=Tm,i=n,j=ts (2.4c)

2.2.3 Top tier problem: Engagement in LTM (Base case)

The base case consists of optimizing the plant production schedule for an extended scheduling horizon Tm. The production target profile that minimizes utility cost (and by extension, maximizes profit since the only revenue stream for the process is the sale of product) is computed based on time-varying LTM electricity price. We utilize model (2.3) to formulate the top tier scheduling problem. To this end, problem (2.5) is solved. This is equivalent to solving the DR-relevant scheduling problem described earlier (Table 1.1). The result of this optimization assuming the price forecast for the LTMs does not change, represents the optimal production profile and cost given no engagement in the STM i.e. no

13 additional DR capacity is oﬀered to the grid in any of the time slots.

Tm,n,ts X LT M in out min πk Pk,i,j + penalty × (Sk,i,j + Sk,i,j) P sp ,Sin ,Sout k,i k,i,j k,i,j k=1,i=1,j=1 (2.5) subject to Model (2.3)

Constraints (2.4)

Base The optimal production target proﬁle from this formulation is labeled Pk,i,j .

2.2.4 Lower level problem: Engagement in STM

Here the maximum DR capacity (in terms of maximum possible reduction in power consumption) per time slot is computed in view of previously implemented scheduling decisions. For every time slot, the energy consumption for the given slot is minimized irrespective of cost (ﬁrst term of objective function in formulation (2.6)) and the utility cost for the rest of the scheduling horizon is optimized, taking into account the fact that power consumed in excess of the committed LTM amount is paid for according to real-time prices.

14 ts Tm,n ts X X STM diff LT M X diff min Pk,i,j + Zk,iπk,i Pk,i + πk ( Pk,i,j − Zk,iPk,i ) P sp ,Sin ,Sout k,i k,i,j k,i,j k=κ,i=ι,j=1 k=κ,i=ι+1 j=1 k>κ,i

Tm,n,ts X in out + penalty × (Sk,i,j + Sk,i,j) k=κ,i=ι+1,j=1 k>κ,i,j=1

ts diff X Base subject to Pk,i = Pk,i,j − Pk,i,j j=1 Model (2.3)

Constraints (2.4)

Production target proﬁle is set to the values implemented in previous time steps

∀k < κ, i < ι (2.6) where κ ∈ set k, and ι ∈ set i represent the beginning of a particular time slot in the scheduling horizon. Zk,i is a binary variable that takes on value of one when power consumed for a time slot is above the base energy consumption for the period, and zero otherwise. To

diff retain a linear formulation, we replace the bilinear term, Zk,iPk,i , with the positive artiﬁcial variable, Ak,i, and add the following constraints

diff Pk,i − ¯ ≤ Zk,iM (2.7a)

−Zk,iM ≤ Ak,i ≤ Zk,iM (2.7b)

diff Ak,i ≥ Pk,i (2.7c) where M represents a large positive number. In formulation (2.6), the artiﬁcial variable, A,

diﬀ diﬀ takes on the value of Pk,i when Z is non-zero. ¯ is a tolerance value that allows Pk,i to be

15 treated as zero for negligible positive values. Since the STM prices are not known ahead

STM of time, price forecasting is required and πk,i in formulation (2.6) represents the forecast price. We elaborate on this in section 2.3.1.

The power consumption proﬁle that corresponds to the optimal solution of problem (2.6) can be construed to consist of two sections corresponding to two periods of time. The

short-term short-term power consumption, referred to as Pk,i,j , corresponds to the power consump- long-term tion for the current time slot. The profile for the rest of the horizon Pk,i,j is dependent on the decision to provide demand response service in the current time slot. If this service is not provided, then the power consumption profile for the rest of the horizon is the base profile computed from (2.5). Else, power use over the rest of the horizon has a modified profile which results from the solution of Problem (2.6) ( see Fig. 2.2).

The DR capacity computed from formulation (2.6) is equivalent to the maximum plant “power discharge” for the given time slot. Drawing parallels with battery operation, the energy-intensive process alleviates the grid peak demand by curtailing its power consumption to the bare-minimum, just like a storage battery supports the grid operation during peak demand periods by discharging power into the grid network. Unlike the LTM scheduling calculation, the optimization problem (2.6) for the STM is formulated in a shrinking horizon fashion. This is done to accommodate the volatility of the STM.

The available “DR headroom” per slot is the area between the previous plant optimal

long-term power consumption proﬁle on a long-term perspective, Pk,i,j , (i.e., based on the LTM and forecast STM prices) and the short-term proﬁle given maximum load reduction for the

16 long-term time slot under consideration, Pk,i,j

ts X long-term short-term DR headroom = Pk,i,j − Pk,i,j (2.8) j=1

This concept of DR headroom is represented graphically in Fig. 2.2.

Power, P (kW) Base (long-term) profile Pmax Short-term profile Modified long-term profile

Pmin

i, j = 1 Time i, j = ts (a) Power, P (kW) Past Present Future Pmax

DR headroom for time slot, ι

Pmin

i, j = 1 i, j = ts (b) Time

Figure 2.2: (a) Power consumption profile as predicted by the production schedule and (b) actual power consumption profile for a hypothetical chemical process in response to time-varying electricity price. The power consumption is minimized for a given time slot, i = ι of length ts. Energy cost is optimized for the rest of the horizon based on LTM and forecast STM prices. The difference between the long-term and short-term optimal power consumption profiles for slot ι represents extra DR capacity referred to as “DR headroom”, that can be deployed in STMs.

DR headroom can be dispatched in the STM if the revenue to be earned by doing so is above a speciﬁed threshold. The threshold is a tuning parameter that can be set to avoid

17 frequent fluctuation of production rate targets for negligible income. It indirectly accounts for the cost associated with frequent production rate changes which reflects e.g., equipment wear and tear - note that quantifying accurately these long-term effects is very challenging and, as a consequence, they are not considered directly in this work.

Including the plant dynamic model and constraints in formulation (2.6) ensures that the power consumed by the process can be curtailed feasibly albeit at a cost, and that product demand is always met. The last constraint of formulation (2.6) conserves the history of implemented decisions from the start of the scheduling horizon. We use this information together with the modified long-term optimal schedule to compute the total profit made by the process over the entire horizon if DR headroom is sold in the given time slot. This new profit, ProfitSTM , is clearly lower than the base profit and is used to compute the cost for providing DR headroom in the STM and in supplying the decision to engage in the STM.

2.2.5 Economics

STMs are structured such that buyers and suppliers of energy submit their bids per unit energy to be supplied or bought during the time slot under consideration. Typically, bids are submitted until a stipulated time, the market gate closure, which is set several minutes before the delivery period. Based on the asking prices and quantity for both demand and supply, electricity demand and supply curves are plotted and clearing prices are set [37, 38]. Energy buyers whose bidding prices are at or above the clearing price are instructed to procure energy during the given time slot. Similarly, energy suppliers whose oﬀers fall at or below the clearing price are allowed to sell during the given period. Regardless of their bidding prices, the energy suppliers are paid the market clearing price [37].

18 In our proposed scheme, the bidding price for short-term DR service to be rendered by the power-intensive plant is determined by the cost of DR headroom. By computing the difference in energy cost incurred (and thus profit) based on previously computed long-term plant production schedule from base case, problem (2.5)) and the current optimal profile

considering maximizing DR in the given time slot (P rofitSTM , from optimization problem (2.6)), the cost of DR headroom per time slot is determined as

DR headroom cost = Base proﬁt − (FMM revenue + ProﬁtSTM) (2.9a) Total DR headroom cost DR headroom cost per unit load = (2.9b) Total DR headroom The FMM revenue in Eq. (2.9a) is a cumulative sum of all revenue earned from providing DR headroom in the STM during previous time slots and the potential revenue to be earned from selling DR headroom in the current operating period. It is computed thus

X FMM revenue = πk,i×DR headroom sold for slot, ι + FMM revenue (previous time slots) k<κ,i k=κ,i<ι (2.10)

The DR headroom cost, like the DR headroom (Eq. (2.8)) is computed in every time slot. It is a function of the past events of the plant and the current event chosen to engage in, based on the choice between implementing previous (modiﬁed) long-term optimal schedule or selling DR headroom when the energy market presents favorable opportunities. The bidding price at the STM is set equal to the DR headroom cost per unit load available until a break-even point when revenue from the STM covers all losses associated with committing to a LTM sub-optimal proﬁle. From this point onward, the bidding price is set to zero. This way, the process is guaranteed participation in the STM whatever the market clearing price may be. Fig. 2.3 summarizes this iterative process.

19 Remark 1. At every given time slot, the bid is being computed for a period, minutes ahead (market gate closure) of the current slot. Hence, the FMM revenue for all slots within this time window is computed based on forecast STM prices, whereas, the revenue for previous

time slots are computed based on published STM price. Similarly, P rofitSTM , which is computed over the entire scheduling horizon, is based on both forecast and actual STM prices. As a result, the DR headroom cost is an estimate of the actual cost that will be incurred in the future. This implies that it is possible for a participating entity to lose from engaging in the STM. However, signiﬁcant gains are realizable (and likely) as we will show in the case study presented below. Furthermore, on a general note, the gains to the chemical process are increased with improved prediction accuracy. Finally, note that the second term of Eq.

(2.10) is zero for the ﬁrst TGC minutes of the scheduling horizon.

2.3 Case study 2.3.1 Model parameters and scheduling implementation

Consider a process described by the model given in formulation (2.2) (Section 2.2.2). The data for the process model are summarized in Table 2.1. We consider a case where the process purchases electricity on the DAM, the LTM, and modulates its production rate to minimize cost accordingly. We show that by tactically deploying DR headroom in the FMM, the STM, the overall proﬁt earned by the process is increased.

Given a 24 h ﬁxed scheduling horizon (Tm = 24 h) discretized on a minute grid (j ) to represent process dynamics and into time slots (i) of size 15 minutes (ts = 15 mins) as in formulation (2.3), the base case scheduling problem is solved at the start of the horizon in response to DAM electricity price. Results from this optimization form the baseline against

20 Compute lower tier production proﬁle; DR headroom − formulation (2.6) & Eq. (2.8) (for slot 75 minute ahead)

Compute base production pro- ﬁle and proﬁt − formulation (2.5)

DR no headroom > 0

yes

Compute cost estimate for DR headroom − formulation (2.9)

Break-even point yes reached. Bid price = 0 Cost ≤ 0

Bid price = Cost of DR headroom per unit load

Bid price no ≤ πSTM Implement previous modiﬁed k,i long-term optimal schedule (forecast)

yes

Compute revenue estimate for time slot

Revenue estimate > λ no

yes

Sell DR headroom; update revenue (add to FMM revenue taking into account actual/published πSTM & forecast πSTM )

Shift lower-level problem forward by one time slot

Figure 2.3: Flowchart for process simultaneous engagement in long and short-term electricity markets. DR headroom is sold in the STM when predicted proﬁtable. 21 Table 2.1: Process model parameters Parameter Value Pmin 1600 kW Pmax 2400 kW D 1000 kg 2kW h/kg Smax 1000 kg Sdrawrate 200kg/hr τ 0.3 h Product price $1/kg Storage penalty $0.25/kg λ 0.1% ∗ Base proﬁt ¯ 10−4 M 105

which proﬁtability of engaging in the FMM will be measured. For each 15 minute time slot,

the process DR for the slot, TGC = 75 minutes ahead (i.e. 5 slots in the future) is maximized (formulation (2.6) is solved). Thus, there is zero engagement in the FMM for the ﬁrst ﬁve slots. Note that the implementation of the resulting schedule will imply transient plant operation since the time constant of the process (18 minutes) is longer than the scheduling period (15 minutes). Since the actual FMM prices are known just before the given operating period, the scheduling is done using forecasts of the FMM price.

Three forecasting scenarios are considered below:

1. Perfect STM price forecast: In this case, we assume perfect knowledge of future prices in the STM. Thus, actual electricity prices for the FMM are employed in the scheduling calculations. This is of course expected to lead to the best results.

2. Same day DAM price as the forecast for STM price (SD DAM): Generally, the day-

22 ahead market price provides a reasonable approximation for the electricity price in real-time [39]. Based on this premise, we employ in this scenario the DAM for each hour as the forecast for the FMM price. For example, if the DAM electricity price for the 17th hour is 0.11$/kW h, then, we predict that the electricity price for all four slots of the 17th hour in the FMM will be 0.11$/kW h.

3. Previous working day/weekend STM price as the forecast for following day STM electricity price (DB FMM): In this scenario, we assume that there exist daily similarities in working conditions in the real-time market e.g. load proﬁles for working days are similar and possibly distinct from weekend load proﬁles. As a result, we use as an approximation of the real-time price for every working day, the STM price for the previous working day and for weekends, the STM price for the corresponding previous weekend day. Thus, the FMM price for Tuesdays - Fridays is dictated by the previous working day, Mondays are set by Fridays of the previous week and Saturdays and Sundays are set by corresponding days of the previous weekend.

We utilize the above scenarios to study the proposed STM participation strategy and illustrate that the proposed approach can lead to increased earnings for a chemical process. It is not the focus of this work to provide a detailed methodology for electricity price forecasting. The interested readers are referred to, e.g., [40, 41] for more details.

Reported ERCOT DAM and FMM electricity prices for the second week in January - July, 2017 (a subset of these data is plotted for illustration purposes in Fig. A.1) are used in this case study. The results are discussed below.

23 2.3.2 Results and discussion 2.3.2.1 Optimal production proﬁles

The scheduling problems were solved using CPLEX 24.4.6 solver in GAMS (GAMS Development Cooperation, 2015). The computation time was approx. 1 minute for each 24 hr scheduling period. A detailed discussion of the results for a sample day is provided in this section. However, the general conclusions drawn are extendable to all days sampled.

Fig. 2.4 illustrates the optimal power consumption proﬁles for the top-level and lower-level problems for the ﬁrst day sampled in January.

The optimal base profile indicates that the DAM price (green curve) by itself incen- tivizes modulation of production rate set points. Process profit is maximized by reducing the production rate when electricity prices are high (hour 6 - 8 ), utilizing available inventory to make up for reduced production. Depleted inventory is refilled during periods of low electricity price (hours 22.5 - 24 ) when production is ramped up in order to satisfy end point constraints.

Results from the lower level problem for all three scenarios (last three plots of Fig. 2.4) reveal that the process is able to provide additional load reduction for multiple time slots in the scheduling horizon according to predicted FMM price. During such times of power reduction, the DR headroom is non-zero and the prices in the FMM are high enough to cover the cost for providing DR headroom in the given time slot as indicated by the bid price (Fig. 2.5). Proﬁles from the lower level problem are implemented and DR headroom is sold in the FMM (shown by red stars in Fig. 2.4). Note that slots with reduced production rates but no sale in the FMM correspond to periods with high DAM electricity price (e.g. hour 7 ). Therefore, production rates for such hours are already scheduled to be at the

24 Figure 2.4: Optimal power consumption proﬁles for top and lower level problems for sample day, 01/09/17. ‘SD DAM’ stands for same day DAM price and ‘DB FMM’ stands for day- before FMM price. Red stars indicate sale of DR headroom in the FMM. The black dashed line is the actual power consumption proﬁle

25 Perfect forecast SD DAM forecast DB FMM forecast 60 60 60

40 40 40 20 20 20 0 0

DR headroom kWh -20 0 -20 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 0.4 0.8 Bid price 0.4 Forecast FMM price 0.3 0.6 0.2

0.2 0.4 0

Price $/kWh 0.1 0.2

-0.2 0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 time, hour time, hour time, hour

Figure 2.5: Available DR headroom (top) and process bid prices as dictated by the DR headroom cost (bottom) for all three STM forecast scenarios for the ﬁrst sample day, 01/09/17. Bid price is set to zero when DR headroom cost is negative minimum and there is no DR headroom available. This is more obvious for the DB FMM prediction scenario. It is worth mentioning that since the FMM is generally more adapted to representing real-time grid conditions compared to the DAM, it is likely that DR headroom will be available for sale when needed in real-time. Our results show that more often than not, DR headroom is non-zero during STM price spikes. However, this is not guaranteed to always be the case.

Also observable from Fig. 2.4 is the power set-point tracking of the process. Recall that the time constant for the process in this example is 18 minutes. As a result, when the power set-point changes from one 15 minute period to the next, the plant operates in transient regime and the actual power consumed (black dashed curve) follows but is not identical to the power set point (blue curve). Employing dynamic models in the scheduling calculations is a way to guarantee that the feasibility and safety of production transitions are not compromised during such transient operation.

26 The DR headroom for each time slot in the scheduling horizon and the cost (as indicated by the bid price) for the optimal profiles presented in Fig. 2.4 are shown in Fig. 2.5. Recall that the DR headroom by definition, is the size of the area between the plant’s previous optimal power profile (‘would have been’ profile) and the new profile that minimizes the energy consumed in the current time slot. Based on the DAM price for the given time slot and whether DR headroom was sold in previous time slot or not, the headroom increases and decreases accordingly, capping at a maximum of approximately 60 kWh. If DR headroom was sold in previous time slots and DAM prices are low for the current slot, the DR headroom will be high, since the natural next move in the previous iteration will have been ramping up production to make up for sold DR load (i.e. the difference between current minimum power consumption and previous recommended consumption profile will be large). Observe that even though DR headroom for the first few hours (0 - 6) is non-zero, no sale is made in the FMM. Recall that no engagement in the STM is possible in the first five slots (hour 1 and first slot in hour 2) due to the stipulated market gate closure. From thereon however, the lack of participation in the STM in the early hours of the scheduling horizon could happen for several reasons. First, it is possible that the FMM prices are not high enough to cover the DR headroom cost for the given time slots. As a result, the bid curve lies above the predicted FMM prices (Fig. 2.5) and no sale is made in the FMM. A similar situation occurs when the revenue to be obtained from selling DR headroom is below the threshold value, λ,(early hours of the DB FMM scenario). Even though the bid price is lower than predicted FMM price, there is no engagement in the STM because expected earnings are not incentivizing enough. The unsold DR headroom from these hours of inactivity would still be available for deployment in other grid ancillary markets e.g. demand bidding and buyback and load as

27 capacity resource. Lastly, the DR headroom is zero when the plant is already running at minimum production level (hour 7.5 ) or when load reduction is not possible due to other constraints e.g. meeting demand, feasibility constraints (hours 12 - 14 for perfect and SD DAM forecast scenarios). Towards the end of the scheduling horizon, DR headroom declines to zero as end point constraint (Eq. (2.4c)) must be satisﬁed.

2.3.2.2 Economics

Table 2.2 compares the average (over the 49 days sampled) cost and proﬁt to the process for the top and lower level scheduling formulations for a 24 hr period.

