Developing a Stochastic Optimization Model for Operating the Manitoba Hydro Multi-Reservoir Hydroelectric Power System

Developing a stochastic optimization model for operating the Manitoba Hydro multi-reservoir hydroelectric power system

By Jacob Snell

A Thesis submitted to the Faculty of Graduate Studies of The University of Manitoba in partial fulfillment of the requirements for the degree of

MASTER OF SCIENCE

Department of Civil Engineering University of Manitoba Winnipeg, Manitoba

The province of Manitoba generates more than 90% of its electric power from hydroelectric generating stations located in the Nelson-Churchill Rivers basins. Prudent management of the major reservoirs in the system is essential for providing value through economic and reliable electricity. Reservoir managers are challenged by the variability of reservoir inflows, the misalignment of electrical energy demands and seasonality of reservoir inflows, and travel time lags between reservoirs and major generating stations on the Lower Nelson River. This thesis examines some of the challenges of current hydroelectric system management and applies a Sampling Stochastic Dynamic Programming algorithm to the operation of the Manitoba Hydro electric system. Direct consideration of variability and uncertainty of inflows are incorporated in the algorithm by generating a water value function based policy that considers multiple inflow scenarios and inflow scenario transition probabilities derived from conditional probability distributions based on a regression relationship between sequential periods of system inflow. The travel time lag is incorporated into the algorithm directly as a lagged inflow state variable to bridge reservoir release decisions between time periods. Storage values and penalties are incorporated to reflect operating license requirements and to prevent depletion of reserve storage that can lead to infeasibilities in the algorithm. The water value function policy is simulated over 38 historical years of inflow scenarios and compared against historical reservoir operation decisions. Model results show significant improvement in economic values and reductions in energy deficits over historical scenarios but are highly sensitive to the calibration of storage benefit and penalty values. Extensions to the model to use hydrological based inflow models, improving the transition matrix by evaluating alternative hydrological variables, and alternative approaches to the storage benefit and penalty method are discussed.

Acknowledgements

I would like to acknowledge the support, advice, guidance and encouragement from my advisor, Dr. Masoud Asadzadeh, to continue with my masters studies and thesis, and my advisory committee for much of the same. Furthermore, I would like to acknowledge my colleagues at Manitoba Hydro, particularly Mr. Kevin Gawne for his avid support and encouragement in perusing a higher education and development as a professional engineer. Lastly, I would like to thank my wife, Kaitlin, for believing in me and encouraging me along the way.

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Table of Contents Abstract ...... ii

Acknowledgements ...... iii

Table of Contents ...... iv

List of Tables ...... vi

List of Figures...... vi

SI Unit Conversions ...... x

1. Overview ...... 1

2. Study Area ...... 3

2.1. Manitoba Hydro Overview ...... 3

2.2. Manitoba Hydro Electric System Description ...... 3

2.2.1. Electric System Economic Dispatch ...... 6

2.3. Manitoba Hydro Reservoir System ...... 7

3. Manitoba Hydro Hydroelectric Reservoir System Operation...... 13

3.1. Reservoir System Operation Objectives...... 13

3.2. Inflow Forecasting ...... 14

3.3. EMMA Model ...... 15

3.4. Ideas for Improving Manitoba Hydro Midterm Operations ...... 16

4. Review of Multi-Reservoir Operation Optimization Approaches ...... 18

4.1. Linear Programming Approaches ...... 20

4.1.1. Stochastic Linear Programming ...... 21

4.1.2. Stochastic Linear Programming Applications ...... 23

4.2. Dynamic Programming ...... 25

5. Methodology ...... 30 iv

5.1. Overview ...... 30

5.2. Dynamic Programming Overview ...... 30

5.3. Sampling Stochastic Dynamic Problem Description ...... 34

5.3.1. Considerations for Flow Travel Time in the Optimization Model ...... 35

5.3.2. State Variable Discretization ...... 36

5.3.3. Subproblem Description ...... 37

5.3.4. Water Value Function Approximation ...... 45

5.3.5. Inflow Scenario Selection ...... 47

5.3.6. Transition Probability Calculation ...... 49

5.4. Analysis Setup ...... 50

6. Results and Discussion ...... 55

6.1. Historic Results ...... 55

6.2. “No-Lag” Approach Results ...... 56

6.3. ”With-Lag” Case Results ...... 64

6.4. Hydrological Results ...... 68

7. Conclusions and Recommendations for Future Work ...... 72

7.1. Summary of Findings ...... 72

7.2. Limitations ...... 73

7.3. Applications ...... 74

7.4. Recommendations for Future Work ...... 75

References ...... 77

Appendix A1: Estimation of the travel time delay in the Manitoba Hydro hydroelectric reservoir system ...... 87

Appendix A2: Hydrologic and energy system results from the simulation of historical releases 89 v

List of Tables

Table 1 - Manitoba Hydro generating station capacity (Manitoba Hydro 2019) ...... 4 Table 3 – Average annual inflows for the aggregated Long Term Flow Data locations ...... 10 Table 4 – Storage bounds and discretization limits ...... 37 Table 5 - Outflow limits on modeled lake outlets ...... 45 Table 6 - Energy deficits from simulated historic operating decisions by flow year ...... 56 Table 7 – “No-lag” approach with varying number of discretization points for the state variables and storage values. The Lake Winnipeg Max Outflow Penalty value was 1.5E+04 for all cases . 58 Table 8 – “No-lag” approach sensitivity to storage values for the 7/3/7 discretization ...... 59 Table 9 – Impact on the “No-lag” approach results due to the number of inflow sequences selected for optimization. The Lake Winnipeg Max Outflow Penalty value was 1.5E+04 in all cases...... 60 Table 10 -“With-lag” approach with varying number of discretization points for the state variables and storage values. The Lake Winnipeg Ma Outflow Penalty value was 1.5E+04 for all cases. .. 66 Table 11 – “With-lag” approach sensitivity to storage values ...... 67 Table 12 - Comparison of different time lags and the Nash Sutcliffe efficiency index for the time lag estimation of Stephens Lake inflows ...... 88

List of Figures Figure 1 - Manitoba electrical energy demand and peak demand by month (Manitoba Hydro 2018) ...... 4 Figure 2 – Map of Manitoba Hydro hydroelectric system facilities. © Manitoba Hydro (Manitoba Hydro 2020) Used with permission...... 5 Figure 3 - Nelson-Churchill Rivers Basin, © Manitoba Hydro (Manitoba Hydro 2020). Used with permission...... 8 Figure 4 - Manitoba Hydro hydroelectric power system and connection to neighbouring power market ...... 9

Figure 5 - Semi-monthly total system inflows for the LTFD locations from 1978-2015 and the average in a black line ...... 10 Figure 6 – Overview of the ISO process. Inflow cases are analyzed independently with deterministic optimization, creating a set of operating results or polices specific to the inflow scenario analyzed. Inferences are made on all sets of operating results through regression or other methods to derive operating rules, which can then be used to simulate operations of the reservoir system...... 18 Figure 7 - Overview of the two-stage problem solved with stochastic linear programming approaches. A two-stage scenario tree is developed with known inflows in the first stage and unknown inflows in the second stage. The problem is solved with a stochastic linear programming and the first stage operating decisions are implemented in the simulation. The problem is advanced to the next stage to repeat the process for the entire planning period...... 22 Figure 8 – Two stage scenario tree with a 3-2 branching structure. Stage 1 inflows are known with certainty whereas stage 2 have multiple possibilities...... 22 Figure 9 - Overview of the Stochastic Dynamic Programming process. Selected inflow cases and the probability to transition between them are determined. The selected inflow cases are included in the discretization of the other state variables. The problem is solved with the backwards dynamic programming procedure for all points in the state space and for all time periods, creating a set of water value functions. The water value functions can then be used to simulate operations of the reservoir system...... 27 Figure 10 - Description of a simple dynamic programming approach for a reservoir with two storage levels in time. The transition description section describes how the release decision would be calculated when transitioning between different states. The bolded equation at the top highlights the calculation for when transitioning from high storage in stage 0 to high storage in stage 1. The value description𝑥𝑥𝑥𝑥 describes how the benefit functions for transitioning to a future state across a stage and shows the future value functions at the end of the final stage . The bolded equation at the top highlights the calculation for determining the future value function𝑇𝑇 for time-period 1 and high storage, 1( )...... 32

𝑡𝑡 − 𝐹𝐹𝐹𝐹 − 𝑆𝑆ℎ vii

Figure 11 - Future water value function, as a function of storage. Solutions to the future water value function at the discretized state variable𝑓𝑓𝑓𝑓 points, such as the low Sl and high Sh level storge points, can be used to approximate a continuous future water value function. Additional state variable discretization points can be used to improve the approximation accuracy...... 33 Figure 12 – Manitoba electrical demand (a) and MISO Power Market prices (b) ...... 39 Figure 13 - Example of water value function cut selection, shown for a single reservoir. The 'true' water value function is shown in blue. Measurements of the water value are calculated at the discretized storage points, shown as dashed lines. Water value function cuts are the tangential lines. The maximum error of the current approximation is shown as the red double-sided arrow. The next water value function cut to be added to the approximation is at the discretization with the highest water value function error, which in this case, is at the red double-sided arrow. ... 47 Figure 14 - Historical inflow traces and average as a black line from 1978/79 to 2015/16 by location and total of all locations...... 48 Figure 15 – Scenario total system inflow volume, highlighting the quantile inflow selection method for 8 scenarios (a) and 4 scenarios(b) ...... 49 Figure 16 – Example of the linear regression approach for determining [ , | + 1, ]. (a)

shows the linear regression between and + 1. Here, four scenarios𝑃𝑃𝑃𝑃 ( 𝑞𝑞𝑞𝑞= 4𝑡𝑡) 𝑞𝑞are𝑞𝑞 included,𝑗𝑗 represented by the highlighted circles.𝑞𝑞 𝑞𝑞(b) shows𝑞𝑞𝑞𝑞 the normal distribution of𝐼𝐼 𝑜𝑜the residuals for the linear regression of the third scenario, , 3, with mean 1, + 1,3 + 0, . The distribution is

divided into segments at the scenario values𝑞𝑞𝑞𝑞 from (a)...... 𝜙𝜙 𝑡𝑡𝑡𝑡𝑡𝑡 𝜙𝜙...... 𝑡𝑡 50 Figure 17 – Overview of the optimization process and simulation process for the “no-lag”, “with- lag” and Historic cases ...... 52 Figure 18 - Conceptual UML Class Diagram describing the association between the MBHydroModel class through to the CLP solver. The CyCLP class is compiled C++ and python code and directly communicates with both the python based lpModel class and the clpSolver class that executes CLP...... 53 Figure 19 - Description of the optimization phase with parallel processing. Process 0 is the master process that completes the non-parallel tasks. The gather and broadcast are MPI calls that allow

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the multiple processes to collaborate by gathering the dual values from the individual processes and then broadcasting the water value function cuts for the next stage or iteration...... 54 Figure 20 - Lake elevation traces from the simulated historic operating decisions for (a) Lake Winnipeg, (b) Cedar Lake and (c) Southern Indian Lake and (d) average inflows for Stephens Lake. Traces are shown for the flow years with energy deficits and the average of all simulated flow years...... 56 Figure 21 – Impact of number of iterations on the economic objective for the ”no-lag” approach ...... 62 Figure 22 - Lake elevation traces from the 8 and 10 iteration results for (a) Lake Winnipeg, (b) Cedar Lake and (c) Southern Indian Lake and (d) average inflows for Stephens Lake. Traces are shown for the average of all flow years...... 62 Figure 23 - Lake elevation traces from the 8 and 10 iteration results for (a) Lake Winnipeg, (b) Cedar Lake and (c) Southern Indian Lake and (d) average inflows for Stephens Lake. Traces are from the 2003/04 inflow year...... 63 Figure 24 - Outflows and lagged inflows during the 2003/04 simulated inflow year for the 10- iteration case. Energy deficits occur in the timestep immediately prior to inflows at Stephens Lake increasing from lagged outflows...... 64 Figure 25 - Impact of the number of iterations on the “With-lag” results ...... 68 Figure 26 - Average results from the simulated history, "no-lag", and "with-lag approach from all simulated flow years for (a) Lake Winnipeg elevation, (b) Cedar Lake elevation (c) Southern Indian Lake elevation, and (d) Stephen's Lake network inflow...... 69 Figure 27 – Lake Winnipeg, Cedar Lake, and Sothern Indian Lake elevation traces the Stephens Lake network inflow results for the 2003/04 flow year through to the 2015/16 flow year for the simulated history, “no-lag” and “with-lag” results...... 71

Figure A 1 – Elevation and network inflow traces for historic inflow years 1978/79 to 1989/90 90 Figure A 2 – Elevation and network inflow traces for historic inflow years 1990/91 to 2002/03 91 Figure A 3 - Elevation and network inflow traces for historic inflow years 2003/04 – 2015/16 . 92 Figure A 4 – Energy supply volumes for the simulated history ...... 93

Figure A 5 – Energy demand for the simulated history ...... 94 Figure A 6 – Energy supply for the ‘no-lag’ approach...... 95 Figure A 7 – Energy demand for the ‘no-lag’ approach ...... 96 Figure A 8 – Energy supply for the ‘with-lag’ approach...... 97 Figure A 9 – Energy demand for the ‘with-lag’ approach ...... 98

SI Unit Conversions SI Name Symbol Quantity SI Unit Symbol Conversion Megawatt MW power Watts W 1 MW = 1000 W Gigawatt hour GWh energy Joules (J) J 1 GWh = 3.6e12 J volumetric cubic meters Thousand cubic feet per kcfs m3/s 1 kcfs = 28.3168 m3/s second flow rate per second cubic meters Thousand cubic feet per kcfs-day volume m3 1 kcfs-day = 2,446,576 m3 second in a day per second Feet ft length meters m 1 ft = 0.3048 m

1. Overview Manitoba Hydro is the electric utility provider for the Province of Manitoba that derives most of its electric power from hydropower sources on from the Nelson- Churchill Rivers basins. The timing of the utility electric demand and water supply over a year are out of phase, where most of the inflow arrives in the system in the spring and summer, while electric energy demand in Manitoba is the largest in the winter months. The uncertainty and variability of inflows present additional challenges to reservoir managers who must plan day-to-day reservoir releases, while considering the impacts on the availability of water in future weeks and months. Lastly, the largest storage resources in Manitoba are located far away from the major generation sources, requiring reservoir managers to consider the multiple week travel time between them.

Manitoba Hydro manages reservoirs in the Nelson-Churchill Rivers basins with the assistance of a Decision Support System (DSS) to assist in making efficient water resource decisions (Manitoba Hydro 2014). The DSS Manitoba Hydro currently employs is a deterministic optimization model based on an updated version originally created by Barritt-Flatt and Cormie (1988) with the objective of determining the optimal operation of the reservoir system over a variable planning period of six to 18 months and then configured on a weekly basis. However, the use of a deterministic model assumes a perfect forecast of inflows over the planning period, ignoring uncertainties in projected future inflow conditions. Drought risks are addressed by testing the operating decisions against a low inflow scenario. The method has been extended to evaluate operating decisions over a range of inflow scenarios to assist in determining optimal operating decisions.

Using stochastic optimization to directly consider the uncertainty of future inflows in the optimization in a probabilistic fashion can increase the quality of solutions by reducing the bias introduced from the perfect knowledge assumption (Labadie 2004). However, stochastic optimization increases the complexity of the multi-reservoir management problem and is computationally highly demanding (Philbrick and Kitanidis 1999) (Cote and Arsenault 2019, Labadie 2004).