Table 2.2: Comparison of process economics for top and lower level scheduling formulations Top-tier Lower-level problem Item problem Perfect SD DAM DB FMM FMM fore- scenario scenario cast Energy consumption kWh 47996 47996 47996 47996 Energy cost $ 13227 13419 13428 13447 Profit from product sale $ 10773 10581 10572 10553 Revenue from FMM $ 0 378 385 354 Total profit $ 10773 10959 10957 10907 Increase in energy cost % 1.46 1.52 1.67 Increase in profit, % 1.73 1.71 1.24

Even though the total energy consumed in both cases is the same, higher energy cost is incurred for all cases of the lower level problem. This will often be the case since the FMM prices are more volatile, often spiking to very high values compared to the DAM electricity price. By sacrificing the DAM optimal power profile and providing DR headroom in the STM, the process incurs higher utility cost. However, the overall profit is higher in all

28 three FMM participation cases due to the new income stream (note that results presented indicate the average savings for a 24 hr time span). This could be a significant increase on a daily basis for a large industrial facility. Further statistical analysis using the paired t-test indicates that the total profits for the top and lower level problems are statistically different for a 95% confidence level. The proposed scheme can lead to a win-win situation for the process and utility. Unlike previous works where the reduction of process utility cost by engaging in LTMs only is emphasized, this result reveals that sacrificing minimum utility cost for participation in STM can lead to increased overall profit for the process. In addition, the process by participating in STMs provides valuable DR to the grid.

2.3.3 Impact of process agility on performance

The process time constant plays an important role in the level of flexibility available in the plant and thus extent of participation in STMs. With higher values of process time constant, the process is more sluggish, rarely attaining steady state before production target changes are made. Studies of the influence of agility on process performance were carried out for a sample day assuming perfect STM price forecast (Fig. A.2). As shown in Fig. 2.6, the larger the difference between the process time constant and the pace of market change, the more pronounced transient plant operation becomes. The actual power consumed does not follow closely the target profile. This further emphasizes the need to employ dynamic models in determining safe and feasible plant operation schedules for participation in STMs.

Also observable from Fig. 2.6 is decreasing rate of participation in the STM as process time constant increases. This will generally lead to reduced total revenue from selling DR

29 τ = 0.3 h τ = 0.4 h τ = 0.5 h 2400 2400 2400 Psp P 2200 2200 2200

2000 2000 2000 Power, kW

1800 1800 1800

1600 1600 1600 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 τ τ = 0.6 h τ = 0.7 h = 1.0 h 2400 2400 2400

2200 2200 2200

2000 2000 2000 Power, kW

1800 1800 1800

1600 1600 1600 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 time, hour time, hour time, hour

Figure 2.6: Effect of process time constant on the agility of a process assuming perfect STM price forecast for 01/09/17. Higher time constants result in sluggish response to rapid set- point changes (actual power consumption, P, does not instantaneously track set points, P sp) leading to a transient (rather than predominantly steady-state) operation. Engagement in the STM (red stars) also diminishes with increasing time constant headroom in the STM as illustrated in Fig. 2.7a. Since higher energy cost is generally incurred with increased STM participation, it follows that the total energy cost (sum total calculated based on both LTM and STM prices) will be lower for a sluggish process. A combination of both effects – increased STM revenue and increased energy cost resulting from active engagement in the STM, will lead to an overall profit that will likely be higher for an agile process as observable from Fig. 2.7b. However, it is also possible for a less agile process to earn more profit than an agile process. For example, consider the rare situation that a highly profitable and unpredicted price spike occur in the STM towards the end of the

30 Figure 2.7: Comparison of (a) top: Energy cost and total revenue earned from the STM (b) bottom: Total profit to the process for top and lower level (perfect forecast, sample day 01/09/17) scheduling formulations, for increasing process time constant scheduling period. In this case, the most agile process may have expended all DR headroom in earlier time slots and currently be in recovery mode, unable to provide DR headroom during this boon period as constraints e.g., demand and end constraints must be satisfied. A less agile process on the other hand, as a result of minimal participation during previous time periods may have available DR to offer during this chance period. This, together with the fact that the sluggish process will intrinsically incur lower energy cost, may end up resulting in a higher overall profit for the slow process compared to an agile process. As mentioned, however, such situations are likely to be rare. It is also possible that a specific

31 price structure will be particularly well suited for a process with a speciﬁc time constant; this is also captured in the cases illustrated in Figure 2.7, where the STM revenue for the process with τ = 0.6h is much higher than, e.g., the processes with τ = 0.5h or τ = 0.7h.

Finally, the total profit for engagement in the STM in addition to LTM is generally significantly higher than the profit from participation in the LTM only. A less obvious reduction of profit with increasing process time constant is observed for involvement in LTM only.

2.4 Conclusions

In conclusion, a novel method of quantifying the extra load reduction called DR headroom available within power-intensive industries was proposed. Deploying such extra DR capacity in STMs has accompanying costs that can be employed to determine bids offered by the chemical industry in this market. The production schedule for the industry providing such service have been computed using dynamic models to ensure safe and feasible operation especially during plant’s transient state. By active engagement in the STM as ‘energy storage’ provider in form of load reduction in addition to participation in LTM, the process industry can earn higher overall profit even though its utility cost may be increased. Therefore, both utility and chemical processes can benefit from this scheme.

32 Chapter 3

Demand response-oriented dynamic modeling and operational optimization of membrane-based chlor-alkali plants

This chapter is based on material published in paper “Otashu, J. I., & Baldea, M. (2019). Demand response-oriented dynamic modeling and operational optimization of membrane-based chlor-alkali plants. Computers & Chemical Engineering, 121, 396-408”[42]. Joannah Otashu was the primary contributor to this paper. 3.1 Introduction

The chemical industry is highly energy-intensive. In the United States, chemical plants account for 39% of the domestic consumption of natural gas and 26 % of the electricity demand [43]. Major contributors to electricity demand in the chemical industry include ethylene and fertilizer production, air separation and chlor-alkali plants [44]. In turn, electricity is a major contributor to the operating cost for most of these processes, a fact that has spurred significant cost-reduction efforts. The latter fall into two broad categories. On the one hand, technology innovation focuses on improving process efficiency via basic science (e.g., new materials, catalysts), new equipment concepts and new plant designs, generally seeking a favorable trade-off between (likely higher) capital expenditure and lower operating cost.

33 On the other hand, the deregulation of electricity markets has encouraged the active participation of chemical plants in the operation of the power grid. These interactions are generally referred to as demand-side management or demand response (DR) and consist of a (typically voluntary) modulation of the electricity demand of a plant in response to time- varying electricity price signals. Engaging in DR programs can lead to significant reductions in operating cost. Loosely speaking, it consists of, i) increasing production rates (and electricity consumption) beyond current demand during off-peak hours, when grid demand is low and energy prices drop, ii) storing the product generated in excess and, iii) using the stored product to (partially) meet demand during peak grid demand (and peak price) hours, when the production rate of the plant is reduced below the instantaneous demand rate. Ev- idently, this mode of operation requires that, i) excess production capacity be available, ii) the product generated in excess can be stored safely and economically and, iii) the plant be sufficiently agile, in the sense that it can modulate its electricity use over a time scale that is relevant to the frequency of fluctuation of electricity prices [34, 45].

To the latter point, it is worth noting that the dynamics of chemical plants are rather slow (with dominant time constants in the order of several hours - or even longer in some cases). Conversely, changes in electricity prices occur over shorter intervals (one hour or less), and it has been shown that engagement in shorter term markets (where prices change every ﬁfteen minutes or faster) tends to be more proﬁtable [30, 4]. As a consequence, scheduling plant operations for providing DR service in fast-changing markets is a non-trivial matter, and scheduling calculations must explicitly account for the dynamics of the process [12, 46, 47].

The body of literature discussing DR scheduling under dynamic constraints has

34 largely focused on “conventional” chemical plants, such as air separation, whose dynamics are suitable for engaging in relatively slow-changing electricity markets, such as the day-ahead market [11, 17, 48, 49, 50, 36]. Here, electricity prices are set –typically in the early afternoon of a given day– for 24h in advance, and change at hourly intervals, a time scale well-matched to the dynamics of the process. We note here several contributions that consider the ability of chemical processes to provide “interruptible power,” i.e., the ability to drastically curtail power use by shutting down the plant. Production scheduling under such circumstances is formulated as an optimization problem under uncertainty. Static plant models are used, and the start-up and shut-down dynamics are approximated using ﬁxed, parameterized transition times [51, 52, 53, 15].

In this chapter, we shift our attention to the engagement of electricity-intensive chemical plants in short-term electricity markets. To this end, we consider a diﬀerent process class, namely, electrolytic processes, whose dynamics are well-suited to this purpose. We focus in particular on the chlor-alkali industry, which is the third-largest consumer of electricity of all chemical plants in the U.S. [44]. Our contribution is twofold:

• We develop a DR-oriented dynamic model of an industrial-size electrolysis plant, incorporating phenomena relevant to DR participation

• We formulate a production scheduling problem that accounts for the process dynamics, and investigate the economic beneﬁts (and potential process limitations) associated with engagement in short-term electricity markets.

The chapter is organized as follows: we ﬁrst present background information on the chlor-alkali industry and existing relevant DR eﬀorts. We then develop a DR-oriented model

35 for the chlor-alkali process, paying attention to intrinsic process attributes that make the chlor-alkali industry amenable to providing DR in shorter term electricity markets. Finally, we discuss the scheduling of chlor-alkali plant operations to provide DR, quantify the economic beneﬁts and provide concluding remarks.

3.2 Background and Literature Review 3.2.1 The Chlor-Alkali Industry

Chlorine and caustic soda (i.e., sodium hydroxide), the two major products of the chlor-alkali industry, rank highly amongst the most important commodity inorganic chemicals [54]. They are widely used as feedstock for other processes, including the production of organic chemicals, pulp and paper, water treatment, plastics, alumina, etc., and support the activities of major industries like construction and manufacturing [54]. The chlor-alkali industry in the North American region generated over 30 million tons1 of chlorine and caustic soda with an estimated value reaching $8 billion in 1998 [55].

Historically, brine electrolysis has been the dominant method for producing chlorine on an industrial scale, [55] and remains the major means of chlor-alkali production today [54]. The process (Fig. 3.1) begins with the pre-treatment of brine to remove impurities and saturate the salt solution [56, 55]. Pretreated brine with dilute sodium hydroxide is then processed in electrolysis units (our focus in this study is on membrane cells as it is the most recently introduced technology) to generate chlorine and hydrogen gas. The exiting “wet” (i.e., relatively high moisture content) chlorine gas is cooled from the cell temperature

1ton as used in this chapter refers to US (short) tons unless otherwise mentioned

36 NTP

Acid dryer

Rectifier 13oC Cooling

Dry Hydrogen Cl Liquefaction Wet 2(g) unit treatment Cl2(g)

Storage Membrane H Evaporator electrolyzer 2 NaOH(aq)

Liquid Brine pretreatment 32-35% Cl2 NaCl - purification (aq) Caustic - filtration - saturation etc. Heat exchange unit Cooling water or steam

Figure 3.1: Process ﬂow diagram for chlor-alkali production through membrane electrolysis. Units in black dashed boxes are modeled in this chapter (as described later). to 13oC [57, 2] in a chilled water heat exchanger and then dried using sulphuric acid [56]. Most of the dry gas, now at ambient or normal temperature and pressure (NTP), is liqueﬁed through a series of compression and cooling stages. The resulting liquid product is stored and can be shipped to users via tanker trucks or rail cars. The rest of the dried gas can be diverted via pipelines for use in other plants [57] or sold as needed.

The brine electrolysis process is highly energy-intensive. The chlor-alkali industry reportedly accounted for 2 % of the total electrical power generated in the United States in the late 90s [55]. Research to improve the energy eﬃciency has focused on the development

37 of energy efficient membranes [58, 59, 60, 61, 62] and advanced electrodes [63, 64, 65, 66, 67]. Process intensification of the membrane technology is being investigated [68]. Engagement in demand response programs has strong potential for reducing operating cost, and relevant efforts are reviewed in the next subsection.

3.2.2 Existing DR Eﬀorts

Initial research revolved around qualitative assessment of the demand response potential of the industry [69, 45]. Babu and Ashok [70] illustrate the provision of load management services by a chlor-alkali facility under time-of-use pricing. Similarly, motivated by variable prices in the European electricity market, Masding and Browning [71] developed semi-empirical dynamic models for a mercury chlorine cell and employed these to study the optimal power consumption of the unit for various cell loads and brine ﬂow rates. A more recent eﬀort involves the demonstration of the load following capability of a chlor-alkali unit connected to a hybrid renewable energy generation system discussed in [72].

We note here that these studies present some gaps, particularly at the modeling level. Specifically, simplified steady-state models are employed in [70]. Masding and Browning [71] did not include a model description for reasons of confidentiality. Finally, Wang et al. [72] focused on demonstrating the provision of demand response by leveraging the availability of supplementary on-site generation rather than via modulating the operation of the plant. These findings motivate our effort to develop a DR-oriented dynamic model for chlor-alkali plants, noting that such a model must strike a balance between representing the DR-relevant dynamics and the level of detail. Detailed dynamic process models are in general too cum- bersome and computationally intensive for use in production scheduling calculations that

38 must be solved in a practical amount of time. This is especially important given our purpose of scheduling (and potentially rescheduling) production for fast changing markets. On the other hand, simpliﬁed/shortcut models are typically computationally parsimonious, but may ignore relevant dynamics and lead to results that are infeasible to implement in practice. We base our model development work on existing literature information, which is also reviewed below.

3.3 A DR-relevant model of the membrane-based chlor-alkali process

Early eﬀorts in modeling chlor-alkali electrolysis processes include the development of steady state [73] and dynamic models [74, 75] describing the evolution of hydroxyl ion concentration in a diaphragm cell under isothermal conditions. Electrochemical models for analyzing membrane performance [76, 77] are also available.

Table 3.1 summarizes some models for the chlor-alkali process and relevant models developed for electrolytic processes in general. The scope of these models runs a gamut, ranging from steady state representation [70] to detailed dynamic models of representative zones of the electrolyzer [79]. As pointed out above, our aim is to make a compromise between model completeness and simplicity; to this end, we identify pertinent cell variables that aﬀect the optimal operating conditions and model these appropriately. In particular, our model will cover:

• The cell temperature: membranes are thermally stable up to temperatures of 105 oC and at too low temperatures, undesirable voids are formed in the membrane jeopardiz- ing membrane stability [56]. Current operation of membrane-based chlor-alkali plants

39 Table 3.1: Review of relevant models for electrolytic processes Reference ElectrochemicalMaterial bal- Thermal model ance model model Van Zee et al., 1986 None Species, dy- None (assumed [74] namic isothermal) Kakhu and None Species, dy- Lumped, dy- Pantelides, 2003 namic namic [78] Ulleberg, 2003 [60] Empirical rela- Overall mass Lumped, dy- tions balance namic Agachi et al., 2007 First principles Species, spa- spatially dis- [79] tially distributed, dy- tributed, dynamic namic Babu and Ashok, None Generic steady None 2008 [70] state Masding and Conﬁdential dynamic models Browning, 2008 [71] Budiarto et al., First principles Species, dy- None (assumed 2016 [80] namic isothermal)

typically limit cell temperatures to a maximum value of 90oC [81]. It is thus crucial to monitor the evolution of the cell temperature especially during non-steady state operation to guarantee safe operation. Additionally, since the cell temperature contributes signiﬁcantly to the cell voltage [56, 82] we model the dynamics of this variable in order to represent more accurately the power load ﬂuctuations attainable by the plant.

• Current-voltage relationships: while ignored in some works (see Table 3.1), neglecting this fundamental correlation hinders the accurate quantiﬁcation of the instantaneous power reductions that can be provided by the chlor-alkali facility under non-steady state operation. In their review, the authors of [83] deduce that employing ﬁrst princi-

40 ple models to describe temperature and pressure effects on current-voltage relationships are necessary to adequately describe the physical response of the electrolyzer. Exper- imental investigation [84] of the transient operation of an electrolyzer has also shown that simplified current-voltage relationships alone, which are widely adopted in literature to describe the electrolysis unit in the studies of hybrid energy systems, are insufficient to characterize the dynamic response of the electrolyzer.

• Material balance: we include dynamic models for the electrolyte concentration for each compartment capturing the rate of reaction and transfer of species across the membrane, with the expectation that the ﬂow rate of material through the cells can also be modulated to impact power consumption.

3.3.1 Mathematical model of the membrane cell

The core of the chlor-alkali production process is the electrolysis unit where the electrochemical reaction occurs. Several electrolytic cell technologies are available [56]. In this chapter, we consider the membrane cell technology (Fig. 3.2) since it is becoming increasingly widespread in the chlor-alkali industry and replacing other technologies[56, 85, 86]. The cell consists of anode and cathode compartments (chambers), separated by a sheet of ion-selective membrane that allows unidirectional transfer of sodium ions and water molecules from the anode to the cathode compartment [79]. The anode chamber is fed with saturated brine and the cathode compartment with dilute sodium hydroxide. Sodium chloride dissociates into sodium and chloride ions in the anode compartment. The chloride ions are then oxidized to produce chlorine gas according to Eq. (3.1a). Water in the catholyte is reduced to hydrogen gas and hydroxyl ions (3.1b). The sodium ions also combine with

41 Current flow + - 2 e-

Saturated Cl 2

Saturated H2

Spent brine 22ClCle H OeHOH22 2 22 Concentrated NaOH NaCl, H2 O Na  . 32 - 35% OH 

.  Anode membrane Cl Cathode NaOHNaOH

NaCl, H O 2 Dilute NaOH Saturated brine NaOH, H2 O

Figure 3.2: Schematic diagram of a single membrane cell electrolyzer.

hydroxyl ions to form caustic soda. This hydroxide solution is collected as the outlet stream from the cathode chamber. The overall cell reaction is given by Eq. (3.1c).

− − Anode 2Cl → Cl2(g) + 2e (3.1a)

− − Cathode 2H2O(l) + 2e → H2(g) + 2OH(aq) (3.1b)

Overall 2NaCl(aq) + 2H2O(l) → Cl2(g) + H2(g) + 2NaOH(aq) (3.1c)

Brine for membrane electrolysis is typically pretreated to high purity levels [56] and the presence of chlorides and chlorates in the caustic product is reduced at current densities above 0.3A/cm2 [76]. Also, since we restrict cell conditions to lie within values yielding high current efficiencies (above 90% [56]), we assume a fixed Faraday efficiency, ηF , of 100%, without loss of generality. The symbols to be used for the electrolyzer model and parameter values used later in the simulation study are summarized in Table B.1.

42 3.3.1.1 Electrochemical model

The cell voltage is tripartite, consisting of the thermodynamic, kinetic and ohmic components [87]. The thermodynamic component, represented by Uequ, is governed by the Nernst equation and is given in Eq. (3.2) [88].

ppano RT Cl2 cat Uequ = Urev + ln ano − 2.302 14 + log(COH ) (3.2) 2F α CCl where Urev is the reversible cell voltage, α is a multiplication factor, F,R are constants and p, C, T stand for partial pressure, concentration and temperature respectively. The above equation approximates the activities of the ions in terms of electrolyte concentrations.