The goal of this thesis is to consider uncertainty in inflow and the flow travel time in the operation of the Manitoba Hydro multi-reservoir hydroelectric system. To this end, a stochastic optimization model of the system is developed and analyzed against the historical operation. The content of the thesis is as follows:

- Chapter 2 describes the Study Area, providing an overview of Manitoba Hydro multi- reservoir hydroelectric system. - Chapter 3 reviews the Manitoba Hydro reservoir decision making process and optimization model and discusses limitations of the current approach. - Chapter 4 is a review of approaches to the optimization of multi-reservoir hydroelectric systems under uncertainty, with attention to linear programming and dynamic programming-based approaches. - Chapter 5 describes the method employed in this research. It includes a description of the stochastic model formulation, a sampling stochastic dynamic programming model, and its application to the Manitoba Hydro multi-reservoir hydroelectric system. - Chapter 6 provides results and discussion of the application of the model. - Chapter 7 is a concluding and summary chapter. Major findings are summarized, limitations of the approach are presented, potential applications are presented, and recommendations for future work are provided.

2. Study Area 2.1. Manitoba Hydro Overview The study focuses on the operation of the Manitoba Hydro hydroelectric reservoir system. Manitoba Hydro is an electric utility company that provides electric service to the province of Manitoba. As of 2019, Manitoba Hydro had 586,795 electric customers and 5604 MW of electric generation capacity (Manitoba Hydro 2019). Annual electrical energy demand in Manitoba was 25.7 TWh in 2018. Over 90% of the energy Manitoba Hydro generates comes from hydroelectric power stations located in the Nelson-Churchill river basins (Manitoba Hydro 2019). The amount of hydropower available year to year varies based on the hydrological system. For example, energy production from hydropower stations in Manitoba varied from 19.3 TWh in 2003/04 to 31.5 TWh in 2004/05 (Snell, Prowse and Adams 2014).

2.2. Manitoba Hydro Electric System Description Figure 2 shows an overview of the Manitoba Hydro electrical system, describing the power stations in Manitoba, major transmission lines internal to the province and interconnections to neighboring power systems. Manitoba Hydro operates 15 hydroelectric generating stations and two thermal stations with combined power capacities shown in Table 1. Energy is also supplied to the electric grid from two wind power stations. There are four isolated diesel generation serving local communities that are not connected to the rest of the electrical system and are excluded from the study area.

Figure 1 highlights the demand for electrical power in Manitoba during 2017/18, showing both total energy used and peak electric demand by month. Both peak and total energy use is largest in the winter months, due to the electric heating demand. In 2017, approximately 40% of residential customers in Manitoba were using electric space heating, whereas all of Canada had approximately 25% of space heating from electric sources in 2013 (Manitoba Hydro 2018) (Natural Resources Canada 2017).

Table 1 - Manitoba Hydro generating station capacity (Manitoba Hydro 2019)

Station Name Type Capacity (MW) Limestone Hydro 1400 Long Spruce Hydro 1000 Kettle Hydro 1220 Wuskwatim Hydro 208 Kelsey Hydro 288 Jenpeg Hydro 90 Grand Rapids Hydro 480 Pine Falls Hydro 88 Great Falls Hydro 137 McArthur Hydro 55 Seven Sisters Hydro 170 Slave Falls Hydro 68 Brandon Thermal 238 Selkirk Thermal 118

Energy demand Peak demand 3,000 5,000 4,500 2,500 4,000 2,000 3,500 3,000 1,500 2,500 2,000 1,000 1,500 Demand (GWh) 500 MW Demand Peak 1,000 500 0 0 1-Jul 1-Jul 1-Jan 1-Jan 1-Jun 1-Jun 1-Oct 1-Oct 1-Apr 1-Apr 1-Feb 1-Sep 1-Feb 1-Sep 1-Dec 1-Dec 1-Aug 1-Aug 1-Nov 1-Nov 1-Mar 1-Mar 1-May 1-May Figure 1 - Manitoba electrical energy demand and peak demand by month (Manitoba Hydro 2018)

Figure 2 – Map of Manitoba Hydro hydroelectric system facilities. © Manitoba Hydro (Manitoba Hydro 2020) Used with permission.

The Manitoba electric system is connected to neighboring power systems in Saskatchewan, Ontario, and the United States, allowing Manitoba Hydro to exchange electric energy to meet Manitoba energy demand or sell excess energy supply. The largest interconnection is between Manitoba and the United States, where power is delivered into the system operated by the Mid- continent Independent System Operation, MISO. MISO operates an electricity market that allows Manitoba Hydro to exchange energy with. The Ontario connection is also into an electricity market operated by the Independent Electric System Operator of Ontario, whereas the connection into Saskatchewan is directly with the SaskPower Corporation system without an electricity market.

2.2.1. Electric System Economic Dispatch The operation of the electric system is primary concerned with ensuring energy supply resources are dispatched to meet electrical energy demand at all times. Economic dispatch generally starts with the supply resources that have the lowest variable costs and dispatches higher cost resources until demand is met. Variable operating costs for plants are composed of operating and maintenance costs, the additional operating and maintenance costs required to generate additional energy, and fuel costs. The operating costs for resources vary. The US Energy Information Administration (2020) has estimated the variable operating costs for hydropower and wind power plants to be $0/MWh. Natural gas thermal stations variable operating and maintenance costs to vary from $1.87-5.84/MWh US. Fuel costs for wind and hydro stations are typically $0/MWh although the Province of Manitoba charges a water use rental for water used for generating hydroelectricity in Manitoba (Water Power Regulation 2010). Fuel cost for gas thermal stations varies based on the price of natural gas and the station performance characteristics. The average fuel costs for gas thermal station was $27.35/MWh US in 2018 (US Energy Information Administration 2020); however, Manitoba Hydro thermal stations have lower than average performance characteristics, resulting in larger fuel costs. Imports from neighboring power markets are also a supply alternative for Manitoba Hydro. Power market prices in MISO, the system Manitoba Hydro has the largest interconnection to, averaged $26.97/MWh at the

Minnesota hub in 2018 and varies through the year (Midcontinent Independent System Operator 2018). Considering all supply options, generally Manitoba Hydro will dispatch hydro and wind generation first, followed by power imports from energy markets, followed by thermal generation. When there is excess hydro and wind generation available, the energy can be exported to the neighboring electricity markets to earn additional revenue for Manitoba Hydro.

If the utility is unable to satisfy the electrical demand, energy deficits occur resulting in customers not receiving electricity, which can have significant socioeconomic consequences. The value of lost load or unserved energy is a measure of the socioeconomic impact of energy deficits and can vary widely depending on the method used to derive its value (Schorder and Kuckshinrichks 2015). North American reliability analyses have placed the value between $9000/MWh - $20,000/MWh range although comprehensive reviews show the value can range from a few hundred dollars per MWH to $300,000/MWh (Newell, et al. 2014) (Carden, Pfeifenberger and Wintermantel 2011).

2.3. Manitoba Hydro Reservoir System Hydroelectric stations are the key facilities that link the Manitoba electrical system to the Nelson- Churchill Rivers basins. The Nelson-Churchill Rivers basins, shown in Figure 3, range from the Rocky Mountains in the West to nearly Lake Superior in the East, draining through the western provinces and northern midwestern states into the Nelson River and onto Hudson Bay. The Churchill River is partially diverted into the Nelson River by the Churchill River Diversion project. All the hydraulic facilities related to the hydroelectric system Manitoba Hydro operates are within the basin. Figure 4 shows a schematic describing the network connections between the facilities. The peripheries of the system Manitoba Hydro operates includes the Pointe du Bois hydroelectric station on the Winnipeg River, Cedar Lake on the Saskatchewan River, Southern Indian Lake on the Churchill River and Lake Winnipeg, which includes the end point of numerous rivers including the Red River, Fairford River, Bloodvein River. The largest reservoirs used for managing seasonal variances in hydroelectric generation potential are Lake Winnipeg, Cedar Lake, and Southern Indian Lake.

Figure 3 - Nelson-Churchill Rivers Basin, © Manitoba Hydro (Manitoba Hydro 2020). Used with permission.

Total system inflow traces for the Manitoba Hydro Long-Term Flow Data dataset are shown in Figure 5. Inflows are largest in the spring and summer and typically decrease into the winter period. Table 3 highlights the average annual inflows from the different inflow points shown on the network schematic, Figure 4. The net local inflows in the record are calculated based on a mass balance of observed upstream and downstream flows and measured storage changes, which encompass real phenomenon, such as evaporation and groundwater flows, as well as errors in measurement and observations. This can result in net negative inflow supply during certain periods.