The kinetic component is expressed in form of the cell overpotential, Uovp, which is a function of temperature, T , and current density, (I/Ae), and is described by the Tafel expression [79].

−5 −4 Uovp = 1.3322 × 10 T + 1.3212 × 10 T log(I/Ae) − 0.02 − 0.11 log(I/Ae) (3.3)

Voltage drop due to ohmic losses, Uohm, is computed in terms of a temperature dependent cell resistance [79] thus

2.6125 × 10−4 − 1.75 × 10−6(T − 273) Uohm = I × (3.4) Ae

The resulting cell voltage, Ucell, is then:

Ucell = Uequ + Uovp + Uohm (3.5)

43 3.3.1.2 Material balance model

By reaction stoichiometry (Eq. (3.1)), the rates of generation of H2 and Cl2 in moles, N˙ , are equal and are deﬁned by Faraday’s law as

η N I N˙ cat,out = N˙ ano,out = F c (3.6) H2 Cl2 2F

where ηF is the Faraday eﬃciency. Note that the above equation represents the rate of

generation from a stack of Nc cells. Eq. (3.6) assumes that there is no accumulation of produced gases in the electrolysis cell. This is a reasonable assumption since the solubility of both gases in the electrolyte solution is negligible at cell operating temperatures [80].

The mass balances for the species in the electrolyte are given for the anode and cathode compartments separately.

The mass balance on the chloride ion in the cell (anolyte only) is

dn Cl = Cano,inV˙ in − Cano V˙ − 2n ˙ ano,out (3.7) dt NaCl NaCl Cl2

The sodium ion balance

dnano D A Na = Cano,inV˙ in − V˙ × nano/V − Na c (nano − ncat ) (3.8) dt NaCl Na δV Na Na dncat D A Na = Ccat,in V˙ in − V˙ × ncat /V + Na c (nano − ncat ) (3.9) dt NaOH Na δV Na Na The hydroxide ion balance is

dncat OH = Ccat,in V˙ in − Ccat V˙ + 2n ˙ ano,out (3.10) dt NaOH NaOH Cl2

Literature sources [88, 56] report that around 0.6 − 1.5 and 0.3 − 0.5 moles of water evapo- rate from the cell per mole of chlorine and hydrogen gas generated respectively at operating

44 temperature range of 70 − 90oC. We consider values of 0.6 and 0.3 moles of water being evaporated from the anode and cathode compartments respectively. In a sense, this represents a “worst-case scenario” in terms of thermal regulation of the cell: it requires the highest cooling duty since it considers the lowest rate of water evaporation and hence the lowest amount of latent heat removed.

The water balance for the system is therefore

dnano D A H2O = (55.56 × 103 − Cano,in)V˙ − V˙ × nano /V + H2O c (ncat − nano ) dt NaCl H2O δV H2O H2O (3.11) − 0.6n ˙ ano,out Cl2

dncat D A H2O = (55.56 × 103 − Ccat,in )V˙ − V˙ × ncat /V − H2O c (ncat − nano ) dt NaOH H2O δV H2O H2O (3.12) − 2.3n ˙ ano,out Cl2 It is assumed that the cell volume is maintained at a constant level and the ﬂow rates of electrolyte solution into and out of a cell compartment are equal.

3.3.1.3 Thermal model

We adopt an overall energy balance to describe the thermal dynamics of the electrolytic cell after the discussions presented in [60, 88, 78]. The production of gas in the anolyte and catholyte causes suﬃcient mixing to assume uniform temperature in the cell [74]. Assuming constant lumped heat capacity, the energy balance is

dT N n C = N [n ˙ inCinT in − n˙ outC T ] − Q˙ rxn − Q˙ evp − Q˙ loss + Q˙ gen (3.13) c t pt dt c t pt t pt

˙ where Cpt is the lumped heat capacity and Q represent heat ﬂows. The energy balance is presented for the entire stack of cells. The terms in parentheses represent the sensible heat

45 loss from circulating electrolyte solution. Q˙ rxn is the energy generated as a result of the overall cell reaction. This is computed based on the total number of moles of Cl2 (or H2) liberated from the stack of cells given by Eq. 3.6.

Q˙ rxn = N˙ ano,out∆H (3.14) Cl2 rxn where ∆Hrxn is the enthalpy of the overall cell reaction calculated from the heats of formation of the components of the total reaction [89]. The overall reaction is endothermic.

A substantial amount of heat is removed from the cell with saturated exiting gases [88]. This is accounted for by the latent heat of evaporation of the associated water vapor.

Q˙ evp = 0.9 × N˙ ano,out∆H (3.15) Cl2 vap

The heat loss from the electrolyzer to the ambient is a factor of the thermal resistance across the stack of cells and temperature gradient between the cells and the environment

˙ loss 1 Q = (T − Tamb) (3.16) Rt where Rt is the thermal resistance of the stack of cells and Tamb is the ambient temperature. The ohmic heat released by the electrolytic system (part of which supplies the heat of reaction [56]) is given by ˙ gen Q = NcUcellI (3.17) for the cell stack.

3.3.2 Liquefaction unit

The Uhde liquefaction system [2] for an industrial membrane electrolytic process is considered in this chapter. The liquefaction process (Fig. 3.3) consists of a single-step com-

46 pression of the dry chlorine gas, followed by two-stage cooling using diﬂuoromonochloromethane.

Vent gas

-55oC

Separator

Stage 1 -15oC Separator

Stage 2 Compressor

Compre- Compre- ssor ssor Collector NTP Collector 0.3 Stage 1 MPa Dry Cl2(g) To demand Liquid Cl2 storage on load cells

Figure 3.3: Chlorine gas liquefaction unit - Uhde system. Chlorine flows from first horizontal heat exchanger at −15oC into the the second liquefier, placed at an angle of 60o where it is further cooled to −55oC. Red dash lines show the refrigerant stream. [2]

The chlorine compressor is electrically driven and directly contributes to the total grid load of the plant. We model the compressor as a single stage centrifugal compressor. Chlorine gas is compressed from atmospheric pressure to 0.3 MPa [2] via a polytropic process

[56, 90]. The exit gas temperature, Tf , can be computed in terms of pressure, P , and the initial temperature thus n−1 hPf i n Tf = Ti (3.18) Pi The polytropic exponent, n is selected to be 1.45 [56]. Assuming ideal gas behavior [56], the

47 th theoretical compression work, Wcomp in J/mol is given as

n−1 th n hPf i n Wcomp = RTi − 1 (3.19) n − 1 Pi where R is the gas constant in JK−1mol−1. To compute the work rate for the compressor, W th is multiplied by the molar ﬂow of chlorine product (N˙ ano,out from Section 3.3.1.2) less comp Cl2

the demand for the gaseous product. The polytropic eﬃciency, ηpc, is expressed in terms of

n and the adiabatic exponent, k (k = Cp/Cv = 1.4 for diatomic gases), as follows [91] (k − 1)/k η = = 0.92 (3.20) pc (n − 1)/n

Thus, the actual work done in compressing one mole of dry chlorine gas, Wcomp, is

th Wcomp = 1.086 × Wcomp (3.21)

The symbols used for the compressor model and the parameter values used later in the simulation case study are given in Table B.2.

3.3.3 Storage unit

Liquid chlorine from the liqueﬁer is stored in storage vessels situated on load cells [56, 2]. We model the storage vessel as an aggregator thus dM s = M˙ − M˙ (3.22) dt in out ˙ where Ms is the total material stored and M denotes mass ﬂow rate. Storage limit equivalent to 13.435 ton of stored liquid chlorine is imposed based on the plant capacity. Assuming a

99% product recovery from the liquefaction process [2], the demand for liquid chlorine, Dliq, is satisﬁed thus D = 0.99 × [N˙ ano,out − D ] + [M˙ − M˙ ] (3.23) liq Cl2 gas out in

48 The demand for gaseous chlorine, Dgas, is entirely supplied directly from production and no liqueﬁed product is re-gasiﬁed to meet this demand i.e., the following holds true

N˙ ano,out ≥ D (3.24) Cl2 gas

The nomenclature for the symbols used is given in Table B.3

3.4 Simulation case study 3.4.1 Steady-state simulation

In order to compare the model results to the numbers reported in literature, a steady state simulation for the Uhde membrane cell described in [56] was carried out in gPROMS (ProcessBuilder 1.2.0) [92]. The unit is able to generate approx. 55 ton of chlorine gas per day at nominal current density of 0.4A/cm2 with a maximum capacity of 80 ton of chlorine per day [2]. The cell specification and input parameters based on normal operating conditions are given in Table B.5. The energy consumption for the cell operating at a constant current density of 0.4A/cm2 and cell temperature of 90oC is 1940 kWh per ton of caustic generated2 (also see Figure 3.5). This compares reasonably to the quoted literature value of 2100 kWh [56] (at the same operating conditions). The discrepancy in values is likely the result of the simplifying assumptions made during the model development, such as a 100% Faraday efficiency employed in the model development (e.g., with a 96 % Faraday efficiency, the energy consumed per ton of caustic goes up to 2020 kWh).

2 Operation of chlor-alkali cells yield approx. 1.1 ton of NaOH per ton of Cl2 produced [93]

49 3.4.2 Dynamic simulation

A dynamic simulation for the model parameters summarized in Table B.5 was per- formed in the gPROMS (ProcessBuilder 1.2.0) process environment [92]. The plant response to a step change in current density from 0.5A/cm2 to 0.6A/cm2 is shown in Fig. 3.4. The

0.6 3.1

3.08 0.58

2 3.06

0.56 3.04

3.02

0.54 Cell voltage, V voltage, Cell

3 Current density, A/cm density, Current 0.52 2.98

0.5 2.96 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Time, hour Time, hour

205 0.342 91 C C NaOH

NaCl fr wt concentration, Catholyte

200 0.34 90

C o 195 0.338 89

190 0.336 88 Temperature, Temperature,

Anolyte concentration, g/l concentration, Anolyte 185 0.334 87

180 0.332 86 0 0.5 1 1.5 2 2.5 3 3.5 4 0 0.5 1 1.5 2 2.5 3 3.5 4 Time, hour Time, hour

Figure 3.4: Plant response to step change in current density from 0.4A/cm2 to 0.6A/cm2 at time, t = 15 min. cell voltage shows the fastest response, increasing instantaneously to 102% of the steady state value then settling to the new steady state in approx 1 h 50 min. The cell temperature and concentration show ﬁrst order dynamics with time constants of approximately 15 min

50 and 19 min, respectively. These ﬁndings agree with the literature [84, 79, 61, 66] whereby the cell concentration and temperature show slower dynamics compared to the cell voltage.

Fig. 3.4 shows that the steady state temperature at current density of 0.6A/cm2 violates the maximum safe operating temperature bound of 90oC. Operating at a current density of 0.4A/cm2 with inlet feed conditions as speciﬁed in Table B.5 yields a steady state temperature of approx. 83oC (Figure 3.5). The cell temperature thus, is sensitive to

C 100 95 o

95 C o 90 90

85 85

80 Feed temperature, temperature, Feed

Steady state cell temperature, temperature, cell state Steady 75 80 0.2 0.3 0.4 0.5 0.6 0.7 0.2 0.3 0.4 0.5 0.6 0.7 Current density, A/cm2 Current density, A/cm2

Figure 3.5: Temperature trends with current density changes for conditions specified in Table B.5. Left: steady state cell temperature for constant inlet feed conditions. Right: required inlet feed temperature for specified constant outlet cell conditions. the current density load and must be monitored carefully during transient cell operation. Regulation of the cell temperature is typically achieved by manipulating the inlet feed temperature, alternating between heating and cooling of the feed stream as a function of the cell temperature [81]. Additionally, the flow rate of the recycled catholyte stream can be manipulated to enhance temperature control [56]. In the sections that follow, we set the feed temperature as an optimization decision variable determined based on the operating current density. Cooling will typically be needed with high current densities while heating is required to maintain the cell temperature at the specified level during operation at low

51 current densities. The optimized value for the feed temperature in turn sets the heat exchanger heat duty for heating or cooling the recycle stream. Thus, the heat exchanger is fed with cooling water or steam as needed. We also assume equal temperatures for the catholyte and anolyte feed as it is recommended to maintain close temperatures on either side of the membrane [56].

3.5 Price-based demand response operation of the membrane chlor- alkali process 3.5.1 Production scheduling problem

We cast the DR operation of the chlor-alkali process as an optimal production scheduling problem. The optimization aims to find a time-varying production level profile (thus, the associated current density, I, and feed inlet temperature, T in), that minimizes the cost of electricity over a fixed production time horizon.

Assuming a 3 day scheduling horizon (TH = 72 h), the optimization problem (subject to the dynamic plant model discussed earlier, and to demand and feasibility constraints) to be solved is formulated as:

Z TH 1 h ˙ ano,out i Min Eprice × NcIUcell + Wcomp × (NCl − Dgas) in ˙ ˙ 2 I,T ,Min,Mout 3600 0 subject to Dynamic plant model, Eq. (3.2)- (3.17) (3.25a) Demand constraints, Eq. (3.23)- (3.24)

Feasibility constraints (given in Table B.5)

Ms,TH ≥ Ms,0 (3.25b)

A multiplying factor of 1/3600 is included in the objective function to account for the

52 discrepancy of time units between the scheduling problem (where time is expressed in hours to account for the longer time horizon) and the dynamic model (where seconds are used). Additionally, an end constraint is imposed on the inventory level ensuring that the product

stored at the end of the horizon, Ms,TH , is at least equal to the initial amount of material inventory, Ms,0.

It is assumed that electricity prices are known with certainty for the entire horizon.

The optimal plant schedule is computed in response to electricity price (Eprice) changes in the fifteen minute electricity market (price profile is shown in Fig. B.1). The scheduling horizon is thus divided into 288 control intervals, each of length 15 min. The current density of the electrolysis unit can change every fifteen minute as electricity prices change. Rather than modulating the plant load by shutting down some cell units, we assume continuous operation of the entire cell stack of 160 cells, with each cell unit operating between minimum and maximum current densities of 0.3A/cm2 and 0.6A/cm2 respectively [56]. We note that even though the quoted lower bound for the current density for the Uhde cell is 0.2A/cm2, we have enforced a lower bound of 0.3A/cm2 in the optimization formulation since at current densities below this value, chloride ions seep through the membrane to the cathode chamber contaminating the caustic product and reducing chlorine yield [56, 81]. Additionally, the choice to maintain a non-zero lower limit on the current density is motivated by previous research [84] which suggests that maintaining some minimum current flow within the membrane cell attenuates the long-term deterioration of cell performance that may be caused by highly transient operation. Other cell parameters was adopted from the Uhde cell specification [56]. The total demand for chlorine is set such that nominal production at 0.4A/cm2 exactly satisfies demand with no need for storage. The storage unit at maximum capacity is

53 able to independently supply the demand for liquid chlorine over a 6 h period. Other input parameters were set based on typical operating conditions of the chlor-alkali plant [56]. All input variables discussed are summarized in Table B.5. The dynamic model of the plant as included in this DR-oriented scheduling problem is a diﬀerential-algebraic equation (DAE) system, having a total of 1780 equations (of which 20 are diﬀerential). Problem (3.25) was implemented and solved in gPROMS [92] on a 64 bit Windows 7 PC with an Intel Core i7, 2.60 GHz processor and 16.0 GB RAM. The optimization was interrupted after approx. 3 days of CPU time and no improvement in objective function value for over 30 iterations.

3.5.2 Optimization results

The operation schedule for a ﬁxed horizon optimization, whereby the ﬁfteen minute market electricity price is assumed known for the entire 72 h period is shown in Fig. 3.6.

The plant can curtail its power consumption from the nominal value of 4, 838 kW to a minimum of 3, 442 kW. Thus, a maximum reduction in power demand of 29% (corresponding to a reduction in current density from 0.4A/cm2 to 0.3A/cm2) is attainable. This is a significant short-term load curtailment for a 2 MW facility, and is achieved –as expected– during peak electricity price periods (e.g., hours 18 and 33). From the electric utility stand- point, this plant has the potential to provide substantial “peak-shaving” and “valley-filling” services by fluctuating current density in response to electricity price signals. Stored product is depleted during these peak-shaving periods to sustain demand for chlorine product. The plant recovers when electricity becomes cheaper. On the other hand, the energy (operating) cost to the chlor-alkali facility is reduced from $94, 451 (at the nominal operating conditions) to $87, 557 leading to a 7.3% savings in energy cost over the 3 day horizon.

54 100

50 cent/kWh 0 Electricity price 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour 10000

5000 Power, kW 0 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour 20 , ton 2(l) 10

0 Stored Cl 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour

Figure 3.6: Optimal schedule showing the actual power consumption (middle) and inventory (bottom) for the chlor-alkali plant over a 3 day horizon.

Naturally, this cost savings will vary based on the electricity price proﬁle. It is worth noting that the current study assumes that accurate predictions/forecasts of electricity prices are available, and the savings will also depend on the accuracy of these forecasts. One potential means to compensate for forecast inaccuracy is to implement the optimization framework described above in a recursive fashion, repeating the calculation periodically over a moving or shrinking time horizon [4]. This approach represents the topic of a future study.

55 Also observable from Fig. 3.6 is the repeated strategy of over-generating product during low electricity price periods (e.g., hours 35 - 42) and storing the excess product for deployment during future electricity price peak periods. Such operation exploits the availability of product storage and under-utilized plant production capacity. Note that the available inventory is not necessarily depleted to zero. Since the optimization formulation is deterministic, production is scheduled such that periods with the most favorable electricity prices in the entire horizon are utilized for recovery as the end constraint on inventory level must be satisﬁed (Eq. 3.25b). Also, the production capacity is constrained such that the minimum production level at any time must satisfy the demand for gaseous product (Eq. 3.24). The beneﬁt of this strategy becomes more obvious with increasing length of the scheduling horizon. Consider the case whereby Problem 3.25 is solved over a 24 h horizon. The storage capacity is only minimally utilized over the scheduling time span (Fig. 3.7). Intuitively, the plant is able to maximally utilize its DR potential when more future information (in this case, electricity price) is available with certainty. When forecast electricity prices are used, improved utilization of plant and storage capacity will be strongly dependent on the accuracy of the price forecast.

The dynamic evolution of the electrolyte concentration and temperature in response to frequent changes in current density is shown in Fig. 3.8. Observe that while the frequent change in current density provides almost instant power fluctuations as a result of the rapid response of the cell voltage, these fluctuations do not instantly propagate in the cell variables. Particularly, we see a smother profile in the cell variables as compared to the step profile for the actual power consumption level. The process exhibits some inertia resulting from its dynamics and impulse power changes of small magnitudes (e.g., the dip in hours 4.5

56 8000

6000

Power, kW Power, 4000

2000 0 4 8 12 16 20 24 Time, hour 14

, ton , 2(l) 10

8 Stored Cl Stored

6 0 4 8 12 16 20 24 Time, hour

Figure 3.7: Optimal production schedule for the chlor-alkali plant for a 24 h horizon and 33) are barely noticeable. With proper control design, it is likely that the cell may be able to tolerate more frequent modulation of current density without adversely aﬀecting the cell variables i.e., cell is robust and “impulse” current changes is not propagated in the cell variables. This will present an opportunity for the chlor-alkali plant to provide fast paced ancillary services like frequency response.

Fig. 3.9 illustrates the fluctuation of feed temperature in order to regulate the cell temperature. As expected, the optimal feed temperature profile suggests introducing cooler feed into the cell when current density values are high and heating up feed to high temperature values when the current density is reduced. It is noteworthy that the constraints on the cell temperature are active for over 50% of the scheduling period and the cell temperature places additional limits on the current density values. Temperature regulation can be enhanced through other means (like varying electrolyte flow rate with current density and

57 8000

6000

Power, kW Power, 4000

2000 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour

92 C o 90

86 Cell temperature, temperature, Cell

84 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour

240 C NaOH 0.34 230 C

NaCl fr wt centration, 0.335 con- Catholyte 220 0.33

210 0.325 Anolyte con- Anolyte centration, g/l centration, 200 0.32

190 0.315

180 0.31 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour

Figure 3.8: Evolution of cell variables for the optimal production schedule implementing diﬀerent catholyte and anolyte ﬂow rates [56]). This will likely further expand the DR capability of the plant, leading to greater savings on operating costs.

3.6 Conclusion

Chlor-alkali production via electrolysis is an energy-intensive industry that can optimize its operations to provide signiﬁcant demand response services. To eﬀectively achieve

C o 88

84 Feed temperature, temperature, Feed 82 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour

92 C o 90

86 Cell temperature, temperature, Cell

84 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour

Figure 3.9: Optimal temperature proﬁle for feed electrolyte (top) and resulting cell temperature for the optimal schedule (bottom). this, the electrolysis plant models used for computing optimal schedules should adequately represent the dynamic evolution of relevant cell variables. The model presented in this chapter can be utilized to study various demand response engagement strategies for membrane- based chlor-alkali plants. The extensive case study presented in the chapter reveals that the inherent inertia of the process allows accommodation of short-term changes in power consumption level. Consequently, such plants are a potential candidate for fast-paced demand response. The study also reveals that several process variables (besides current/current density and associated power consumption) evolve during such transitions, and must be accounted for when scheduling demand response operations. Particularly, the cell temperature constrains the speed of demand response and available demand response capacity, and should be carefully controlled. While the demand response capabilities of a industrial-size

59 membrane-based plant are substantial, the benefits accrued by the process operators do depend on the accuracy of electricity price forecasts, and further work is required for dealing with price uncertainty. To this end, a moving horizon approach appears like a reasonable option, and will constitute the topic of future work. Additionally, it should be noted that even though the chlorine storage capacity considered in this study is well within the storage limits for liquid chlorine [94], in general, flexible operation for demand response may be further constrained by guidelines regarding on-site chlorine storage. Finally, at the fundamental level, more research is required to assess and quantify the effects of cycling operation on the cell performance and its degradation when fast demand response operation is implemented.

60 Chapter 4

Scheduling Chemical Processes for Frequency Regulation

This chapter is based on material published in paper “Otashu, J. I., & Baldea, M. (2020). Scheduling chemical processes for frequency regulation. Applied Energy, 260, 114125 ”[95]. Joannah Otashu was the primary contributor to this paper.

4.1 Introduction

The transition towards higher dependency on renewable energy sources for electricity generation and the advent of the “prosumer” (producer-consumer) era, whereby users of electricity can simultaneously generate power through distributed resources like rooftop solar panels, have elevated the concern of ensuring the stability of the power grid. The challenge of matching the supply and demand for electricity is thus exacerbated by the new variability stemming from the inclusion of renewable energy sources in the power generation mix. While the influx of wind and solar power generation promises to mitigate pollution, it also threatens grid robustness, subjecting it to significant disturbances such as large, sudden capacity losses and frequency deviations [96, 97]. In this context, discussions of modifying energy use patterns of residential, commercial and industrial users (i.e., demand-side management/demand response) to alleviate such stress on the grid have amplified. Their scope

61 is twofold. From the end-user perspective, energy consumption schedules can be designed to minimize costs incurred by the users, who are incentivized to do so via time sensitive electricity tariﬀs. Alternatively, from the utility operator perspective, the goal is to utilize demand-side management to address short-term (i.e, preserving grid stability) and/or long-term (e.g., minimizing transmission losses or generation costs) goals. Several papers on both perspectives have been published in the last decade and reviews are provided in [98, 99, 100, 101, 102].

Specifically, for industrial demand response, the user-centric paradigm largely involves price-based response. Users can bid power consumption/generation capacity in energy markets or alternatively, voluntarily schedule plant operation in response to variable electricity prices. Zhang and Hug [103] and Ramin et al. [104] develop such optimal bidding strategies employing mixed integer formulations for aluminum smelting and metal casting processes, respectively. Zhao et al. [105] and Zhang et al. [106], exemplified the production scheduling problem for steel plants under time varying electricity prices. Such voluntary responsive process scheduling has also been studied for air-separation processes with various degrees of model complexity [46, 11, 52, 107] and considering various plant configurations [10]. Fur- thermore, Wang et al. [72] and Hadera et al. [108] explore hybrid power generation sources for expanding the flexible plant capacity deployed in similar demand response activities for chlor-alkali process and steel plants, respectively. We note that in these circumstances, the influence of the grid operator on the functioning of the load is at best indirect: users are likely to want to take advantage of, e.g., lower off-peak electricity tariffs or avoid penalties, but they are neither obligated nor can they be forced to do so. As a consequence, the impact of price-based demand response on the operation of the grid can be difficult to anticipate

62 exactly. Instances where several users adhere to the demand response paradigm can have negative consequences, such as the advent of rebound peaks, i.e., large increases in electricity consumption at times that are under normal circumstances considered to be oﬀ-peak (e.g., as observed in the California pilot study [109]), are also a concern. This has provided the impetus and motivation for seeking a grid-centric paradigm, whereby users are in general contractually obligated to follow directions formulated by the grid operator. Example studies include frequency regulation discussed for aluminum smelting by Todd et al. [22] and Zhang and Hug [110]. Chau et al. [111] and Zhang et al. [112] provided similar demon- strations for cement manufacturing. Alternatively, demand response through load following can alleviate some uncertainty arising from the demand side, as illustrated in [113]. More disruptive methods like load curtailment [114] and directly integrating industrial loads in grid frequency control architecture [115] have also been proposed.

Much headway in implementing the grid-centric paradigm has been made with a speciﬁc category of electricity users, namely, residential and commercial buildings [116, 117, 118], which make up 74.7% of electricity use in the US [43]. The proposed solutions include, e.g., invoking a social objective function in determining optimal consumption patterns for responsive users, and explicitly incorporating models of the related loads in solving the grid-optimization problem. The beneﬁt of considering supply and demand in an integrated fashion is twofold: demand schedules are determined with explicit models of the loads, and the optimal dispatch for the responsive end-users is feasible and can be closely followed. Secondly, from the supplier perspective, improved estimates of available demand response capacity enhance the ability to balance supply and demand in the face of uncertainty, with the latter originating both on the supply and demand sides.

63 Such an integrated approach has, however, not been extensively studied for industrial demand response [119, 120] with few examples presented in [121]. Industrial users (our focus hereafter will be on chemical processes) present the advantage of having large, concentrated electricity loads, and are therefore prime candidates for engaging in demand response programs that support the grid, as pointed out by multiple authors [69, 122, 4, 123]. However, their operation is subject to strict safety and product quality constraints (both static and dynamic). Violating these restrictions may lead to severe economic penalties and in extreme cases, jeopardize operational safety.

The currently preferred means of capturing operational restrictions placed on load/ consumers in grid-level models (e.g., optimal power ﬂow models) involve the use of ramp rate and capacity constraints [124]. These represent (typically time-invariant) bounds on the (rate of change of) power demand by an entity, and are based on the dynamics of the physical system. Given that the dynamic response of a chemical process is generally nonlinear (and thus its “true ramp rate” will in eﬀect change as a function of the current operating point), such ramp rate constraints must be chosen very conservatively to be applicable to the entire operating range and avoid the aforementioned violations. In turn, this unnecessarily diminishes the demand response potential of a chemical process. Conversely, a “looser” choice of ramp rate constraints may lead to infeasible process operation [11, 119, 46, 36] or overestimation of the demand response capacity of industrial end-users [119, 125]. An Alcoa aluminum smelting facility for example, which currently provides frequency regulation in the mid-western grid, implements ample control strategies including pot line rotations in extreme cases to satisfy process constraints while providing regulation [22].

A potential solution to this conundrum is to provide grid operators with detailed and

64 accurate models of the dynamic behavior of chemical processes as grid loads. This is an unlikely path from the point of view of process operators, who are reluctant to share such data lest they divulge sensitive business information. Thus, by the present modus operandi, whereby “exogenous” dispatch signals for responsive loads are externally generated by the grid operators using static models, participating chemical processes face the risk of infeasible (in the sense of not being able to meet the load curtailment limits of the grid) or unsafe operation, especially when schedule changes for demand response occur at a fast pace or such grid support services are to be provided continuously for an extended period of time.

Motivated by the above, in this paper we propose a converse “endogenous” approach: we introduce the means to incorporate grid-level considerations in the optimization of the production schedule of a chemical plant. We focus on the response of chemical processes over fast time scales and in reaction to changes in the frequency of the electricity supply (i.e., frequency regulation).

Specifically, we begin by formulating the production scheduling problem for a chemical process as a dynamic optimization problem that explicitly accounts for the dynamics of the plant. The (base case) problem considers a time horizon that spans several hours and for which (e.g., day-ahead) electricity prices are available at hourly intervals and known in advance. We connect this problem to the goal of providing frequency regulation service by defining a new quantity, the frequency-adjusted price, as correction to the day-ahead price based on the instantaneous deviation of the grid frequency from its nominal value. A new optimization problem is then formulated to compute the optimal process behavior as a deviation from the base case. We provide a classification of chemical processes in terms of a responsiveness factor, which relates their behavior in response to frequency changes to

65 the aforementioned base case. We use an industry-relevant chlor-alkali process model to illustrate the theoretical concepts.

4.2 Problem Formulation

In the sections that follow, we discuss the formulations for the base case and frequency- adjusted production scheduling. Following this, we define a new incentive for scheduling for frequency regulation, namely, the frequency-adjusted price. Finally, we discuss the notion of process flexibility with regards to the grid system conditions through the defined measure – process responsiveness factor. We begin by introducing the description of time that serves as a basis for defining the scheduling problems.

4.2.1 Time description

A discrete-time representation is employed. The scheduling time horizon for the base case, τB, spans multiple hours while the frequency-adjusted scheduling problem is formulated for a shorter time horizon, τF . Additionally, the discrete control intervals for the frequency- adjusted scheduling problem are set such that the optimal schedule is flexible enough to track the frequency changes reflected in the modified price signal (i.e., control interval of several seconds - a few minutes). While the control interval for the longer-term scheduling problem (i.e., the base case) can generally span the length of the rate of change of the chosen base case electricity price (e.g. one hour for hourly changes in electricity prices), a natural approach is to synchronize the base schedule with the frequency-adjusted formulation by implementing the same control intervals for both problems. This allows for greater flexibility to provide frequency response in the short-term. For example, consider the case where the production

66 target for the base schedule varies hourly in response to the day-ahead electricity price. For every hour with optimal base production level set to the maximum or minimum bounds, no “room” (i.e., additional modulation capacity) is available for frequency response. However, if the production schedule is allowed to vary more frequently (i.e., sub-hourly) albeit in response to an hourly signal, the optimal production levels may have diﬀerent values within the same operating hour, thereby providing “room” to reschedule for frequency regulation. Finally, we assume that the process model incorporated in the scheduling formulations describes the transient evolution of the process variables over a continuous time domain.

4.2.2 Demand response scheduling and a correction based on grid frequency

In the course of providing demand response, the industrial entity (chemical process) schedules its production to minimize the costs incurred for electricity consumption. The power consumption rates, P , are varied by leveraging storage, S, in response to a given set of electricity prices. This constitutes the base case (Problem (4.1)).

Z τB min Energy cost P,S 0 s.t. dynamic process model

inventory model (4.1)

base case price proﬁle

end-constraint on inventory The base case electricity price can be any set of prices known with some certainty over a time span that is typically much larger than the sample time of the grid frequency signal. For example, the base proﬁle may be a ﬂat price rate, critical peak prices or even the day-ahead electricity prices, as these are published at least 24 hours before the delivery period. Here, the

67 product inventory is employed to “store” electricity in the form of chemical products. During peak electricity demand periods, when production is lowered in a demand response activity, stored product is used to compensate for reduced production levels. Conversely, production is increased above the nominal level during oﬀ-peak periods and the excess product generated is stored. The schedule computed using (4.1) is in essence independent of short-term power imbalances, which are reﬂected in e.g., the grid frequency.

A second scheduling formulation is developed to deploy any additional demand response capacity of the process in the provision of frequency regulation. Problem (4.2) computes an updated near-term optimal production schedule in response to a modiﬁed price signal that accounts for the current state of the grid (the frequency-adjusted price discussed in 4.2.3). As the frequency of the grid varies quickly and in real-time, we compute this scheduling problem over a shorter time horizon, τF τB.

Z t+τF min Energy cost P,S t s.t. dynamic process model

inventory model (4.2)

frequency-adjusted price proﬁle

end-constraint on inventory

A short scheduling horizon is suitable for the given deterministic formulation since prediction accuracy (of system frequency in this case) diminishes with a longer prediction horizon [126, 127]. In order to minimize losses associated with the uncertain grid frequency ﬂuctuations, an additional constraint on the inventory level is imposed, stipulating that the amount of product stored at the end of the scheduling horizon τF , must at least be equivalent to the

68 inventory anticipated to be available in the base case for the corresponding time period.

Besides ensuring suﬃcient product generation in the near term (τF ), to meet the long term inventory/demand constraint (at the end of τB), imposing this constraint also achieves the following: (a) an equitable comparison of the outcomes of both optimization problems (e.g., energy consumption rates) can be made (b) the diﬀerence in total energy consumption over

τF for the frequency-adjusted schedule compared to the optimal values scheduled in the base operation for the same time period is kept to a minimum. This latter outcome is beneﬁcial since intuitively, it is desirable not to deviate too signiﬁcantly from the overall consumption levels determined in the base case. Rather, the idea is to reschedule production rates within

τF in the most eﬃcient manner to help restore the grid frequency to the nominal value despite the limitation of being only aware of short-term market conditions (i.e., frequency-adjusted prices for the present τF not the entire horizon, τB).

4.2.3 Frequency-adjusted price

Currently in most energy markets, load resources contribute to frequency regulation by responding to real-time signals generated by the balancing authority. These dispatch re- quests specify the time dependent/instantaneous power reduction or increment to be achieved by the load resource within its ramp limits in order to maintain overall grid stability. As the calculation of such dispatch signals does not typically reflect the nonlinearities and other intricacies in the process dynamics, as well as other production constraints the process may be subject to, it may be difficult to closely follow the given signals. For the purpose of generating feasible power modulation patterns for frequency regulation, we develop a modified electricity price i.e., the frequency-adjusted price as an internal surrogate for dispatch signals

69 from the system operator. This new price reﬂects the current state of the electric grid and aims to curtail the mismatch between the demand and supply of electricity thus stabilizing the grid system frequency around its nominal value (taken to be 60 Hz).

Deﬁnition 1 (frequency-adjusted price). Given the instantaneous grid system frequency, f, and the nominal price signal, πnom, the frequency-adjusted price, πf is given as

h 60 − f i π = π 1 + α . (4.3) f nom 60

The adjusted price can be construed as a correction factor that reﬂects the relative state of the grid (i.e., deviations of the instantaneous grid frequency from normal conditions scaled by a factor, α) weighted by the corresponding base case price for the given time period.

A sample adjusted price profile and corresponding frequency values are shown in Fig. 4.1. When the grid is in an overload state, the frequency falls below the nominal value and, as a result, the frequency-adjusted price is higher than the base price. This will lead to new power consumption targets for the load that are lower than the base values thus relieving some stress from the grid. The converse is the case when generated power is underutilized and the system frequency rises above 60 Hz. The tunable parameter α is a positive real number (ideally greater than or equal to 60) to avoid diminishing the effect of the frequency deviations. Note that low α values lead to new prices that slightly fluctuate around the base price profile. This may lead to minimal response by the process to grid frequency fluctuations. The impact of the value of alpha is further discussed later in this chapter. The above formulation allows a chemical process to carry out production scheduling and determine power consumption profiles for frequency regulation locally.

70 60.3

60.2

60.1

59.9 grid frequency, Hz frequency, grid 59.8

59.7 0 0.5 1 1.5 2 2.5 3 time, h 0.8 base price frequency adjusted price, = 60 frequency adjusted price, = 600 0.6

0.4

0.2 electricity price electricity

-0.2 0 0.5 1 1.5 2 2.5 3 time, h

Figure 4.1: Top: System frequency values over a three hour period Bottom: Corresponding frequency-adjusted prices compared with the nominal base prices for two diﬀerent values of α

4.2.4 Process responsiveness

To quantify the amount of ﬂexible process load available for frequency regulation with respect to the volatility of the grid frequency, we deﬁne a new metric called process

71 responsiveness factor.

Deﬁnition 2 (Process responsiveness factor). The process responsiveness factor, Rf , is a relative measure that deﬁnes the extent of power consumption changes made relative to the process base consumption schedule in response to frequency deviations observed within

the scheduling period, τF , as a result of solving the modiﬁed production scheduling problem.

Mathematically, the instantaneous value of Rf is computed as

∆Pt Rf = (4.4) ∆ft

where ∆Pt and ∆ft are the normalized values of the deviation of the instantaneous power consumption from the base proﬁle and the grid system frequency from the nominal value respectively.

Note that the instantaneous power consumption of the given industrial end-user is constrained by the bounds on the process variables and inherent inertia of the plant. Thus, the evolution of the responsiveness factor over time is inﬂuenced by the individual dynamics of the given industrial plant. This is further discussed in the subsections that follow.

Let PF,t, be the plant power consumption schedule for the frequency-adjusted opti-

mization problem, (4.2) and PB,t be the optimal base case power proﬁle resulting from (4.1).

With f as the grid system frequency, the instantaneous power deviation, δPt and frequency

deviation, δft, are computed as

δPt = PF − PB (4.5a)

δft = f − 60 (4.5b)

72 δPt and δft are scaled (relative to values within horizon, τF ) such that their absolute values lie between 0 − 1 and 1 − 2 respectively. i.e.,

δPt ∆Pt = (4.6a) max | δPt |

 sgn(δf ) 1 + |δft| if δf 6= 0 t max|δft| t ∆ft = (4.6b) 1 if δft = 0

According to Eq. (4.6a), | ∆Pt | is equal to one when the power deviation for the given period is at its maximum level. Other power changes made within the period are represented as a fraction of this maximum deviation, with positive values indicating higher power consumption with respect to the base level and vice-versa. Similarly, | ∆ft | is a scaled value of the frequency deviation relative to the maximum value observed during the scheduling period. | ∆ft | is equal to one when the grid frequency is at its nominal value (minimum absolute frequency deviation) and two when the frequency deviation observed is maximum.