Figure 4 - Manitoba Hydro hydroelectric power system and connection to neighbouring power market

450 400 350 300 250 200 150 INFLOW (KCFS) 100 50 0 4/1 7/8 8/5 9/2 -50 1/6 2/3 3/3 4/15 4/29 5/13 5/27 6/10 6/24 7/22 8/19 9/16 9/30 12/9 1/20 2/17 10/14 10/28 11/11 11/25 12/23 MONTH/DAY

Figure 5 - Semi-monthly total system inflows for the LTFD locations from 1978-2015 and the average in a black line

Table 2 – Average annual inflows for the aggregated Long Term Flow Data locations

LTFD Location Average Annual Inflow, 1978-2015 (kcfs)

SF 32.9

SRIAO 20.3

CRIAO, BR1 33.5

LWPIAO, NR0 27.2

NR1, NR2 3.1

BR2 1.4

BR3, BR4, NR3, NR4 5.4

Total 123.8

The lakes and outlets of the reservoir system are licensed by the Province of Manitoba through the Water Power Act to ensure the facility makes beneficial use of the resource and is within the public interest (The Water Power Act 2014). The operation of all facilities are important to consider in reservoir operations planning, but Southern Indian Lake, Cedar Lake and Lake Winnipeg is the focus of this study as these reservoirs encompass the majority of the active

storage Manitoba Hydro controls and are large enough to significantly influence the volume of water reaching the larger, lower Nelson power plants at the seasonal timeframe (Kuzyk and Candish 2019) (Manitoba Hydro 2020).

Lake Winnipeg is operated under the Lake Winnipeg Regulation Water Power Act license. The license allows Manitoba Hydro to manage the water level of the lake between 711.0 ft and 715.0 ft for hydroelectric power management purposes, providing approximately 15,000 kcfs-days or 37.2 km3 of active storage (Province of Manitoba 1972). If the elevation goes above 715.0 ft, Manitoba Hydro must set the outflow out of the lake to the maximum possible discharge. Outflow from the lake is managed from the Jenpeg hydroelectric generation station, as the Jenpeg station forebay has a backwater effect on outflow from Lake Winnipeg. Management of the Jenpeg forebay level allows reservoir managers to change the hydraulic gradient between the forebay and Lake Winnipeg and control the outflow. The arrangement is simplified to two outlets in the system network schematic Figure 4, where Nelson West Channel represents flow from Lake Winnipeg towards Jenpeg and Nelson East Channel represents flow from Lake Winnipeg around Jenpeg directly into Cross Lake.

Southern Indian Lake is operated under the Churchill River Diversion (CRD) Water Power Act (WPA) license and CRD Augmented Flow Program (AFP) authorization. The Churchill River Diversion allows Manitoba Hydro to divert flows from the Churchill River system into the Burntwood and Nelson River systems to increase water flow at the Lower Nelson generating stations. The license and program allows Manitoba Hydro to operate Sothern Indian lake between 843.0 ft and 847.5 ft, providing approximately 1300 kcfs-days or 3.2 km3 of active storage (Province of Manitoba 1973) (Province of Manitoba 2020). Outflow from the lake is controlled by two control structures. The Missi Falls control structure outflows into the lower Churchill River. The Notigi control structure outflows into the Burntwood River, joining the Nelson River at Split Lake. The Notigi control structure is connected to Southern Indian Lake by way of the South Bay channel. The license and other operational requirements stipulate minimum outflows for Missi Falls to be between 0.75 and 4.25 kcfs, depending on the time of

year. Notigi maximum outflows are also stipulated in the WPA license and AFP to be 33.9 kcfs for 34.9 kcfs, depending on the time of year.

Cedar Lake is operated under the Grand Rapids Water Power Act license. The license requires the lake level to be below 842.0 ft and the minimum operating level is 831 ft, providing 3000 kcfs- days or 7.7 km3 of active storage. Outflow from the lake is through the Grand Rapids Generating Station and associated spillway structure (Province of Manitoba 1975).

The flow travel time between the seasonal storage reservoirs and the major generating sources on the Lower Nelson River requires reservoir managers to account for the travel time lag when determining reservoir releases. Manitoba Hydro has estimated that, the flows take 15 days to arrive at Kettle, the first station in the Lower Nelson cascade, from Lake Winnipeg and 9 days for from Southern Indian Lake to arrive at Kettle for average inflow conditions, based on internal studies (Magura 2013).

3. Manitoba Hydro Hydroelectric Reservoir System Operation 3.1. Reservoir System Operation Objectives Manitoba Hydro has identified the objective for the operation of the Manitoba Hydro hydroelectric reservoir system to be “to plan for the secure and economic operation of Manitoba Hydro’s system of reservoirs and generating stations while considering the effects on stakeholders and the environment” (Manitoba Hydro 2014).

Manitoba Hydro employs the following management practices to achieve their objective (Manitoba Hydro 2014)):

- Maintaining a dialogue with communities to inform of operating decisions - Accommodating requests for flow reductions for certain emergency scenarios - Updating forecasts each week and assessing forecast accuracy - Maintaining energy reserves for use in droughts to avoid energy deficits and customer load reductions - Compliance with Water Power Act license requirements - Using a computer-based Decision Support System to help ensure water resources are used as efficiently as possible

The hydroelectric system operations planning process is an adaptive planning process that takes places on a weekly basis that looks at managing the water resource from one week up to two years out (Manitoba Hydro 2020). This planning period is referred to as mid-term operations planning that focuses on managing the variability of inflows and timing the flow releases to best match electrical demand. Long-term planning looks at periods longer than one to two years and is more focused on planning future energy supply than on the operation of the current system. Short term operations examines hour-to-hour and day-to-day operations of the coordinated system and is generally focused on economic operation, less on the management of the available water resource over a hydrological year (Manitoba Hydro 2020) (Manitoba Hydro 2014). This is partially driven by the properties of the reservoirs within the system. The active storage of a reservoir relative to the inflow, controllability, and downstream generating capacity as well as

the flow travel time within the system lead to the mid-term planning horizon to be the time horizon used for hydroelectric system operations planning (Manitoba Hydro 2014).

The weekly operations planning process includes four main steps:

1. Observe system state and update forecasts 2. Assess conditions and adjust plan, incorporating stakeholder input and feedback 3. Communicate operations plan 4. Execute operating decisions

Manitoba Hydro employs the in-house developed Hydro Electric Reservoir Management Evaluation System (HERMES) decisions support system (DSS) to assist in completing the weekly operations planning process. HERMES includes modules for forecasting, simulating operations of the hydrological system, and assessing the operations of the combined hydroelectric power system. Forecasts are prepared for system inputs including inflows, ice conditions, energy demand, energy market prices and plant maintenance schedules. The main model used to assist in developing and assessing the weekly operations plan is the in-house developed Energy Management and Maintenance Analysis (EMMA) model, a hydropower system energy management optimization model (KPMG 2010). The forecasts and model outputs are reviewed alongside external input and feedback at weekly resource planning meetings. Expected water levels and flows are published to an internet website for external communication.

3.2. Inflow Forecasting The HERMES inflow forecasting module uses time-series modeling tools to forecast future inflows similar to those described by Bender and Simonovic (1994). The models project future flows based on antecedent flows using linear or log relationships (Manitoba Hydro 2020). Reservoir managers select the best fit model by linear regression for a given time-period. The best fit model can be adjusted weekly, monthly, or seasonally at each location forecasted. The statistical forecasting model can be used to generate inflow forecasts at different exceedance percentiles, allowing the reservoir manager to assess impacts of higher or lower inflow forecasts. Reservoir

managers can also generate a set of forecasts that trend towards historical flows by blending their forecasts into multiple historical traces. When blending, the statistical forecast linearly transitions into a historical trace over a period defined by the reservoir manager for each location and historical trace.

3.3. EMMA Model The EMMA model is a successive linear programming deterministic optimization model used for determining the optimal operation of the reservoir system over the mid-term planning period. The model was originally developed by Barritt-Flatt and Cormie (1988) in the late 1980s and has been continuously updated since (KPMG 2010).

The objective function of the model is to maximize net revenues and is subject to constraints that reflect limits of the hydro stations, thermal stations, hydrological system characteristics, license limits, electric system characteristics, Manitoba energy demand, and extra provincial energy markets (Manitoba Hydro 2020). Hydro station output is represented as a piece-wise linear station flow versus power output curve. Thermal station output includes a representation of fuel costs and conversion efficiency. The interconnected hydrological system is represented as a set of reservoir storages and outlet flows with stage and flow limits included and relationships between stage and outflow approximated linearly. Hydrological facility license restrictions on stage, outflow, and outflow rate of change are incorporated. Electric system characteristics requiring the provision of electric system reserves and limiting power system flows to physical transmission limits are included. Energy demand must be met by Manitoba Hydro facilities or the import of power from interconnected systems. A representation of energy markets to interconnected systems is included.

Successive linear programming is a process of solving non-linear optimization problems by continuously solving a series linear programming optimization problem and updating the non- linear components of the problem until the problem converges. The EMMA model uses successive linear programming to approximate non-linear relationships in the head versus power

output of hydro stations and the stage versus outflow relationship of hydraulic outlets (Manitoba Hydro 2020). Changes in the objective function value are monitored for convergence.

The EMMA model is a deterministic optimization model, assuming perfect knowledge of all inputs, ignoring the uncertainty inherent in inflow forecasts (Manitoba Hydro 2020). Two approaches are used to limit the impact of assuming perfect foreknowledge. First, the adaptive planning process is employed, where the most recently available inflow forecasts are used and the analysis is recomputed weekly. Second, low flow risks are evaluated and considered when determining DSS recommended reservoir release decisions. Recommended operating decisions from the ‘expected’ inflow scenario are inputs into a ‘low flow’ scenario and if drought storage criteria are violated, the recommended operating decisions are transitioned to those from a ‘low flow’ scenario analysis. Analysis for a range of inflow scenarios are also conducted and are used by the reservoir manager to understand variability of inflows on optimal reservoir decisions qualitatively. Model execution for a one-year horizon and single inflow scenario is approximately five minutes. Multiple scenarios can be evaluated using parallel techniques increasing the execution time to approximately 15 minutes when 40 inflow scenarios are evaluated.

3.4. Ideas for Improving Manitoba Hydro Midterm Operations The HERMES suite of tools has been continuously updated and improved since inception. Nevertheless, three areas for improvement in the forecasting and optimization approaches are identified below.

The first area for potential improvement is the use of advanced inflow forecasting tools, such as hydrological models. Autoregressive based prediction models that Manitoba Hydro currently uses have been a popular approach for reservoir inflow forecasting in the past and is often used as a comparison for newly developed methods (Sveinsson, et al. 2008) (Lohani, Kumar and Singh 2012). Past external reviews of the Manitoba Hydro systems simulation have suggested that statistical regression is a “reasonable” approach for inflow forecasting for the Manitoba Hydro system, give the topography and availability of upstream inflow observations (KPMG 2010). Using

simpler, statistical regression tools can be advantageous compared to the complexities involved and data requirements for calibrating hydrological models (Veiga, Hassan and He 2015). However, there have been numerous advances in the use of hydrological models to generate multiple streamflow forecasts, also known as ensemble streamflow predication, for use in reservoir operations planning (Faber and Stedinger 2001) (Cote and Leconte 2015). Several hydrological models have been developed or adapted for use in the Nelson-Churchill Rivers basin (Boluwade, et al. 2018) (Vieira 2016) (Bajracharya, et al. 2020) (Pokorny, et al. 2020) . Manitoba Hydro is currently investigating adapting such hydrological models use in for reservoir operations (Manitoba Hydro 2020).

The linear approximation of the non-linear hydropower efficiency relationship is another potential for improvement. The EMMA model approximates the non-linear hydropower equation as a piece-wise linear relationship between station flow and power generated (Manitoba Hydro 2020). A successive linear programming approach is used to update the piece-wise linear relationship to account for changing forebay elevations in the model. Barros et al. (2003) and Feng et al. (2018) have both found the successive linear programming approach to be a suitable approximation that maintains the advantages of using linear programming. However, multiple studies have shown that using non-linear approaches have advantages for short-term scheduling by providing more accurate hydropower generation forecasting and more efficient dispatch (Nguyen, Vo and Truong 2014) (Catalao, et al. 2006). The benefits of using non-linear techniques for midterm planning is less clear. Modelers need to choose between more accurate representation of the hydropower system and efficient techniques for representing uncertainty in inflow forecasts (Labadie 2004) (Warland and Birger 2016).

Stochastic optimization methods are another potential area for improvement for the Manitoba Hydro reservoir system operations process that has not yet been perused. Stochastic optimization models use a probabilistic description of future inflows allowing the optimization to determine optimal reservoir releases without the bias of perfect foreknowledge which may increase the quality of solutions from the decisions support system (Labadie 2004). Approaches for multi-reservoir management under uncertainty are reviewed in the following chapter.

4. Review of Multi-Reservoir Operation Optimization Approaches Labadie (Labadie 2004) provides a general classification of stochastic optimization techniques for multi-reservoir operations and management into two categories:

- Implicit Stochastic Optimization (ISO) methods, and - Explicit Stochastic Optimization (ESO) methods.

Implicit stochastic optimization is analogous to a Monte Carlo analysis, where the optimization scheme considers uncertainty through multiple, independent analyses of different inflow scenarios. As the scenarios are independent of each other, deterministic optimization techniques can be applied. Operating decisions from the individual scenarios are used in a regression analysis to develop operating rules conditioned on observable information, such as reservoir storage or past period inflows (Labadie 2004). Alternatives to regression have been applied to develop operating rules from the individual scenario results, including the use of artificial neural networks by Chandramouli and Deka (2005) and Lee and Labadie (2007), and interpolation by Celeste et al (2009). The inferred operating rules can then be used to simulate operations. Figure 6 shows an overview of the ISO process.

Deterministic Operating Operating Inflow Cases Inference Simulation Optimization Results Rules

w1, X1, Regression, xt = f(Yt), w2, X2, interpolation, Where Yt can be … … storage, wn Xn past inflows, or other observable information

Figure 6 – Overview of the ISO process. Inflow cases are analyzed independently with deterministic optimization, creating a set of operating results or polices specific to the inflow scenario analyzed. Inferences are made on all sets of operating results through regression or other methods to derive operating rules, which can then be used to simulate operations of the reservoir system.

The largest drawback of ISO approaches are that the optimization results determined are unique to the inflow scenario employed (Labadie 2004), as the operating decisions are made using perfect foreknowledge (Philbrick and Kitanidis 1999), that is without consideration that future

18 inflows are not known with certainty. Philbrick and Kitanidis (1999) note that using a single inflow scenario is inadequate to determine appropriate reservoir release decisions. Labadie (2004) further notes that the process of developing operating rules from regression techniques may invalidate the operating rules and other methods may require an extensive trial and error approach.

Explicit stochastic optimization approaches use stochastic optimization to directly consider uncertainty in the optimization scheme (Philbrick and Kitanidis 1999). In the context of hydropower reservoir management problem, inflow is taken as the uncertain component and is described in a probabilistic fashion. Reservoir operating decisions are made considering that multiple, possible future inflow scenarios could occur, as opposed to a single inflow scenario used in the optimization process in ISO techniques. Two of the most common explicit stochastic optimization approaches, stochastic linear programming and stochastic dynamic programming are described further below.