Negative ∆ft values indicate decline in grid system frequency and positive values denote higher grid frequency levels compared to the nominal value. Note that δPt and δft have been scaled as shown, so that the absolute value of the responsiveness factor deﬁned in Eq. (4.2.4) lies between zero and one. Some insight can be inferred from these numerical values as depicted in Fig. 4.2. This is discussed in the subsections that follow. It is easy to see that

positive Rf values denote desired process response i.e., decreasing power consumption when grid frequency falls below the nominal values, and increasing power consumption to absorb

some of the excess power during an underloaded grid state. Conversely, negative Rf values indicate a reverse (undesirable) plant response.

73 maximum flexibility, proportional response

Figure 4.2: Range of process responsiveness factor, Rf (positive), values. Black curve indicates proportional power change to grid frequency deviations. The red star denotes the point where maximum ﬂexibility is oﬀered for maximum frequency deviation observed

Below, we discuss the frequency regulation capabilities of a process in terms of responsiveness factor. The general discussion focuses on positive Rf values.

4.2.4.1 Case 1: Rf u 0 (minimum ﬂexibility)

When the power deviation is very small for a given frequency deviation, Rf will be close to zero (blue area of Fig. 4.2). This indicates very little flexibility of the given process schedule. Generally, this inflexibility is undesirable. However, if the frequency deviation is also small (| ∆f |≈ 1), this response even though inflexible can be considered proportional

74 to the frequency deviation and thus acceptable. Such inflexible response could occur when the process power consumption level stipulated by the base case profile is at its bounds and thus no room for new adjustments (depending on ∆f) is possible. This situation of high inflexibility may also be observed when process feasibility or end inventory constraints are active.

4.2.4.2 Case 2: Rf u 1 (maximum ﬂexibility with low proportionality)

Rf will have its highest values (≈ 1) when very large power changes are observed in response to the smallest frequency deviations for the given time period τF (red region in Fig. 4.2). This response, even though highly flexible, is generally not the best process response. Noting that the process is constrained to generate at least the same amount of product pre-planned in the baseline schedule, the above described situation may lead to inflexibility during periods of larger frequency deviations. Like the previous limit case (Rf u 0), a situation like this may arise due to rigid baseline schedules i.e., process is operating at bounds and active constraints. The process thus performs its best to feasibly regulate the system frequency with the highest power change not necessarily provided when the worst frequency fluctuation occurs.

Remark 2. It is noteworthy that the two limit cases discussed above represent circumstances when exogenous dispatch signals would be diﬃcult to track. These instances which are mainly induced by nonlinearities and saturation are otherwise challenging to integrate into grid-level representations of the process models.

75 4.2.4.3 Case 3: Rf u 0.5 (ﬂexible and proportional response)

The third scenario describes the case of proportional process response to grid frequency ﬂuctuations. In this instance, average Rf values around 0.5 will be observed, resulting from moderate to high power consumption modiﬁcations for low to high frequency deviations

(green region of Fig. 4.2). This value is generally more desirable than Rf values of 1, as it indicates a ﬂexible schedule that is also proportional to the magnitude of frequency deviations within the given period. Additionally, the best process response achievable, whereby the maximum change in power draw by production rate modulation is scheduled for the highest frequency deviation observed (point marked with a star in Fig. 4.2), falls within this region. As will be seen in the case study that follows, this occurrence is in general a fortuitous case due to process limitations and the uncertain nature of grid frequency ﬂuctuations.

4.2.4.4 Case 4: Negative Rf values (reverse and undesirable response)

The responsiveness factor will take on negative values if the rescheduled power consumption is higher than the base case levels during periods of low grid frequency. This will also happen if the power consumption rate is lower than the scheduled base level during an underload grid state. This scenario even though undesirable, may be unavoidable. This is because the process must recover (e.g., reﬁll depleted inventory, relax active process constraints) from any rescheduling that may have been previously carried out to provide frequency regulation.

Table 4.1 summarizes the discussed Rf values and ranks them in order of their desirability. The last row presents the least desired response.

Remark 3. As the responsiveness factor is time varying, the behavior of a given process can

76 Table 4.1: Interpretation of process response in terms of the responsiveness factor (decreasing desirability down the rows) Range of Rf value Rf value Process response Average ≈0.5 flexible and proportional Low (under stable ≈0 (∆f ≈ 1) proportional grid conditions) High ≈1.0 flexible and disproportionate Low ≈0 inflexible – – negative reverse (undesirable) response

fluctuate between various levels of flexibility, based on the planned base schedule, the specific dynamics of the involved process and the evolution of the frequency. Process economics and the requirement to satisfy product delivery are other factors that affect this quantity.

Additionally, the extent of flexibility of the process and associated Rf values also depend on the scaling factor, α. This can be tuned to optimize the response of the given process. Finally, the responsiveness factor is an evaluation tool for the demand response activity of the related end user rather than a design target. The varying (and generally, nonlinear) flexibility expressed in the responsiveness factor is not reflected in the current methods employed for frequency regulation. Note that the ‘≈’ sign represents some tolerance level around the given numerical values. A tolerance of ±0.02 is employed in the case study discussed in succeeding sections.

77 4.2.5 Measure of desired process response

The aggregate terms, Rfs and Rfp, measure the degree of ﬂexibility and proportionality respectively. Rfs is computed as the integral of all Rf values over the time horizon

Z t+τF Rfs = Rf dt (4.7) t and Rfp reflects how often the regulation capacity offered falls within the desired proportional category defined earlier i.e., when Rf is either 0.5 or 0 given | ∆f |≈ 1. Let z be an indicator variable defined thus  +/− 1 if Rf ≈ 0.5 or Rf ≈ 0 | ∆f |≈ 1 z = (4.8) 0 otherwise then, Rfp is computed as Z t+τF Rfp = z dt (4.9) t

Note that the proportionality metric Rfp defined above does not indicate strict proportional response i.e., some instances of Rf that are smaller than 0.5 could indicate proportional response (black curve in Fig. 4.2). However, we judge that moderate proportionality with high flexibility (green region) is more desirable than strictly proportional response with lower flexibility. Also, an inflexible response to nominal grid frequency is a desired proportional response. Lastly, while the process responsiveness factor is a more general notion, the numerical values of Rfs and Rfp are not absolute and will vary from process to process and from time frame to time frame depending on the intrinsic dynamics of the system.

78 4.3 Case study

We illustrate the concepts introduced in the previous section using a chemical process – chlor-alkali production – that is in general, well-suited to providing frequency regulation service. A general process description is provided, followed by the plant model and implementation of the production scheduling problems. Lastly, the source of the data used in the numerical study is discussed.

4.3.1 Chlor-alkali process description

The generation of chlorine is a power-intensive process accounting for about 2% of the total electricity consumption in the United States [55]. The most common production method is via the electrolysis of brine, with the membrane electrolytic cell replacing older technologies [54].

The production process begins with the pretreatment of brine, saturating the salt solution to high concentrations and sometimes acidifying it to improve current efficiency [56]. The pretreated brine is then sent to the membrane cells where, together with dilute caustic soda (in the cathode chamber), a pair of redox reactions takes place, generating chlorine and hydrogen from the anode and cathode compartments, respectively. Cell temperatures of 85 − 90oC [81] and pressures of 200 − 240 mbarg [56] are maintained. The gases exiting the cells are saturated with water and undergo treatment including cooling and stage-wise drying steps to generate dry gas [55, 56]. Chlorine (which is sold as gas or liquid) is then liquefied for sale or diverted to gas pipelines. The process flow is summarized in Fig. 4.3.

This electrochemical method for chlorine production is suitable for fast demand response for several reasons. First, as with aluminum smelting [22], fast power changes can be

79 (NTP)

Acid dryer

Rectifier H2 product

(13oC) Dry Liquefaction Hydrogen Cl2(g) unit treatment

Cooling H Wet 2 Storage Cl2(g) Evaporator Membrane electrolyzer NaOH(aq)

Liquid Cl2 Brine pretreatment - purification 32-35% Brine feed - filtration Caustic - saturation etc. NaCl(aq) Heat exchange Cooling unit water

Figure 4.3: Chlorine production through brine membrane electrolysis. The liquefaction and storage units (in orange box) provide leverage for engagement in demand response activities achieved by directly modulating the voltage or current through the cells. Secondly, the relatively slow response of the cell temperature to current fluctuations provides a thermal inertia advantage, allowing for short-term current changes to be absorbed without an immediate significant impact on other process variables. Additionally, the liquefaction unit together with available storage allows flexibility to modulate production rates, over-producing and storing excess product and then depleting storage and reducing production as needed in a demand response activity. However, scheduling chlor-alkali production for demand response must be carefully carried out, since product generation directly depends on the current den-

80 sity ﬂowing through the cells. Also, as chlorine is a hazardous material, the stored quantities and storage conditions must comply with strict regulations [94, 56]. Chlor-alkali plants have been identiﬁed early on as potential industrial DR resources [69] and are currently engaged in slower-paced demand response initiatives e.g., tertiary reserves in the German market [122].

4.3.2 Dynamic model

The chlor-alkali process described above is modeled by a diﬀerential algebraic equation system describing the electrochemical reactions within the cell and the inventory of the storage system. This model is presented in detail in [42] and summarized below.

For an electrolysis cell of speciﬁed dimensions and with given electrode and membrane properties, the cell is modeled as a continuous stirred tank reactor with dynamics described by a set of equations of the following form: ˙ in Ci = Φ(Ci ,Ci, Q, I) ˙ 0 in T = Φ (Ci,I,T ,Ta, cp) (4.10)

00 U = Φ (Ci,I,T ) where X˙ indicates the first-order time derivative of variable X, the superscript ‘in’ represents inlet streams, subscript ‘i’ indexes all chemical species within the system, and the variables C, Q, I, U, T, cp,Ta represent concentration, volumetric flow rate, current, voltage, temperature, lumped heat capacity and ambient temperature, respectively. Φ in general, is a continuously differentiable function describing the nonlinear relationship between the given variables.

Some of the generated chlorine gas is liqueﬁed and stored. It is this inventory of liquid

81 chlorine that is exploited for the demand response activity of the plant. We assume that the demand for chlorine gas is only satisﬁed directly from production and liquid chlorine is not re-gasiﬁed to meet this demand. The management of the liquid inventory is governed by the following set of equations

S˙ = Sin − Sout

˙ tot out in (4.11) Dl = NCl2 − Dg + S − S ˙ tot NCl2 ≥ Dg where Sin and Sout are the ﬂow of material into and out of the storage unit, N˙ tot is the total Cl2 chlorine generation from the entire cell stack, Dg and Dl represent the demand for gaseous and liquid products. At a nominal current density of 0.4A/cm2, the plant generates 56 ton (“ton” refers to short tons throughout this chapter) of chlorine gas per day, with a power demand of 4800 kW (at cell temperature of 90oC). However, the plant power consumption can be lowered to approx. 3500 kW or increased to 8000 kW by manipulating the current density and feed temperature. We assume constant daily demand rates of 6.25 ton/d and 53.75 ton/d for gaseous and liquid chlorine products, respectively. With a maximum storage capacity of 13.435 ton, the storage unit can independently meet the liquid chlorine demand for 6 hours.

The energy consumption is a sum of the electric power consumed in the electrolytic cells and the power required to compress and liquefy excess product

˙ ˙ tot (4.12) E = nIU + Wc(NCl2 − Dg)

where n is the total number of electrolysis cells and Wc is the compressor load.

82 4.3.3 Implementation of production scheduling problems

The scheduling formulations introduced earlier ((4.1) and (4.2)) were implemented for the chlor-alkali facility. An optimal production schedule in response to hourly electricity prices published in the day-ahead Electricity Reliability Council of Texas (ERCOT) market defines the base case operation for the process (Formulation (4.1)). This base schedule is computed for a three hour horizon with scheduling intervals of one minute to enable flexibility in the schedule. While the three-hour scheduling horizon was chosen for computational tractability, it should be noted that this formulation is extendable to longer time horizons. The base case production profile will define the amount of “room” available for frequency regulation in real-time. Problem (4.2) is then solved for every hour within the base case scheduling period to determine new optimal schedules that minimize the energy cost while taking into consideration the grid system frequency. We have assumed that the frequency

values are known for the entire τF (one hour) but not τB (three hours) scheduling horizon. This in essence, represents the best case scenario for the frequency-adjusted scheduling problem. The frequency-adjusted schedules are then executed in real-time as a demand response activity. Both scheduling formulations were implemented as dynamic optimizations in gPROMS Process Builder 1.2.0 [92]. The NLP consists of 955 algebraic and 23 diﬀerential variables. Problems (4.1) and (4.2) solve in ≈ 34000 s and 1600 s (CPU time) respectively on a 16 GB RAM, Intel Core i7 personal computer with Windows 7 operating system.

4.3.4 Frequency data

Fifty diﬀerent scenarios of system frequency values sampled every minute for three hours were generated using MATLAB random number generator. Each set of data has a

83 mean of 60 and standard deviation of 0.0857 (99.9% of values lie within a ±0.3 bound as suggested in [128]). The data were filtered through a low-pass filter with a time constant of 90 s, chosen to be approximately 10% of the dominant process time constant [129]. The frequency-adjusted prices for the optimizations were computed from the resulting filtered frequency values.

4.4 Results and discussion

The discussion presented in this section focuses on the results obtained with the value of the tunable parameter, α in Equation (4.3), set to 600. Insights are drawn from extensive evaluation of the outcome of a single scenario and then average results over all scenarios are presented. Finally, a study of the eﬀect of the value of α is provided.

4.4.1 Optimal production schedules

The optimal base case and frequency-adjusted schedules are illustrated in Fig. 4.4 for the aforementioned single scenario (of reference frequency values). The base operation strategy is – intuitively – to generate excess product during the lowest-price hour and minimize production in the most “expensive” hour, hour 3. In addition, the production schedule is recomputed within every hour in response to the fluctuating frequency-adjusted prices. When the modified prices are low as a result of an underload grid state, production rate is ramped up above the base level if production for the base case is not already scheduled at the maximum limit of the plant and other process constraints (discussed later) are not violated. The converse also holds true. For example, in the first half of the third hour, during which the base production level is at the minimum, very little change in the schedule

84 0.8 base case schedule frequency adjusted schedule 0.6

0.4

0.2 normalized price normalized 0

-0.2 0 0.5 1 1.5 2 2.5 3 time, h 9000

8000

7000

6000

power, kW power, 5000

4000

3000 0 0.5 1 1.5 2 2.5 3 time, h

Figure 4.4: Optimal base case and frequency-adjusted schedule for a sample set of frequency data; α = 600 occurs in response to high frequency-adjusted prices. Equivalently, within this hour, multiple instances of production ramping up during underload grid conditions are observed, although also curtailed by the process feasibility constraints. More signiﬁcant power ramp up and down is generally achieved when the base production rate lies between its upper and lower bounds.

85 Other important factors inﬂuencing the process response are the inventory level and cell temperature (Fig. 4.5). Process ﬂexibility depends not only on the power consumption

7.6 91 base case schedule base case schedule frequency adjusted schedule frequency adjusted schedule 90 7.4

C 89 7.2 o 88 7 87

stored product, t product, stored 6.8

cell temperature, temperature, cell 86

6.6 85

6.4 84 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 time, h time, h

Figure 4.5: Evolution of stored product, liquid chlorine (left) and cell temperature (right) over time capacity limits of the unit, but also on the evolution of the process variables like the cell temperature (and hence, concentrations). For instance, assuming the cell temperature is at its upper bound from a previous production increase, regulation up will not be feasible even if the plant is not operating at the maximum production level or storage limits. Further increasing the current through the electrolysis cells will lead to a higher cell temperature which is unacceptable. Similarly, the end-point constraint imposed on the inventory further limits the regulation capacity of the process. The end-point storage constraint helps meet the contractual obligation to supply chemical product (chlorine) in a timely manner, ensuring it is not violated by the process as a result of engaging in frequency regulation. While the

86 hour-end constraints are exactly met for the ﬁrst and second hour, a slight over generation of product is observed in the third hour. This is because the new initial plant state does not allow for a feasible evolution of the cell temperature to support exact product generation achieved in the base case. A lower production level will lead to further decline of the cell temperature, potentially leading to a violation of the lower bound on this variable. Such consideration of the dynamic evolution of the process variables as a result of rescheduling for demand response is one pertinent aspect typically ignored in several reports [6, 70]. We emphasize that the dynamics of the chemical process contributes to the capacity available for demand response and the feasibility of rescheduling for demand response. The speciﬁc dynamic evolution of the process variables should thus be modeled and taken into account in evaluating the demand response capacities of chemical plants. This is typically not the case when the process dynamics are represented in grid-level models.

Lastly, even though the temperature profile for the frequency-adjusted schedule generally tracks the base profile, the more aggressive production rate modulation implemented in the frequency-adjusted case leads to a less efficient cell temperature evolution that incurs higher heat losses. A more detailed discussion on this is presented in Section 4.4.3.

4.4.2 Flexibility and proportionality of process response

The responsiveness factor, Rf , and resulting ﬂexibility and proportionality measures,

Rfs and Rfp for each hour were computed for the fifty frequency scenarios studied. Fig. 4.6 shows an example of the evolution of the process responsiveness for the first and third scheduling hours. A large decrease in grid frequency motivates reduction of the power consumption levels of the chemical plant which is sustained for longer periods during the first

87 1 0.6

0.5 0.4

f f

R R 0 0.2

0 -0.5 0 0.25 0.5 0.75 1 2 2.25 2.5 2.75 3

1 1 P 0 P 0.5

-1 0 0 0.25 0.5 0.75 1 2 2.25 2.5 2.75 3

2 2 f -1 f -1

-2 -2 0 0.25 0.5 0.75 1 2 2.25 2.5 2.75 3 time, h time, h

Figure 4.6: Process responsiveness factor values with scaled power and frequency deviations for the first (left) and third (right) hour for a scenario of frequency data hour. However, this flexible operation necessitates a recovery period during which zero and sometimes reverse power changes to grid conditions are observed. Thus, negative Rf values are observed at more time points of the given hour. The total flexibility observed for the fifty scenarios is depicted in Fig. 4.7. The process flexibility is highest in the first hour and lowest in the third hour. Also, with higher flexible operation comes larger required recovery period, as reflected in the total negative flexibility.