The key benefit of explicit stochastic optimization approaches is that the optimization is performed without the presumption of perfect foreknowledge by operating directly on probabilistic description of random variables (Labadie 2004) (Kim, et al. 2007) (Cote and Leconte 2015), eliminating the largest drawbacks of ISO approaches. However, this makes the problem more complex with a corresponding increase in the computational requirements. The problem size in stochastic linear programming approaches may become extremely large as the number of time steps and possible inflow scenarios increase, requiring the use of decomposition, scenario tree reduction, or progressive hedging approaches to maintain computational tractability (Labadie 2004) (Cote and Leconte 2015). Stochastic Dynamic Programming (SDP) approaches suffer from the ‘curse of dimensionality’ (Bellman and Dreyfus 1962)(Labadie 2004) (Philbrick and Kitanidis 1999), where the number of sub-problems to be solved is subject to combinatorial explosion as additional reservoirs are added or the discretization of the inflow probability space is increased. The Stochastic Dual Dynamic Programming procedure by Peiria and Pinto (1991) avoids much of the dimensionality problem in SDP approaches, but is an approximate methods

that require simplifications to the problem space (Cote and Leconte 2015) (Philbrick and Kitanidis 1999).

4.1. Linear Programming Approaches Linear programming (LP) continues to be a popular approach for solving the reservoir management problem.

Linear programming models are formulated as:

Equation 1

min = 𝑇𝑇 𝑥𝑥 𝑍𝑍 𝑐𝑐 : 𝑥𝑥

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡

Where is the objective function, is a cost𝐴𝐴𝐴𝐴 ≥coefficient𝑏𝑏 vector, is a decision variable vector,

is the constraint𝑍𝑍 matrix, and is vector𝑐𝑐 of limits on the constraints𝑥𝑥 . Applied to the reservoir𝐴𝐴 management problem, the decision𝑏𝑏 vector typically represents a vector of reservoir releases for all outlets in the system but can also take on other variables as required by the specific problem benefit function.

The dual problem to the linear programming primal problem is formulated as:

Equation 2

max = 𝑇𝑇 𝑦𝑦 𝛽𝛽 𝑏𝑏: 𝑦𝑦

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑇𝑇 In the dual problem, each variable in the𝐴𝐴 primal𝑦𝑦 ≤ 𝑐𝑐 becomes a constraint in the dual, and each constraint in the primal becomes a variable in the dual (Rardin 1998). Duality theory also shows that min = is equal to max = (Rardin 1998). Where the primal problem is 𝑇𝑇 𝑇𝑇 formulated𝑥𝑥 𝑍𝑍 to 𝑐𝑐allocate𝑥𝑥 the use of reservoir𝑦𝑦 𝛽𝛽 𝑏𝑏 storage𝑦𝑦 in the reservoir management problem, the

dual problem can be thought of determining the value or shadow price on the use of reservoir storage.

Linear programming approaches as outlined above have several advantages. Firstly, the algorithm solves a convex problem which is known to converge to a global optimum contrary to certain non-linear programming problems: however, it may have challenges with extremely large problems with large numbers of decision variables and or constraints (Boyd, Boyd and Vandenberghe 2004). Recent benchmarks show both commercial and open source linear programming solvers are capable of solving problems with tens of thousands of constraints and millions of decision variables in under an hour, indicating that free solutions are available that can solve problems of this size (Mittelmann 2020). Algorithms and performance improvements have continued over several decades and are expected to continue to improve as chronicled by Bixby (Bixby 2012). The duality theory of linear programming is another advantage that helps in decomposition approaches used for the reservoir management problem (Labadie 2004).

4.1.1. Stochastic Linear Programming Representation of the stochastic case is generally accomplished by representing the reservoir operation problem as a two-stage problem, where inflows are assumed to be known with certainty in the first stage but are unknown in the second stage (Labadie 2004). The problem is formulated to minimize total costs in the first stage and expected total costs in the second stage. Only the first stage decisions are implemented as they are not conditioned on a specific but unknown future inflow scenario occurring. Simulation of the entire planning period can be completed by solving the two-stage problem, implementing the first stage results, incrementing to the next period, and then repeating. Figure 7 highlights the process for Stochastic Linear Programming.

Solve Two- Implement Develop Stage Inflow Cases first stage Simulation Scenario Tree Stochastic decisions Optimzation

Advance to next stage

Figure 7 - Overview of the two-stage problem solved with stochastic linear programming approaches. A two-stage scenario tree is developed with known inflows in the first stage and unknown inflows in the second stage. The problem is solved with a stochastic linear programming and the first stage operating decisions are implemented in the simulation. The problem is advanced to the next stage to repeat the process for the entire planning period.

Figure 8 shows an example of a scenario tree. Each branch of a scenario tree represents different possible inflows occurring with a probability of occurrence at that node. While solving a deterministic equivalent problem is possible, the computational burden can become intractable as the dimensionality of the scenario tree grows (Pan, et al. 2015). Most stochastic linear programming approaches found in the literature use some form of decomposition or scenario reduction approaches.

1 2

𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆

1 2

3 4

5 6 = 1 = 2 = 3 =4

𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 Figure 8 – Two stage scenario tree with a 3-2 branching structure. Stage 1 inflows are known with certainty whereas stage 2 have multiple possibilities.

Decomposition approaches aim to break up intractable, large-scale problems by temporarily reducing the problem into a more tractable form. For the reservoir management problems, the decomposition is typically performed over time stages and branches on the scenario tree. Traditional decomposition approaches move between a master problem that is solved with a current approximation, and a subproblem that is used to update and improve the approximation (Galati 2010). Benders decomposition is a popular, traditional cutting-plane decomposition approach that leverages duality theory to generate bounds on the objective function value at given variable approximations known as Benders Cuts (Hooker and Ottosson 2003).

Scenario tree generation attempts to generate a discrete and adequate approximation of the stochastic inflow process with a limited number of outcomes. Scenario trees that best represents the stochastic process are generated and then reduced to a practical size (Latorre, Cerisola and Ramos 2007). Hoyland and Wallace (2001) describe a method of generating a discrete number of scenarios that have statistical properties close to or equal to the properties of the underlying uncertain element, using optimization approaches to minimize the difference between the statistical properties of interest in the discrete sampling of the random process and the underlying uncertain element. Clustering based method is another approach that aims to sample directly from observations to generate the scenario tree (Xu, et al. 2015). Scenario tree reduction approaches such as those by Growe-Kuska et al (2003), Heitsch and Romisch (2009), and da Costa et al (2006) aim to reduce the size of the scenario tree while maintaining certain statistical properties.

4.1.2. Stochastic Linear Programming Applications The SOCRATES model by Jacobs et al (1995) is one of the first examples of a multi-stage stochastic linear programming solution method applied to the multi-reservoir management problem. The model uses a nested Benders decomposition to break the master problem down into individual problems for each stage and streamflow scenario. Each sub-problem has a unique previous scenario but multiple future scenarios. The sub-problem is solved, and the information is either 23

passed in the forward direction to derive the starting storage conditions for the next sub- problem, or, together with other scenarios in the same stage and same pre-ceding scenario, compute a Benders cut to pass back to the pre-ceding sub-problem as an approximation of the future cost function. Different ‘tree-traversing’ strategies, the order in which the sub-problems are solved, and other Bender cut enhancements were employed to improve the computational efficiency.

Carpientier et al (2012) formulated a multistage stochastic linear programming formulation for the Hydro Quebec system using a progressive hedging algorithm. The inflow representation is a scenario tree, with scenarios branching out at nodes, such that each node has a common preceding scenario but multiple subsequent scenarios. The main problem is disaggregated into sub-problems for each time stage and inflow scenario using Lagrangian relaxation on non- anticipativity constraints. These non-anticipativity constraints require that the decision variables for all sub-problems at a given node are the same but include a penalty value that allows for violation of the constraint at a cost. Dual values on the non-anticipativity constraint are passed back to the master problem. The progressive hedging algorithm portion is the method of solving all the sub-problems and updating the node-wise decision variable vector and non-anticipativity constraint dual values until a stopping criteria is met.

Helseth et al (2017) show an implementation of a two-stage stochastic linear programming approach to the long-term hydro thermal scheduling problem for a Nordic system with 1265 hydro stations and reservoirs. Application of the model was for evaluating transmission line and pumped storage expansion as opposed to operations planning, where the run times of over 50 hours observed are less of a concern. The overall scheduling problem is solved as a sequence of two-stage stochastic linear programming problem, where the decisions from the first stage are implemented, the system is updated, and the problem is re-optimized for the next stage. Benders decomposition is used to decompose the second stage branches of a scenario tree into value cuts that approximate the future value water in the first stage.