Fig. 4.8 illustrates the average distribution of process responsiveness according to

Rf values (numerical classiﬁcation was done with a tolerance of ±0.02). It is observed

88 total positive flexibility total negative flexibility 0 55

-5 fs

fs 45

40 -10 35

30 -15

25 flexibility measure, R measure, flexibility

flexibility measure, R measure, flexibility 20 -20 15

10 -25 hour 1 hour 2 hour 3 hour 1 hour 2 hour 3

Figure 4.7: Aggregate positive (left) and negative (right) process responsiveness, Rfs, for each hour

1% 1% < 1% < 1% 2% 2% 11% 15% 19% 27%

41% 47%

35%

37% 61% hour 1 hour 2 hour 3

flexible & proportional proportional inflexible others (positive) negative

Figure 4.8: Average distribution of process responsiveness according to classiﬁcation presented in Table 4.1. Average proportionality (Rfp) for each hour is about 2 - 3%. Flexible but disproportionate response (Rf value of 1) was not observed

89 that even though unavoidable process recovery period may sometimes lead to reverse process response, this occurrence happens less than 30% of the time. Also, the most desired process response, flexible and proportional response, whereby the maximum possible power consumption modifications are implemented for the highest frequency fluctuations observed occurs infrequently. Even though flexible operation is significant, it is not guaranteed that the highest modification to power draw levels will be implemented during the most critical grid state. This is a disadvantage of the proposed method for frequency regulation. More intrusive (from the process perspective) demand response programs like direct load control can ensure that load modifications from the participating entity are carried out at the instant the response is needed by the grid. The trade-off is that this sudden load change may lead to infeasible process operation for a chemical plant e.g., generation of unsellable product, failure to meet product demand, and in extreme cases, unsafe process operation - all of which are undesirable.

4.4.3 Energy consumption

The total energy consumption and product generation for each hour and the entire horizon were computed for all frequency-adjusted optimal schedules and compared to the base case optimal values. The energy consumption for all scenarios are higher than the base case schedule (Fig. 4.9). Futhermore, the increase in energy consumption trends positively with the amount of additional chlorine product generated in the frequency-adjusted cases. The non-zero positive intercept observed indicates that there is some eﬃciency loss inherently associated with the altered production schedule. The frequency-adjusted power proﬁle is suboptimal in terms of energy consumption. This is due to the fact that losses (such as

90 2 7 hr 1 6 hr 2 hr 3 1.5 5

1 3

0.5 1

% increase in energy consumption energy in increase % 0

% increase in total energy consumption energy total in increase % 0 -1 0 0.5 1 1.5 0 1 2 3 4 5 6 % increase in total Cl generated % increase in Cl generated 2 2

Figure 4.9: Percent change in total (left) and hourly (right) energy consumption rates versus increased product generation for the optimal schedules generated for ﬁfty scenarios of frequency-adjusted prices current loss, thermal loss) are higher for the new production schedules. For example, it is observed for frequency scenario 1 that the ohmic heat generation per ton of chlorine produced is 0.52%, 0.19% and 1.10% higher than the base case optimal values for the three consecutive hours. Note that this is the case even though the amount of product generated in the ﬁrst two hours is the same as the base case scenario (as previously indicated in Fig. 4.5).

Another noteworthy observation is the higher deviation of the energy consumption rates from the base case values for the individual hours (Fig. 4.9) compared to the total energy consumption levels. In a few instances, lower energy consumption values are observed for the frequency-adjusted schedule. This does not suggest that the baseline result is suboptimal, since the initialization of the process variables (i.e., the end values from the previous

91 operating hour) is not necessarily the same as in the base case or the same between scenarios. Of course, these negative values are not observed in the first hour given that the base case and frequency-adjusted scheduling problems for the first hour are initialized with the same values. The cumulative variation in energy consumption is however positive and generally lower than the highest values for the individual hours. This observation suggests that reducing the scheduling horizon for the frequency-adjusted problem (e.g., to ten minutes) may drive the aggregate energy use for the hour closer to the base case values. The disadvantage of this however, is the accompanying increased constraint on the process flexibility.

4.4.4 Choice of α

The value of α determines the weighting of frequency deviations relative to the base case electricity price. It prescribes the frequency dependency of the industrial load. Large α values generally lead to larger swings in frequency-adjusted prices and thus more aggressive process response compared to the more smooth responses for low α values (Fig. 4.10). Occasionally, this less aggressive response may be sustained for longer periods leading to a higher overall ﬂexibility for the hour e.g. hour 2 for α = 60 (Fig. 4.11). This less aggressive response may, however, be less desirable for frequency regulation as this service requires fast and frequent load changes to be made in response to quickly changing grid conditions.

We note that the responsiveness factor, Rf , bears similarity with the droop property of power generation units. The droop characteristic of a generator controls the power output of the generating unit according to the grid frequency. Generators with higher droop settings accomplish larger change in the generator operating state (resulting in larger deviations from their nominal output power) for a given frequency deviation compared to generators with

92 9000 = 0 (base case schedule) = 60 = 600 = 60000

8000

7000

6000 power, kW power, 5000

4000

3000 0 0.5 1 1.5 2 2.5 3 91

90 C o 89

87 cell temperature, temperature, cell 86

84 0 0.5 1 1.5 2 2.5 3 time, h

Figure 4.10: Optimal production schedule and cell temperature evolution for diﬀerent values of α

lower droop [130]. This relationship between Rf and the droop characteristic of generators can be exploited for a rigorous method of deﬁning α. For example, α can be tuned to motivate desired “droop” property, as reﬂected by Rf values. Additionally, rather than a

93 60

50 60 600 6000 60000

fs R 40

flexibility measure, 10

0 hour 1 hour 2 hour 3

Figure 4.11: Hourly (positive) flexibility for different values of α time-invariant variable, α can be set to be time-varying, evolving as the process response changes. However, it is important that this variability of α is limited (e.g., to avoid distorting the original shape of the motivating price profile).

Finally, although in theory, α can be tuned for desirable droop properties, the process response is still signiﬁcantly controlled by the governing dynamics of the process. Thus, the ultimate droop qualities desired may not be attainable. However, a compromise can be made. Developing a robust algorithm for tuning this parameter will require the integration of models of the electric grid and co-optimization of the grid and process objectives, for instance. This will be explored in our future studies.

4.5 Conclusions

The provision of frequency regulation by industrial plants can be beneﬁcial to improving electric grid robustness especially since industrial loads oﬀer a distinct capacity

94 advantage. Unlike electricity end-users in e.g., the residential sector, a single chemical plant can provide megawatt-scale power demand changes in a demand response activity. Addition- ally, constraints such as human comfort/behavior that typically limit the demand response capabilities of buildings, do not play a significant role in industrial operations. However, to accurately estimate the extent of regulation that can be safely provided by rescheduling the operation of industrial plants (and complex chemical processes in particular), realistic plant models must be employed. Consequently, in this chapter, dynamic models that are representative of brine membrane electrolysis have been employed to study the provision of frequency regulation by an industrial chlor-alkali facility. The results show that the process is often responsive to frequency fluctuations providing variable regulation capacity every hour. The capacity for frequency regulation is constrained by the dynamics of the process, and it was found that is not guaranteed that the maximum load change achievable by the facility can be safely offered at the instant of the grid’s greatest needs. Nonetheless, significant load changes can be deployed in frequency regulation with minimum disruption to the process. Additionally, frequency regulation services can be provided with a small increase in the total energy consumption of the plant. As the provision of grid support services is typically financially rewarded (e.g., capacity payments), increased costs to the process resulting from this engagement can likely be easily recovered. Future research should consider the fact that, since industrial facilities contribute a significant share of the total power demand on the grid, the impact of such regulation services on overall grid stability needs to be assessed. Furthermore, since frequency regulation is generally implemented in real-time and based on grid frequency measurements, additional work is required to determine the effect of uncertain grid conditions on the capacity available for frequency regulation.

95 Chapter 5

An Integrated Approach to Demand Response: Study on the Complete Integration of Chemical Processes in Grid-Level Operations

The material in this chapter is in preparation for publication.

5.1 Introduction

The power sector is witnessing an increased focus on improving the electric grid stability. This can be attributed to the recent (r)evolution of the electricity generation landscape. As fossil fuel-based generation is gradually but significantly displaced by greener sources of electricity, the power grid is becoming “smarter” and the roles of the grid nodes become increasingly fluid. Traditional end-users of electricity can now act as suppliers, either actively generating power or reducing their demand at times when this is beneficial for grid stability. In this context, maintaining grid stability has become more challenging, as both generation rates and power demand can fluctuate more readily than ever before. Solutions such as demand response (DR), whereby users of electricity modify their daily (and sometimes seasonal) demand in order to close the power demand-supply gap have been widely investigated, with a plethora of available literature reporting both theoretical concepts and applications to end-users in the residential, commercial (popularly, buildings)

96 and industrial realms.

While DR is generally a beneficial grid support service, DR activities carried out independently by end-users are essentially “selfish” (i.e., primarily aimed at improving user-level economics) and their aggregate effect can become detrimental at the grid level. For instance, new power demand peaks (“rebound peaks”) can emerge when several users modify their load patterns concurrently and in the same way, following the same time-varying electricity price information [109]. Tailored pricing schemes such as the coincident peak pricing offered by the Texas grid operator, ERCOT, aim to incentivize power demand peak clipping [131]. However, this approach currently addresses seasonal peaks and does not resolve the rebound peak problem. Pooling DR resources and managing them centrally together with generation sources can help ensure appropriate allocation of reserves while avoiding deleterious effects. Such concerted load control of, e.g., residential loads, was also shown to lead to generation cost savings [117]. At the industrial level, a hub consisting of 150 MW of industrial loads (7% allocated for DR) was recently certified for frequency regulation in the French power grid [132] (even though quantified grid benefits have not yet been reported in literature).

In spite of the aforementioned progress, evaluating the benefits and limitations of col- laboratively managing (large-scale) flexible demand alongside generation resources remains an important issue. Specifically, such coordinated management of the entire power system requires (1) information sharing among stakeholders while preserving business confidentiality for industrial entities and privacy for consumers (2) appropriate telemetry for fast information retrieval and dissemination and, (3) models that closely represent the response of each party to relevant changes in the state of the power grid. The present study focuses on the latter point. Our main contributions are as follows:

97 • Comprehensive load models that are suitable for grid-level computations are developed for a class of electricity end users, namely, electricity-intensive chemical processes.

• We propose a framework for scheduling power supply and flexible operation of the end-user chemical process to meet two goals: ensure optimal operation of the power grid and a production schedule that meets the chemical product demand and abides by all safety constraints of the chemical plant. To this end, we discuss incorporating the models of the dynamics of the end-user chemical plant and the optimal power flow problem, with the goal of creating a unified optimization framework.

• We present an extensive case study concerning coordinated demand-response operation for a relevant electricity-intensive chemical process, chlor-alkali production, integrated with a canonical OPF model from the literature.

• We benchmark the results of applying our new concepts against the demand response program popularly studied in literature for chemical processes, i.e., independent price- based demand response (referred to as self-directed price response by the Texas grid operator [133]).

5.2 Background 5.2.1 Power system model

Power system modeling broadly addresses two classes of problems: (1) the long-term (daily or longer-term) schedule of power generation units for a given power demand, called the unit commitment problem and (2) the short-term (less than a day) reliable operation of the grid under transmission constraints and dynamically varying load, called the optimal

98 power flow (OPF) problem [134]. OPF is crucial for sustaining the robustness of the (constrained) power grid under changing conditions, such as fluctuations in renewable energy generation, variable end-user load, and unplanned generator outages. The associated power system model is often used in evaluating the impact of demand response programs on the power network at different levels. Some examples include demonstrating DR dispatch cost minimization under uncertain conditions [135], evaluating rebound effects of aggregate DR loads [136], studying system reliability under load curtailment [137]. The general form of the OPF can be compactly represented as:

minimize f(x,xˆ) xˆ subject to g(x, xˆ) = 0 (5.1)

h(x, xˆ) ≤ 0

The objective, f, of the power ﬂow problem may include minimizing transmission losses, the active power operating cost or the transition period from a violated to un-violated network state [134]. This cost is a function of the set of decision variables,x ˆ, such as the generated power, nodal power injections, bus voltages, transformer and shunt taps, and the dependent state variables, x, e.g., reactive power, line parameters. The cost function can be expressed as a quadratic or linear combination [138, pp. 69–72] of these variables. The equality constraints, g(x, xˆ), encompass the nodal active power balance (based on Kirchoﬀ’s laws, Eq. 5.2), load models (mostly steady state representations, see Section 5.2.2) or load data, battery state of charge Eq. (5.3) and transmission line, transformer, and shunt constraints Eq. (5.4). The power balance stipulates that the net power injected at every node is zero.

Pi, j = Pgen + (Pdischarge − Pcharge) + Pload shed − Pdemand (5.2)

99 where Pgen stands for the active power from all generator types, Pcharge and Pdischarge represent the charge and discharge rates of grid connected batteries, Pload shed is the demand curtailed

(under extreme grid circumstances), Pdemand represent all power consumption from the grid, and Pi, j is the active power ﬂow between buses. The battery state of charge, SOC, is given as

d(SOC)/dt = Pchargeηcharge − Pdischarge/ηdischarge (5.3) where η represents the cycle eﬃciency.

The OPF model grows in size as the number of generators, loads and buses in the grid topology increases. Additionally, the transmission system model further complicates the security-constrained OPF. However, tractable formulations can be derived by e.g., adopting DC, (rather than AC) power ﬂow model [138]. The DC formulation of the transmission line model is given as no.buses X Pi, j = Bi,j(θj − θi) (5.4) i=1 i6=j where θ is the voltage angle in radians and Bi,j is the circuit susceptance matrix in per unit. Finally, the operating limits are represented by the inequality constraint, h(x, xˆ). These limits typically incorporate constraints on the voltage angle, transmission line capacity, generator capacity and ramp rates as shown in the expression, 5.5.

min max P? ≤ P? ≤ P? (5.5a)

ramp up ramp down dPgen/dt ≤ Pgen ; dPgen/dt ≤ −Pgen (5.5b)

SOCmin ≤ SOC ≤ SOCmax (5.5c)

min max θi,j ≤ θi,j ≤ θi,j (5.5d)

100 In the above, the ? subscript denotes any and all subscripts (i,j, gen, charge, discharge and load shed) introduced earlier. The OPF can be implemented explicitly as-is (some algorithms are discussed in [139]) or reformulated as a conic [140], linear or quadratic program and solved using standard commercial solvers.

5.2.2 Transient load models for chemical processes

An important input of the OPF problem is the power demand of grid-connected loads. Explicit power consumption data or load models (often derived from historical data) are typically used to provide this information for power ﬂow computations. However, in the case of ﬂexible end-user demand (e.g., demand response operation), it is necessary to adequately represent the variable power consumption rate of the respective loads [141]. This is particularly relevant for industrial loads, since they typically represent high single-point power demands that can potentially considerably improve (or worsen) the state of the power grid at any point in time.

The models employed to describe the demand response behavior of end-users may be (1) static or dynamic and (2) linear or non-linear. The conventional practice for representing industrial loads, including chemical process loads, in grid-level computations is to model them as “generators” with capacity limits and ﬁxed ramp rates. Widely used approximations involve describing the ﬂexible operation of industrial loads solely based on price elasticity [142, 137], or employing historical data to model their price sensitivity [143]. While these approaches may be relevant for understanding trends [144], there exist several limitations. First, such models are static and ignore any dynamic limitations of an industrial load. As a result, power demand schedules computed using these models may be dynamically infeasible

101 – their implementation may lead to constraint violations (such as exceeding operating or safety limits on diﬀerent pieces of equipment) during transitions from one operating level to the next [16, 11, 4]. Additionally, uncertainty is a concern and the realization of the end-users’ power demand level may lead to increased instability of the power grid [145].

The obvious alternative is to develop ﬁrst principles models of the industrial loads of interest, and to incorporate them in OPF models and calculations. However, such models are complex and likely to increase the computational cost of the resulting problem to the point where solutions cannot be obtained in a practical amount of time. Moreover, it is almost certain that industrial entities are reluctant to share such detailed levels of information with the power grid operator (or any other third party).

Thus, in the context of embedding (more) detailed representations of industrial loads in the OPF problem, it is necessary to achieve a balance between model simplicity and the level of (dynamic) detail. Linear and mixed integer linear representations are tractable with desirable optimality properties. Consequently, in the present study we will focus on two classes of such models (1) linear state space (SS) models and (2) Hammerstein-Wiener (HW) models. These will be discussed below, and later on embedded in the OPF problem.

5.2.2.1 Linear state space models

The general form of a linear state space model is given in Eq. 5.6.

Z(˙t) = AZ(t) + BU(t) (5.6) Y (t) = CZ(t) where z represents the state variable, u, the system input, y, system output and uppercase symbols denote (in this case, constant) matrices or vectors. State space formulations of

102 dynamical systems ﬁnd widespread use in control systems and correspondingly, power system modeling [146].

5.2.2.2 Hammerstein-Wiener models

Hammerstein-Wiener (HW) models consist of a linear dynamic block, ﬂanked by two static nonlinear functions: one (Hammerstein) transforming physical system inputs before they are presented as inputs to the dynamic model and the other (Wiener) transforming outputs of the dynamic model to outputs of the physical system. Mathematically, this can be represented by the following set of equations: v(t) = Φ(u(t))

w(t) = Π(v(t)) (5.7)

y(t) = Ψ(w(t)) where Φ, Π and Ψ are the input nonlinear, dynamic linear and output nonlinear functions respectively. v, w are the transformed input and output variables and u, y are the physical input and output variables respectively. The dynamic model represented by Π can take any form including, but not limited to state space models and transfer function formulations. Similarly, the nonlinear transformations can be piecewise linear or polynomial functions.

HW models are versatile and have found useful applications in control theory and, recently, demand response studies. For example, Hammerstein-Wiener models developed for an air separation process were employed to demonstrate independent DR scheduling of the process [11, 147] and for emission reduction [148].

The two model types discussed above describe the dynamic evolution of the variables associated with the process. For a complete description of industrial chemical process loads,

103 bounds on variables, states and outputs should be incorporated into the load model prefer- ably, using linear inequalities. In the next section, two problem formulations that employ the transient load models discussed above in the OPF problem are introduced.

5.3 Problem deﬁnition

Given the power system model (Eq. (5.1)) and load model (of type Eq. (5.7) and/or Eq. (5.6)), two main problems are formulated:

• Problem 1 (P1) – monolithic formulation: determine the power flow schedule consisting of power generation levels, flow from batteries, and the “controllable” load – in this case, flexible industrial load – power demand schedule (along with other decision variables necessary to maintain feasibility) required to satisfy the power demand on the constrained network at minimum cost. In this formulation, the OPF is solved with an explicit model of the industrial load in an integrated fashion, and its solution includes a (time-varying) vector of decision variables at the level of the load (for a chemical process, this will include, e.g., material flow rates, operating temperatures and pressures). The electricity cost for the end-user is computed from the resulting nodal electricity prices.

• Problem 2 (P2)– sequential formulation: determine the power flow schedule by solving the OPF for a given base profile of the flexible industrial load. Then, independently compute the variable power consumption schedule (at the end-user level) of the industrial load based on resulting nodal electricity price from the OPF problem.

Both formulations are summarized in Fig. 5.1.

104 Monolithic formulation (P1) Sequential formulation (P2) Solve OPF Solve OPF Input: – end-user (industrial chem- Input: – end-user power demand ical process) load model data Output: – generation schedule Output: – generation schedule Utility level – ﬂexible schedule (power de- – nodal electricity price mand and end-user decision variables) ∗utility observes end-user load – nodal electricity price schedule

Minimize electricity cost Implement ﬂexible schedule Input: – nodal electricity price End-user level Output: – ﬂexible power demand schedule

Figure 5.1: Monolithic and sequential formulations addressing the optimal power ﬂow and demand response scheduling problem

The problem formulations above represent, in eﬀect, two limit cases in terms of information sharing and operational control. In P1, total control of the industrial load is ceded to the grid operator, which is assumed to be “omniscient” of the dynamics of the load and related constraints. This scenario mimics direct load control whereby the utility controls the demand of end-users as required for grid stability, with some diﬀerences: (1) the dynamics of the load is fully considered at the utility level (in order to guarantee the feasibility of operating schedules for the chemical process load) and, (2) rather than shutting down loads completely, continuous dynamic load scheduling of the process is carried out. Thus, the process load serves as an additional “generator” that is available for dispatch by the utility [4].

In P2, the end-user selﬁshly carries out demand response as a price-taker and the utility observes the resulting power demand schedule. In P2, the activity of the end-user is assumed to have no impact on the electricity price within the period of operation considered. This is price-based demand response and is typically proposed in literature for industrial

105 chemical process loads. In this case, any mismatch in the scheduled power generation and observed demand is compensated by the utility via ancillary services, fast ramping (and expensive) generators, and energy storage.

Concerted scheduling of power supply and demand in the manner described will require information sharing between the end-user and utility operators. However, as noted above, explicit dynamic plant models contain details such as plant efficiency, process agility and conversion rates which may be considered business sensitive. Thus, plant operators may be unwilling to share such plant models. To address this, bridge models that convey plant limitation in terms of dynamic agility while concealing business relevant information may be developed for use in power calculations. Such bridge models may for instance, sacrifice some level of accuracy (while still representing plant dynamics) for confidentiality. Study of the above two limit cases is still however crucial e.g. to provide insights on the benefits and limitations of such integrated DR scheduling approach.

5.4 Case study 5.4.1 Industrial process: chlor-alkali production

Chlor-alkali production involves the electrolysis of brine (concentrated sodium chloride solution) to generate chlorine and caustic soda (sodium hydroxide). The process is power-intensive with the electrolysis cells accounting for about 50% of the electricity consumption [56]. Chlorine is commonly sold and shipped in liquid form and additional power is needed to operate a liquefaction unit which typically comprises of pumps, compressors and condensers. Accounting for this, electricity makes up 50 - 70% of the operating cost for this industry [149] and there is therefore a strong incentive to minimize expenses.

106 Electrolyzer cells can be run at varying current density levels [42] or in diﬀerent operating regimes [150] in a DR context, which contributes to minimizing operating cost. Additionally, a production facility may shut down (some of the) electrolysis cells during peaks in power demand, thereby providing ancillary services that generate an additional revenue stream. In the present study, we consider the DR operation of the chlor-alkali process by continuous modulation of the cell current density level (as opposed to shutting down cells), and the relevant model of the facility is discussed below.

5.4.2 Transient load model for chlor-alkali process

The starting point here is the detailed model of a chlor-alkali plant derived in chapter 3. This model is highly nonlinear and relatively high dimensional, and the previous results indicate that it is computationally expensive. Motivated by this, HW and SS models of the form discussed in Section 5.2.2 were identified for this plant. In simplifying process models for scheduling calculations, it is important to consider the scheduling relevant variables and active constraints for dynamic feasibility [11, 147]. Thus, dynamic models are developed only for the plant power consumption and the cell temperature, as the latter limits the agility and demand response capabilities of the plant under variable operation [42]. The power consumption model (whether HW or SS) is denoted PM and predicts plant power use (output) based on the current density (input), while the temperature model (again, whether in SS or HW form) is denoted TM predicts the cell temperature (output) as a function of the current density and inlet temperature (inputs). Identification for the two models proceeded as follows: first, input-output data for a 72 hour period were obtained from a simulation experiment where the current density was changed in a stepwise fashion. Then, HW and

107 SS models were identiﬁed using the System Identiﬁcation Toolbox in MATLAB [151]. The structure of the resulting models is summarized in Table 5.1. Nonlinear input and output functions in the HW models were expressed using piecewise linear (PWL) functions.

Table 5.1: Structure of data driven SS and HW models for chlor-alkali plant variable number of seg- dynamic number of seg- NMSE ments in input model ments in output nonlinearity [order, ts] nonlinearity Nonlinear HW model PM none 5, 450 s 10 0.8674 TM [current density, [4, 3] 7, 45 s 5 0.9877 inlet temperature] Linear SS model PM n/a 1, 450 s n/a 0.9993 TM n/a 2, 45 s n/a 0.9938

In table 5.1, both the SS models and the dynamic block of the HW models are expressed in the linear state-space form of equation (5.6), in discrete time with sample time ts. The sample times for both PM and TM follow the guideline of at least ten samples per process time constant. TM is sampled more frequently for numerical stability of the optimization. The identiﬁed models ﬁt closely the validation data as represented by the normalized mean square error (NMSE), as illustrated in Fig. 5.2.

Finally, the liquefaction unit and inventory ﬂow through the system are described using the models previously developed [42]. There is no need to identify HW and SS models for these unit operations as the given models are rather simple.

108 8 full order model HW model 7 linear SS model

5 power, MW power,

0 2 4 6 8 10 12 14 16 18 20 22 time, h

C o 88

86 temperature, temperature,

84 0 2 4 6 8 10 12 14 16 18 20 22 time, h Figure 5.2: Reduced-order model versus validation data responses for the chlor-alkali plant

5.4.3 Power grid topology

We demonstrate the theoretical concepts discussed on the basis of a modified version of the IEEE reliability 24 bus network presented by Soroudi [3] (Fig. 5.3). The branch characteristics, load and generator capacities are consistent with the IEEE test system. In addition, the network includes two batteries of capacity 200 MW and 100 MW located at bus 19 and 21, respectively. Wind turbines of maximum rating 200 MW, 150 MW and 100 MW are stationed at bus 8, 19 and 21, respectively. Bus 13 is the designated slack bus. The chlor-alkali plant hub with a total capacity of 71 MW represents 2.5% of this grid load. We note that the power demand of an industrial chlor-alkali facility may be higher than this value (e.g., the Dow-Mitsui plant in Freeport, TX [152] has an anticipated power draw of about 170 MW. Generally, the fraction of system-wide load represented by the chemical plant will vary based on e.g., the plant location (industrial or residential zone). We provide a discussion on the effect of this flexible load size (as a fraction of the total power system

109 Figure 5.3: Modiﬁed IEEE 24 bus reliability test system. Small red and large green circles represent loads and wind turbines, blue crosses represent generators and white rectangles represent batteries [3] load) later. It should be noted that the aforementioned plant model [42] represents a chlor- alkali plant with a maximum demand of around 8 MW. We scale up this capacity linearly to represent the total designated plant load of 71 MW. This is appropriate as large chlor- alkali plants typically consist of several identical cell stacks (repeated modules) operating in parallel to meet the required chlorine production capacity.

5.4.4 Optimization implementation

We adopt the GAMS [153] multi-period DC OPF model (which represents the topology discussed previously) developed by Soroudi [3] with some modifications. We implement the OPF program for a 24 h period divided into 15 minute time slots. This is done to reflect the fifteen minute market run by ERCOT, the Texas grid operator. Additionally, we implement wind generation and power demand fluctuations for a typical fall day in Texas (Fig. 5.4). It

110 1 wind generation power demand 0.9

0.8

0.7

0.6

0.5 fraction (of maximum) (of fraction 0.4

0.3

0.2 0 2 4 6 8 10 12 14 16 18 20 22 time, h Figure 5.4: Wind generation and power demand pattern for Houston hub, 09/13/18 - 09/14/18 (scaled relative to the maximum values observed). Data source: www.ercot.com is assumed that all loads on the grid excluding the 71 MW plant load, follow this demand proﬁle. The objective function for the OPF problem is a linear combination of the generation cost and penalties for any load shed or wind spilled during the operating period.

Problems P1 and P2 described above were implemented in GAMS using the reformu- lation of the HW models proposed by Kelley et al. [147], which results in a mixed-integer linear program (MILP). The resulting MILPs could be solved to zero optimality gap eﬃ- ciently (ca. 20 minutes on a 64 bit Windows 7 PC, Intel Core i7, 2.60 GHz processor, 16.0 GB RAM).

In the sections that follow, we discuss the results from four cases (described in Table 5.2) under normal and congested grid conditions. Congestion is simulated by assuming that 24% of generation i.e., the largest generator and 56% of the second largest generator (G2) are unavailable.

111 Table 5.2: Scenarios considered in solving P1 and P2 Case label Description Sequential (P2) Nom base case. OPF is solved with constant industrial load PDR process independently computes DR schedule (using Hammerstein-Wiener process model) based on price signals from the grid and resulting schedule is communicated to OPF (“Process Demand Response - PDR”) Monolithic (P1): GPDR dynamics of industrial load represented in utility schedules the the OPF formulation using a nonlinear ﬂexible industrial HW process model (“Grid-optimized Pro- load cess Demand Response - GPDR”) linGPDR dynamics of industrial load represented in the OPF formulation using a linear SS process model (“Grid-optimized Process De- mand Response with linear process model - linGPDR”)

5.5 Discussion of results 5.5.1 Flexible operation of concentrated end-user power demand

Under normal grid situation, independent flexible operation of the industrial load leads to similar benefits as when grid operators are in control of scheduling the power demand of the plant. This is shown in the result summary provided in Table 5.3 (nominal values shown for the unscaled plant in this table and similar result tables). In this case, the power grid is not necessarily strained to its limits and no load is shed nor is any transmission line congested. Thus, the benefits from demand response are completely characterized by the reduction in the power flow cost. More power from cheaper generation units can be utilized to satisfy increased power demand from the industrial load during off-peak hours (here, the second half of the day), reducing the aggregate demand needed to be satisfied earlier in the day. This is depicted in the optimal production schedules shown in Fig. 5.5.

112 Table 5.3: Results for plant load concentrated on bus 7 (data indicate percent change from nominal value) Grid-related Chemical plant-related Scenario Congestion Load Optimal Energy Electricity cost, $ shed, ﬂow cost, utiliza- cost, $ MWh $ tion, MWh Normal Nom - - 1, 813, 696 117, 298 1, 260 PDR - - −0.4% 1.5% −13.2% GPDR - - −0.5% 1.4% −12.0% linGPDR - - −0.4% 1.0% −11.4% Congestion Nom 7, 059, 305 29.4 2, 129, 493 117, 298 1, 831 PDR 0.0% 0% −0.6% 1.6% −16.1% GPDR −100% 0% −0.7% 1.5% −11.3% linGPDR −100% 0% −0.6% 1.0% −10.8%

PDR PDR 8 90 90 20 7

15 88 88 C o

6 C

10 o T, 5

86 86 T, inlet power, MW power,

5 $/MWh price, 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24

GPDR GPDR 8 90 90 20 7

15 88 88 C o

6 C

10 o T,

5 86 86 T, inlet power, MW power,

5 $/MWh price, 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24

linGPDR linGPDR 8 90 90 20 7

15 88 88 C o

6 C

10 o T,

5 86 86 T, inlet power, MW power,

5 $/MWh price, 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 time, h time, h Figure 5.5: Power demand proﬁle and plant output variables for a single chlor-alkali plant under normal grid conditions, plant load on bus 7

113 For the monolithic formulations P1 (scenarios GPDR and linGPDR), the industrial load is additionally kept as low as feasible for an extended period (hour 10.5 - 12) with some of the makeup deferred to hour 19. From the end-user perspective, the best savings on electricity cost results from the sequential problem in this scenario. With the monolithic formulation, the benefit of flexible operation of the industrial plant is distributed amongst all end users in the form of a “flatter” price structure. For example, the price “spike” in hour 6 for PDR is absent in GPDR and linGPDR (Fig. 5.5, left three plots). Similarly, the high price period between hours 9 - 10 under nominal condition is shortened with the monolithic approach. This results in a slightly lower average electricity price over the operating period for the monolithic problem (average price is $10.74 under nominal & sequential operation and $10.72 for the monolithic formulation). Accordingly, the question regarding the incentive, if any, for the end-user to engage in such collaborative effort arises. We address this matter in the next section.

The load deferment to later hours with the monolithic formulation is more obvious under system congestion (Fig. 5.6). In this case, 71 MW of flexible load available at bus 7 (which is also the node with the congested line) does not have an impact in terms of congestion management with independent DR. However, with the monolithic approach, congestion cost is completely eliminated along with load shedding. It is worth mentioning that the savings to the industrial end-user in this case is still high (comparable to the values achieved under normal grid situation). The constraints that impact (and limit) plant dynamic agility are also shown (Figs. 5.5 and 5.6). Product inventory is depleted and refilled with the final constraint being satisfied in all cases (Fig. 5.7). Similarly, on the grid side, energy storage is utilized to optimize operations.

114 PDR PDR 8 40 90 90

7 30

88 88 C o

6 C

20 o T,

5 86 86 T, inlet power, MW power,

10 $/MWh price, 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24

GPDR GPDR 8 40 90 90

7 30

88 88 C o

6 C

20 o T,

5 86 86 T, inlet power, MW power,

10 $/MWh price, 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24

linGPDR linGPDR 8 40 90 90

7 30

88 88 C o

6 C

20 o T,

5 86 86 T, inlet power, MW power,

10 $/MWh price, 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 time, h time, h

Figure 5.6: Power demand proﬁle and plant output variables for a single chlor-alkali plant under grid congestion, plant load on bus 7

5.5.2 Eﬀect of ﬂexible load location

In the previous section, the plant was located on the “cheapest” node with favorable transmission properties and high renewable supply. We now consider the case where the load representing the chemical plant is located on a load-only bus, bus 5. The results for P1 and P2 under normal and congested grid conditions are summarized in Table 5.4. The beneﬁt of an integrated approach to industrial end-user management becomes apparent during congestion. The operating cost savings that result from the monolithic formulation are over twice as high as the savings achieved through independent DR. The industrial user is now directly aﬀected by the high electricity prices resulting from grid congestion (Fig.

115 bus 19 1 0.8 PDR PDR GPDR 0.8 GPDR linGPDR 0.7 linGPDR 0.6

SOC 0.4 0.6 0.2 0.5 0 2 4 6 8 10 12 14 16 18 20 22 24 time, h 0.4 bus 21 1

0.3 stored product fr product stored

0.2 0.5 SOC

0.1

0 0 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 time, h time, h

Figure 5.7: Sample product inventory for single chlor-alkali plant (left) and state of charge of grid connected batteries (right) under normal grid conditions

Table 5.4: Results for plant load concentrated on bus 5 Grid-related Chemical plant-related Scenario Congestion Load Optimal Energy Electricity cost, $ shed, ﬂow cost, utiliza- cost, $ MWh $ tion, MWh Normal Nom - - 1, 813, 696 117, 298 1, 260 PDR - - −0.4% 1.5% −13.2% GPDR - - −0.5% 1.3% −12.1% linGPDR - - −0.4% 1.0% −11.5% Congestion Nom 8, 765, 245 30.5 2, 141, 216 117, 298 3, 453 PDR −99.6% −100% −15.1% 1.6% −20.8% GPDR −100% −100% −15.1% 1.5% −46.7% linGPDR −100% −100% −15.1% 1.0% −46.7%

5.8).

Additionally, as the transmission lines directly connected to this node are not at full capacity, the ﬂexible operation of the chemical plant not only curtails congestion, but averts the load shedding (nominally occurring at bus 13) that would be otherwise necessary to

116 PDR PDR 8 300 90 90

200 88 88 C o

6 C

o T,

5 100 86 86

inlet T, inlet

power, MW power, price, $/MWh price, 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 GPDR GPDR 8 60 90 90

40 88 88 C o

6 C

o T,

5 20 86 86

inlet T, inlet

power, MW power, price $/MWh price 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 linGPDR linGPDR 8 60 90 90

40 88 88 C o

6 C

o T,

5 20 86 86

inlet T, inlet

power, MW power, price $/MWh price 4 0 84 84 0 2 4 6 8 10 12 14 16 18 20 22 24 0 2 4 6 8 10 12 14 16 18 20 22 24 time, h time, h Figure 5.8: Power demand proﬁle and plant output variables under grid congestion, plant load on bus 5 maintain system reliability. This is true even in the PDR case where P2 (sequential) is solved, and the end-user is a price-taker. This level of congestion management was unattainable with either formulation P1 or P2 in the previous scenario where the plant was located on the congested bus (bus 7), mainly due to the physical constraints on the relevant transmission line.

In an alternate scenario, we consider the case where the ﬂexible chemical plant load is distributed across the power network. Note that this is in principle possible due to the modular nature of the electrolysis process (although the total capital investment would be higher than in the case of a single concentrated facility of the same total capacity). For instance, let the total 71 MW plant load be distributed on buses 5, 7 and 8, representing 0.46, 0.25 and 0.04 parts of the total loads on the given buses respectively. In this case, we

117 compare the average cost savings across all buses. Additionally, note that the nominal cases are diﬀerent in the concentrated and distributed load scenarios. We report savings attainable relative to the relevant base case operating conditions. The results for the distributed load study under grid congestion indicate high percent savings in costs as summarized in Table 5.5. These savings fall between the values achieved for the concentrated plant load on a

Table 5.5: Results for distributed plant load (data indicate percent change from nominal value) Grid-related Chemical plant-related Scenario Congestion Load Optimal Energy Electricity cost, $ shed, flow cost, utiliza- cost, $ MWh $ tion, MWh Congestion Nom 7, 059, 969 30.0 2, 134, 644 117, 298 2, 912 PDR −99.2% −100% −14.7% 1.6% −19.8% GPDR −100% −100% −14.8% 1.5% −36.3% linGPDR −100% −100% −14.7% 1.0% −35.6% highly effective, bus 5-type bus and a less effective, bus 7-type bus. In a different scenario with the plant load distributed on buses 5, 6, and 10, up to 49.8% cost reduction was achieved (plots not shown in the interest of brevity). While we cannot claim a general conclusion in that distributed plant loads will lead to high generation cost savings, it is intuitive that a distributed load configuration improves congestion management. More broadly speaking, this indicates that engaging multiple (smaller) electricity intensive end-users may be more effective than relying on a single large load for grid management purposes.

Finally, while the cost savings to the end-user and the grid varies based on the designated plant node, it is important to point out that, in all cases considered, the monolithic formulation consistently leads to the lowest power generation cost (or in the worst cases, equal savings as the sequential approach) (Fig. 5.9). This trend is also observed with the

118 plant electricity cost except for the case where all the plant load is on the cheapest node. In this case, independent DR leads to higher beneﬁt for the end-user.