4.2. Dynamic Programming Dynamic programming, as applied to sequential decision problems, is an optimization approach that uses discretization and decomposition to break an optimization problem down into a set of sequential, stages and discretized states of the system in each stage, allowing the use an efficient search to determine the optimal solution. The original dynamic programming method was developed by Bellman (1957) and since then has seen numerous applications and advances. The dynamic programming approach for the reservoir management problem has the time domain of the problem decomposed into stages and reservoir storage discretized into a set of states (Labadie 2004). A state transition function is used to define the benefit of transitioning between a state in one stage to a state in a subsequent stage. A “cost-to-go” or “benefit-to-go” function is used to incorporate the state transition function and a future value function that incorporates all future state transitions. With known state values in the final stage, a maximum benefit-to-go (or minimum cost-to-go) function can be solved for each state discretization, starting in the final stage and moving backwards. The optimal path through the stages would follow the states with the largest benefit-to-go (or smallest cost-to-go) function. An example of dynamic programming as applied to reservoir management is further explored in Chapter 5.

It is often impractical to discretize the end storage levels of a sub-problem and thus the cost-to- go functions are interpolated between the discretized points of the foregoing bellman function (Tejada-Guibert, Johnson and Stedinger 1995), or approximated with different functions such as the spline stochastic dynamic programming algorithm described by Johnson et al. (1993). The sub-problem function is evaluated at all discretized values of the state variables in a recursive method, generating a set of water value functions that act as a value-based policy that can then be used in a simulation phase for determining reservoir releases.

For the reservoir management problem, a cut typically represents an approximation of the future value of storage at a given storage level, stage, and future inflow scenario(s). To illustrate, the linear programming formulation in Equation 1 is viewed to be representative of minimizing operations costs in a single stage as in Equation 3, based on Pereira and Pinto’s two stage process example (Pereira and Pinto 1991).

Equation 3

min = + ( ) 𝑇𝑇 1 1 1 𝑥𝑥1 𝑍𝑍 𝑐𝑐 𝑥𝑥 𝛼𝛼 𝑥𝑥

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 1 1 1 Where ( ) represents the future𝐴𝐴 value𝑥𝑥 ≥ in𝑏𝑏 subsequent stages resulting from decision . The

future value,𝛼𝛼1 𝑥𝑥1 ( ), itself is also represented as a linear programming problem in a subsequent𝑥𝑥 stage, shown 𝛼𝛼in1 Equation𝑥𝑥1 4, where the previous stages decisions are carried through as ( ). 1 1 Equation 4 𝐸𝐸 𝑥𝑥

( ) = min = 𝑇𝑇 1 1 2 𝛼𝛼 𝑥𝑥 𝑥𝑥2 𝑍𝑍 𝑐𝑐 𝑥𝑥

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑐𝑐𝑡𝑡 𝑡𝑡𝑡𝑡 ( ) 2 2 2 1 1 Equation 5 shows the dual formulation𝐴𝐴 𝑥𝑥 ≥ of𝑏𝑏 Equation− 𝐸𝐸 𝑥𝑥 4. Solving Equation 5 for a given

determines the dual values, . 𝑥𝑥1

Equation 5 𝜋𝜋

( ) = max ( ) 𝑇𝑇 𝛼𝛼1 𝑥𝑥1 �𝑏𝑏2 − 𝐸𝐸1 𝑥𝑥1 � 𝜋𝜋 𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝑇𝑇 2 2 Solving Equation 5 for an set 𝐴𝐴of values𝜋𝜋 ≤ 𝑐𝑐 of provides a set of dual values, , to approximate

the future water value function.𝑁𝑁 The dual 𝑥𝑥values1 can be passed back into𝜋𝜋 𝑛𝑛Equation 3 as an approximation of the water value function.

Equation 6

min = + 𝑇𝑇 1 𝑥𝑥1 𝑍𝑍 𝑐𝑐 𝑥𝑥 𝛼𝛼

𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 𝐴𝐴1𝑥𝑥1 ≥ 𝑏𝑏1 26

( ) 𝑇𝑇 ( ) 𝛼𝛼 ≥ �𝑏𝑏2 − 𝐸𝐸1 𝑥𝑥1 � 𝜋𝜋0 𝑇𝑇 …𝛼𝛼 ≥ �𝑏𝑏2 − 𝐸𝐸1 𝑥𝑥1 � 𝜋𝜋1 ( ) 𝑇𝑇 𝛼𝛼 ≥ �𝑏𝑏2 − 𝐸𝐸1 𝑥𝑥1 � 𝜋𝜋𝑁𝑁

Stochastic dynamic programming expands on the dynamic programming approach to include a representation of randomness in future inflows similar to the scenario tree structure as in the stochastic linear programming approaches (Labadie 2004). The likelihood of specific inflow cases occurring both in the current and future periods is represented probabilistically. A common implementation has inflows represented as a lag-1 process, where the inflows in a stage act as a predictor of inflows in a subsequent stage, capturing the serial nature of the natural inflow processes and has been seen as adequate for monthly or weekly stages (Yeh 1985) (Tejada- Guibert, Johnson and Stedinger 1995). Other hydrological variables can be used in place of the lag-1 inflows that may improve results (Cote, Haguma, et al. 2011). The hydrologic state variable and total inflows are discretized allowing calculation of transition probabilities between the discretized points (Tejada-Guibert, Johnson and Stedinger 1995). Figure 9 gives an overview of the SDP process.

Determine Inflow Cases Discretize state Complete SDP Water Value Simulation and Transition space Procedure Functions Probabilities

Figure 9 - Overview of the Stochastic Dynamic Programming process. Selected inflow cases and the probability to transition between them are determined. The selected inflow cases are included in the discretization of the other state variables. The problem is solved with the backwards dynamic programming procedure for all points in the state space and for all time periods, creating a set of water value functions. The water value functions can then be used to simulate operations of the reservoir system.

For multi-reservoir systems, inflows are required for multiple locations, and the question of how spatial correlation between locations in the inflow model arises. Kelman et al (1990) introduced the sampling stochastic dynamic programming (SSDP) method, where streamflow scenarios used in the stochastic programming framework are sampled directly from an empirical distribution of inflows generated from a set of inflow scenarios, where each scenario includes inflows for all

locations. Here, the cross-correlations across all sites are implicitly assumed. Scenario sets can be historical inflows as originally used by Kelman et al (1990), a generated scenario set that aims to preserve the cross-correlation structure such as in de Costa et al (2006), or can be based off of ensemble streamflow prediction models as shown in several studies (Faber and Stedinger 2001) (Cote and Leconte 2015) (Cote, Haguma, et al. 2011).

Adding additional reservoirs to dynamic programming and most of its successors exponentially increases the computational requirements, known as the curse of dimensionality. Reservoir systems with more than four reservoirs are viewed as intractable for SDP and SSDP approaches that do not use advanced strategies for reducing the computation time (Turgeon 2005) (Cote and Arsenault 2019). Multiple reservoirs can be aggregated together to reduce the dimensionality, but this approach comes at a cost of reduced accuracy of system details and challenges in disaggregating the results (Tilmant and Kelman 2007).

The Stochastic Dual Dynamic Programming (SDDP) approach avoids the state variable discretization approach in SDP and SSDP, thereby avoiding the curse of dimensionality by calculating a subset of the “interesting” water value function cut parameters to approximate the water value function (Pereira and Pinto 1991). The SDDP approach combines dynamic programming and linear programming into a single, efficient algorithm. The full optimization problem is broken down into sub-periods, as in dynamic programming, and the sub-problems are solved with linear programming. For a sub-problem in stage , the future cost function is represented using Benders cuts on the storage variables from stage𝑡𝑡 + 1. Multiple Benders cuts are amalgamated from different future inflow realizations into a single,𝑡𝑡 future cost function, . By utilizing multiple cuts, the future cost function is approximated as a piece-wise linear𝑎𝑎 function of storage at the end of the current stage. The𝑎𝑎 procedure first simulates operation in the forward direction, with no Benders cuts, then computes and passes back Benders cuts in the backwards direction at the “interesting”, simulated storage levels, each time adding additional cuts to the future cost function. The procedure ends when the difference between the approximated future cost function and the calculated benefit approach a reasonable tolerance.