5.5.3 Increasing the contribution of chemical plant to overall load

The trend discussed previously, whereby the monolithic model leads to lower cost of generation (and thus a lower electricity cost to the end-user) is also observed when the demand of the chemical plant represents a higher proportion of the network load. Figure 5.9 shows the increase in costs for the sequential approach (P2) compared to the corresponding results obtained from the monolithic P1 for simultaneously increasing plant load on buses 9, 10 and 14 evenly, demonstrating that the benefits of P1 increase (vs. P2) as the flexible end- user makes up an increasing proportion of the grid load. A similar congestion cost reduction was observed in all cases for both the sequential and monolithic formulations – over 99% congestion cost reduction and 100% curtailment of load shedding was achieved. However, as noted, congestion management is dependent on the plant location and a unified management of flexible load and the power utility system improves the opportunity for appropriate load and generation scheduling required for a reliable network.

5.5.4 System-wide power demand

The overall power demand on the (congested) network with a concentrated flexible load is shown in Fig. 5.10. We focus the discussion in this section on the case of grid congestion since the overall system load profile does not differ significantly for P1 and P2 under normal grid conditions. The results show that modulation of the industrial demand generally leads to a load flattening effect induced by “peak reduction” and “valley filling.”

119 9,10,14 120

100

12.5 80

10 60 7.5 2.5 5

40 % increase increase % inplant cost 20

0 0 0.2 0.4 0.6 0.8 1 1.2 % increase in generation cost

Figure 5.9: Costs for sequential formulation P2 compared to corresponding values for monolithic P1. Bubble size denotes total chemical plant capacity as a percent of the total system load

Approximately 1% and 2% of the maximum peak load and valley are clipped and filled respectively. This positive impact is observed for both P1 and P2. However, the monolithic P1 results in smoother load shifting events. Under price-based demand response of the end- user (sequential P2), power modulations are less smooth see, e.g., hours 2 - 3, 11 and 19 while a new peak may emerge at hour 11. This is an important observation, as the monolithic approach shows no indication of rebound peaks. Thus, from grid reliability perspective, scheduling supply and flexible loads in an integrated approach is beneficial.

120 demand pattern 3000 nominal demand PDR demand 2800 GPDR demand linGPDR demand

2600

2400 load, MW load,

2200

2000

1800 0 2 4 6 8 10 12 14 16 18 20 22 time, h Figure 5.10: Overall grid load under congestion, concentrated load on bus 7

121 5.6 Conclusions

Academic and industrial studies indicate that electricity-intensive chemical plants can potentially provide various power grid support services. One of such service is industrial demand response. In this paper, we showed that coordinating the flexible operation of the process load along with generation capacity at the utility level presents distinct benefits over demand-response operation of the process in response to time-varying price signals. A unified approach for integrating suitable models of chemical processes in power flow calculations was discussed. Extensive case study results for a chlor-alkali plant demonstrated superior benefits of the integrated/monolithic management approach. In addition to supporting grid operations, the industrial end-user can achieve high cost savings, especially during periods of power flow congestion. Although the location of the chemical plant on the power network affects both the end-user benefit and available leverage for congestion management, a unified approach rather than independent operation often leads to greater overall savings. The integrated approach presented hinges on the assumption that the grid operator possesses an accurate model/representation of the dynamics of the flexible load as reflected in the two models developed (and thus, similar results obtained for the monolithic studies). This may not be the case in practice, where chemical companies may be reluctant to share such information with third parties. Thus, to support the implementation of this unified management framework, future research must address methods and models for information sharing and protection for all parties involved.

122 Chapter 6

Conclusions and Recommendations for Future Work

This dissertation presents an approach that considers process dynamics in the scheduling of power-intensive chemical processes for demand response. Our discourse comprised of comprehensive model development (chapter 3) and applications in diverse demand response paradigms. A method for evaluating power consumption curtailment and trading this capacity profitably in multiple electricity markets was developed (chapter 2). Furthermore, the agility of a chemical plant in providing critical grid support service, namely, frequency regulation was evaluated (chapter 4). Combining the interests of both the end-user and utility (electricity) industry, we proposed a novel unified approach for managing both flexible demand and supply of electricity (chapter 5). The proposed integrated approach incorpo- rates a dynamic load model and quantitative studies revealed the benefits of such unified approach to power demand management. Our work contributes to the effort to realistically include chemical processes in demand response activities aimed at improving the operation of the power grid. Multiple aspects of this dissertation can be explored further.

DR scheduling framework. The approach to bidding load reduction capacity in electricity markets, that fully considers the chemical process dynamics directly lends itself to other DR services that rely on self-scheduling by the participants e.g., load shedding, regulation reserves. While the framework discussed in chapter 2 focused on optimizing

123 the supplementary earnings from a single demand response program, regulation up (i.e., benefiting by decreasing load during the operating period), modifications can be made to allow for participation in multiple DR programs simultaneously. For instance, alternative market structures may require enrollment in both regulation up and down (dual qualification) or participants may voluntarily elect for similar multi qualifications. Accordingly, the DR framework should reflect the relevant market restrictions e.g., reserving at least minimum capacity for each program, submitting multiple bids ordered by priority levels while satisfying all constraints of the chemical process. One challenge to be addressed includes developing tractable models that preserve sufficient details about the chemical process and also reflect complicating market rules. Constraints may be modeled using disjunctive programming. However, such programs can quickly become intractable and suitable reformulations may be necessary. The alternate class of demand response programs whereby the end-user power consumption is directly controlled following utility generated triggers may be fashioned after the modified approach to frequency regulation discussed in chapter 4 – using proxies to grant some control to the end-user while achieving utility relevant objective. Some of such DR programs include load following, direct load control and frequency response. Studies may be carried out to incorporate feedback mechanisms for the different programs. Thus, the performance of this modified approach in terms of achieving the desired effect on the grid side e.g., impact on grid frequency can be evaluated. Finally, it is paramount that the DR frameworks, whether under a self-scheduling scheme or otherwise preserve the dynamics of the chemical process being studied.

Real-time implementation. Another area that may be explored in future research is the real-time (fast) demand response operation of chemical processes. For such

124 applications, the trade-off between computation efficiency and model complexity becomes even more critical. Dynamic real time optimization and rolling horizon scheduling techniques may integrate low order models that closely approximate the nonlinear process response (e.g., those developed in chapter 5) in their formulations for improved solution times. Additionally, the challenge of designing stable algorithms that reflect market changes in real-time increases as the rate of change of the market conditions/variable forcing function increases e.g., in DR programs like frequency response, primary ancillary reserves. Advanced optimization archi- tectures should represent the multiple time scales (plant dynamics, tiered electricity market structures) implied by the problem statement. Forecast accuracy of uncertain parameters e.g., electricity price, electric grid frequency and other dispatch signals may also contribute significantly to the performance of such frameworks.

Cooperative demand response. Beyond independent demand response by industrial electricity end-users, we have discussed some benefits of a unified approach to industrial DR. More variability on the supply end of electricity requires more flexible electricity demand (or greater energy storage). Large scale flexible demand will be most beneficial if such resources can be synchronized with the variable supply of electricity in real-time. A unified management system shows promising potential for achieving this needed synchrony. However, studies resulting in cooperative demand response strategies that ensure information confidentiality and fair reward to all involved parties are necessary. Some argument can be made for sharing partial information or less accurate plant models with the utility operator in favor of preserving business sensitive details. However, achieving such discreetness using models that still closely represent the chemical process behavior under highly transient operation remains an unanswered question. To the second issue raised, defining fair pricing

125 schemes for stakeholders under cooperation is non trivial. From an economic perspective, studies that assess and quantify the impact of a highly dynamic operation of chemical process e.g., maintenance and replacement costs, capital investment for any required retrofits will be necessary to fully evaluate the economic significance of the non-steady state operation of industrial chemical processes. Such studies can help unveil the fair share of benefits to be paid to the end-user.

Other application areas. Finally, even though demand-response has been the focus of this research, the modeling approach and scheduling frameworks can be extended to other application areas where dynamic plant operation may be capitalized on for benefits. Chemical processes that have direct or indirect (upstream or downstream) interactions with commodity-type markets, which are typically volatile may benefit from similar scheduling frameworks. Some example applications include the distribution and production scheduling problem of oil and gas networks and the water-energy nexus. Slight scheduling improvements in such tightly integrated and energy-intensive systems can lead to disproportionately large financial rewards and/or waste minimization.

126 Dissemination of Research Journal publications

1. Otashu, J. I. & Baldea, M. A study on the complete integration of chemical processes in grid-level operations. In preparation

2. Otashu, J. I. & Baldea, M. (2020). Scheduling chemical processes for frequency regulation. Applied Energy, 260, 114125 [95]

3. Otashu, J. I. & Baldea, M. (2019). Demand response-oriented dynamic modeling and operational optimization of membrane-based chlor-alkali plants. Computers & Chemical Engineering, 121, 396-408 [42]

4. Otashu, J. I. & Baldea, M. (2018). Grid-level battery operation of chemical processes and demand-side participation in short-term electricity markets. Applied energy, 220, 562-575 [4]

Conference publications

Otashu, J. I. & Baldea, M. (2018, June). A Two-tiered shrinking horizon framework for participation of chemical processes in short-term electricity markets. In 2018 Annual American Control Conference (ACC) (pp. 5902-5907) IEEE [19]

Conference proceedings

AIChE – American Institute of Chemical Engineers ∗ indicates the presenting author

127 1. Otashu, J.I.∗ & Baldea, M. Electrochemical processes in fast demand response. In- augural DuPont Gold seminar series, National Organization for the Professional Ad- vancement of Black Chemists and Chemical Engineers Annual Meeting, St. Louis, MO, Nov. 2019

2. Otashu, J.I.∗ & Baldea, M. Assessment of frequency regulation capacity of chemical process industries. AIChE Annual Meeting, Orlando, FL, Nov. 2019

3. Otashu, J.I. & Baldea, M.∗ Demand response-oriented modeling and production scheduling optimization for chlor-aklali processes. Computing and Systems Technology Divi- sion Plenary, AIChE Annual Meeting, Pittsburgh, PA, Nov. 2018

4. Otashu, J.I.∗ & Baldea, M. Grid-level “battery operation of chemical processes with engagement in short-term electricity markets. AIChE Annual Meeting, Minneapolis, MN, Nov. 2017

5. Otashu, J.I.∗ & Baldea, M. A Two-tiered shrinking horizon framework for participation of chemical processes in short-term electricity markets. American Control Conference, Milwaukee, WI, June 2017

6. Otashu, J.I.∗, Kumar, A., Baldea, M. Synergistic demand response between grid and process industry. AIChE Annual Meeting, San Francisco, CA, Nov. 2016

128 Appendices

129 Appendix A

Nomenclature and data – chapter 2

Nomenclature Table A.1: Nomenclature and abbreviations

Symbol Description x Vector of state variables u Vector of input variables y Vector of output variables τ Process time constant h P Instantaneous process power consumption kW S Product accumulation kg D Instantaneous product demand kg Plant conversion factor from energy consumed to product generated π Electricity price $ per kWh Tm Length of scheduling horizon in hours n Number of time slots ts Period for which set point targets are ﬁxed in minutes (length of time slots) penalty Penalty for storing material $/kg Z Binary variable A Positive artiﬁcial variable e¯ Small arbitrary positive number M Large arbitrary positive number TGC Time for gate closure (minutes) λ Short-term market revenue threshold value for a single time slot Subscripts k Grid points on hour time grid Continued on next page

130 Table A.1 – Continued from previous page Symbol Description i Time slots corresponding to changes of production target j Grid points on minute time grid κ, ι A designated time point in the scheduling horizon (corresponding to hour, k, & time slot, i, respectively) min Lower bound max Upper bound draw rate Limit on draw rate of process variable Superscripts sp Set point target in Flow into the system out Flow out of the system STM, Short-term market and Long-term market LTM Abbreviations DR Demand response LTM Long-term electricity markets STM Short-term electricity markets ERCOT Electricity reliability council of Texas DAM Day-ahead electricity market SBM Scale bridging model FMM Fifteen minute electricity market PJM The Pennsylvania, Jersey, Maryland Power Pool SD Same day DB Day before

Electricity price data

131 Jan Feb Mar Apr 800 800 300 1000

250 600 600 800

200 400 400 600 150 200 200 400 100

Electricity price cents/kWh 0 0 200 50

-200 -200 0 0 0 24 48 72 96 120 144 168 192 216 240 0 24 48 72 96 120 144 168 192 216 240 0 24 48 72 96 120 144 168 192 216 240 0 24 48 72 96 120 144 168 192 216 240 time, hour time, hour time, hour time, hour May Jun Jul 1500 500 500 LTM - day ahead market price STM - fifteen minute market price 400 400 1000

300 300 500

200 200

Electricity price cents/kWh 100 100

-500 0 0 0 24 48 72 96 120 144 168 192 216 240 0 24 48 72 96 120 144 168 192 216 240 0 24 48 72 96 120 144 168 192 216 240 time, hour time, hour time, hour

Figure A.1: Electricity prices for day ahead market and ﬁfteen minute market for ten consecutive days (2nd week and previous weekend) in the months of January - July, 2017. Data source: www.ercot.com

40 LTM - day ahead market price STM - fifteen minute market price 35

Electricity price cents/kWh 10

-5 2 4 6 8 10 12 14 16 18 20 22 24 time, hour

Figure A.2: Day ahead market and ﬁfteen minute market electricity prices for sample day, 01/09/17. Data source: www.ercot.com

132 Appendix B

Nomenclature and data – chapter 3

Nomenclature and parameter values for model equations

Table B.1: List of symbols for electrolyzer model

Symbol Description Value ∆Hrxn Enthalpy of reaction, kJ/mole of Cl2 447 [89] ∆Hvap Heat of vaporization of water, kJ/mole 41 [91] 2 Ac Membrane eﬀective area area, m 2.7 [2] 2 Ae Electrode eﬀective area, m 2.7 [56] j Ci Molar concentration of species i, in electrolyzer compartment j, mol/m3 Cpw Heat capacity of water, J/molK 75.3 Cpt Lumped heat capacity of electrolyte, J/moleK

DH2O Diffusivity of water molecules across membrane, 2e − 10 [80] m2s−1 DNa Diffusivity of sodium ions across membrane, 4e − 9 [80] m2s−1 F Faraday constant, Cmol−1orAsmol−1 96485 I Current, A n˙ Molar flow rate, mol/s N˙ Total molar flow rate for entire cell stack, mol/s n Number of moles, mol p Partial pressure, bar Q˙ Heat flow rate, J/s or W o −1 Rt Thermal resistance for stack of cells, CW 0.167 [60] T Temperature, K U Voltage or electromotive force, V Urev Reversible cell voltage, Volts 2.255 [88] Continued on next page

133 Table B.1 – Continued from previous page Symbol Description Value V˙ Volumetric ﬂow rate, m3/s V Cell volume, m3 0.2 [56] Greek symbols α Multiplication factor 0.5 [80] δ Membrane thickness, m 2.5 × 10−4 [80] η Eﬃciency, % Superscripts ano Anode cat Cathode evp Evaporation in Flow into the system out Flow out of the system gen Generation loss Loss to surroundings rxn Reaction Subscripts cell Unit cell in electrolyzer stack Cl Pure chloride ions Cl2 Pure chlorine gas cw Cooling water E Energy F Faraday H2 Pure hydrogen gas H2O Pure water molecules Na Pure sodium ions rev Reversible t Total vap Vaporization

134 Table B.2: List of symbols for compressor model Symbol Description Value k Adiabatic exponent 1.4 [90] n Polytropic exponent 1.45 [56] P Pressure, MP a R Universal gas constant, J/molK 8.314 T Temperature, K W Work done, J/mol Greek symbols ηpc Polytropic eﬃciency, % Superscripts th Theoretical Subscripts comp Compressor f Final i Initial

Table B.3: List of symbols for storage tank model Symbol Description Value M˙ Molar flow rate, mol/s Ms Liquid chlorine hold up, mol Dliq Demand for liquid chlorine product, mol/s Dgas Demand for gaseous chlorine product, mol/s N˙ ano,out Chlorine gas production rate, mol/s Cl2 Subscripts in Inlet flows out Outlet flows t Time in seconds

Calculation of lumped heat capacity

The lumped thermal heat capacity is computed as a sum of the contributions of the individual components to the mixture.

X Ct = xiCp,i (B.1) i

135 where xi is the mole fraction of component i, and Cp,i, the corresponding heat capacity. The heat capacity values are given in Table B.4.

Table B.4: Heat capacity at 298K [1] Compound Heat capacity (J/mol K) NaOH 60 NaCl 50 H2O 75

Input variables for scheduling problem formulation

Data for the scheduling formulation are presented in Table B.5

Electricity price data

136 Table B.5: List of model input variables Variable Value Electrolyte inlet flow, V˙ 8.33e − 5 m3/s ˙ 3 Cooling water flow rate, Vcw 1.8e − 4 m /s Inlet temperature of electrolyte, T in 85oC in o Inlet temperature of cooling water, Tcw 18 C o Ambient temperature, Tamb 25 C ano,in Inlet anolyte concentration, CNaCl 300 g/l cat,in Inlet catholyte concentration, CNaOH 31% wtfr Demand for gaseous chlorine, Dgas 0.926 mol/s Demand for liquid chlorine, Dliq 7.96 mol/s Cell specification for the Uhde membrane cell technology [56] Number of cells, Nc 160 2 Operating current density, I/Ae 0.3 − 0.6 A/cm Anolyte & catholyte volume per unit cell, V 100 L Operating pressure, P (constant for anode 1.213 bar and cathode compartments) Feasibility constraints Cell temperature 85 − 90oC Flow rate into and out of storage 0 − 4.5 mol/s Maximum storage capacity 13.435 ton Cell outlet conditions for Figure 3.5 [56] Exit temperature of electrolytes 87oC Catholyte exit concentration 32%wtfr Anolyte exit concentration 210g/l

137 90

30 Electricity price, cent/kWh price, Electricity

10 0 4 8 12 16 20 24 28 32 36 40 44 48 52 56 60 64 68 72 Time, hour

Figure B.1: Electricity price for the ﬁfteen minute electricity market, January 06-08, 2017. Data source: www.ercot.com

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158 Vita

Joannah Igbe Otashu was born in Kaduna, Nigeria, the daughter of Abel Otashu and Esther Aboje Otashu. She received the Bachelor of Engineering degree in Chemical Engineering from Ahmadu Bello University, Nigeria. She started her graduate studies in Chemical Engineering at the University of Texas at Austin in the fall of 2015 and was funded by the Schlumberger Faculty for the Future grant for four years. While at UT, she earned a coursework M.S. degree in Chemical Engineering in 2019 and completed a graduate internship at ExxonMobil in the Upstream Research Company in the summer of 2018.

Permanent address: [email protected]

This dissertation was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a special version of Donald Knuth’s TEX Program.

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