The SDDP approach has seen significant use by Brazilian and Scandinavian hydropower agencies and power system operators, locations where hydropower is a significant portion of the electrical energy production. Tilmat and Kelman (2007) apply the SDDP approach to operation of the Southeastern Anatolia Development project, using the water value function approximations from the last SDDP iteration to simulate operations over 50 synthetic hydrological sequences (Tilmant and Kelman 2007). Gejlsvik et al (2010) combine the SDDP and SDP approaches to simulate the operations of a single hydropower company. SDP is applied to represent stochasticity in energy market prices independently in each stage. Maceira et al (2014) solve the long term hydro scheduling problem for the Brazilian system using a risk aversion scheme to avoid energy deficits during critically low flow years by minimizing a combination of the expected value of system operating costs and the conditional value at risk.

Cote and Aresenault (2019) describe efficient solution techniques applied to a SSDP approach that are able to solve operating policies for a four-reservoir system in less than 5 minutes. A subset of water value function cuts was selected based on reducing the maximum error in the approximation of the water value function in a cut removal algorithm to reduce the size of the linear programming sub-problem. A state space grid reduction technique was used that reduces the size of the grid analyzed by reviewing state variable trajectories from interim simulations.

5. Methodology 5.1. Overview For the purposes of this thesis, the sampling stochastic dynamic programming (SSDP) approach with each sub-problem solved using linear programming was utilized in the development of a stochastic optimization model for the Manitoba Hydro multi-reservoir hydroelectric system. SSDP was selected for the following reasons:

o SSDP has been successfully applied to numerous systems as described in Chapter 4 o SSDP generates a set of water value functions that can be reused to simulate operation of the system without needing to re-complete the optimization, provided the other system elements do not change significantly

o SSDP utilizes scenarios that preserve the cross-correlations across the study area o Linear programming approaches can be employed to solve the sub-problems, which is a technology currently in use and familiar to Manitoba Hydro reservoir system operators

o An SSDP model platform was available for use and adaptation from Manitoba Hydro (Manitoba Hydro 2020).

This chapter starts with an overview of the dynamic programming procedure. A description of the SSDP problem formulation as applied to the Manitoba Hydro system follows, including considerations for flow travel time, a description of the sub-problem, a description of the water value function approximation method, and the inflow scenario selection method. The chapter concludes with a description of how the analysis was setup and executed for two methods of considering the travel time lag as well as a comparison to historic operating decisions.

5.2. Dynamic Programming Overview Dynamic programming is an optimization approach that uses discretization and decomposition to break an optimization problem down into a set of sequential, stages and discretized states of the system in each stage to use an enumerative search to determine the optimal solution. The system state for dynamic programming applied to the reservoir management problem typically represents reservoir storage at the beginning of a stage. Stages represent sequential blocks of time. The system state is discretized to a set of storages and transitions between system states

occur between adjacent stages. The transition between stages is represented as the storage mass balance of transitioning between and , with known inflows, . The associated reservoir

release volume for the transition 𝑆𝑆ca𝑡𝑡 n be 𝑆𝑆calculated𝑡𝑡+1 as = 𝑄𝑄𝑡𝑡 + . A benefit function, ( , ), is calculated for all possible state transitions𝑥𝑥𝑡𝑡 between𝑆𝑆𝑡𝑡+1 − 𝑆𝑆the𝑡𝑡 current𝑄𝑄𝑡𝑡 stage and the 𝐵𝐵subsequent𝑡𝑡 𝑆𝑆𝑡𝑡 𝑆𝑆𝑡𝑡+1 stage, + 1. For a reservoir system, this could represent the satisfaction 𝑡𝑡of water demands, the value𝑡𝑡 of hydroelectric generation, or other values placed on water utilization during the stage. The future value function in a given stage, and state is the maximum of the both the transition benefit function and the future value function in the subsequent stage, shown in Equation 7.

Equation 7

( ) = max( ( , ) + ( )

𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑡𝑡+1 𝑡𝑡+1 𝑡𝑡+1 The recursive nature of Equation𝑓𝑓 𝑆𝑆 7 is exploited𝐵𝐵 𝑆𝑆to solve𝑆𝑆 the𝑓𝑓 optimal𝑆𝑆 path through all stages. When the future value of the system at the last stage, ( ) is known; Equation 7 can be solved for all states and stages by starting in 1 and proceeding𝑓𝑓𝑇𝑇 𝑆𝑆𝑡𝑡 backwards to = 1. Figure 10 provides a description of the problem for a𝑇𝑇 reservoir− with two discretization levels,𝑡𝑡 high storage and low storage . In the example, the optimal future value for each state in a given stage𝑆𝑆ℎ can be determined𝑆𝑆𝑙𝑙 by comparing the combination of the transition value, , and future value for both possible options, either maintaining the same storage level or𝐵𝐵𝑡𝑡 −transitioning1 to the other𝑓𝑓𝑡𝑡 possible storage level, and selecting the maximum. The optimal path from the first stage follows the states with the largest future value function, , in each stage.

𝑓𝑓𝑡𝑡

Transition Description Value Description

( ) = {

( , ) = + 𝑻𝑻−(𝟏𝟏 𝒉𝒉 ) ( ) 𝒇𝒇 𝑺𝑺, +𝐦𝐦𝐦𝐦𝐦𝐦 , 𝟏𝟏 𝒕𝒕 𝒕𝒕+𝟏𝟏 𝒉𝒉 𝒍𝒍 𝒕𝒕 𝑻𝑻−𝟏𝟏( 𝒉𝒉 𝒉𝒉) 𝑻𝑻( )𝒉𝒉 𝒙𝒙 𝑺𝑺 𝑺𝑺 𝑺𝑺 − 𝑺𝑺 𝑸𝑸 𝑩𝑩 𝑺𝑺 , 𝑺𝑺 + 𝒇𝒇 𝑺𝑺 } ( , ) ( ) 𝑻𝑻−𝟏𝟏 𝒉𝒉 𝒍𝒍 𝑻𝑻 𝒍𝒍 High storage, 𝑩𝑩 𝑺𝑺( 𝑺𝑺, ) 𝒇𝒇 𝑺𝑺 1 ℎ ℎ 𝑻𝑻−𝟏𝟏 𝒉𝒉 ( ) ℎ ℎ 𝑥𝑥 𝑆𝑆 𝑆𝑆 ℎ 𝒇𝒇 𝑺𝑺 𝑇𝑇−1 ℎ ℎ 𝑆𝑆 𝑆𝑆 𝑆𝑆 𝐵𝐵 𝑆𝑆 𝑆𝑆 𝑓𝑓𝑇𝑇 𝑆𝑆ℎ

System State System Low storage, ( , ) ( ) 𝑙𝑙 𝑙𝑙 𝑙𝑙 ( , ) 𝑆𝑆 𝑆𝑆 𝑆𝑆 ( ) 𝑇𝑇 𝑙𝑙 1 𝑙𝑙 𝑙𝑙 𝑓𝑓 𝑆𝑆 𝑥𝑥 𝑆𝑆 𝑆𝑆 𝐵𝐵𝑇𝑇−1 𝑆𝑆𝑙𝑙 𝑆𝑆𝑙𝑙 𝑓𝑓𝑇𝑇−1 𝑆𝑆𝑙𝑙 Time = 0 = 1 … = 1 =T 𝑡𝑡 𝑡𝑡 𝑡𝑡 𝑇𝑇 − 𝑡𝑡 System Stage

Figure 10 - Description of a simple dynamic programming approach for a reservoir with two storage levels in time. The transition description section describes how the release decision would be calculated when transitioning between different states. The bolded equation at the top highlights the calculation for when transitioning from high storage in stage 0 to high storage in stage 1. The value description describes how the benefit functions for transitioning to a future state across a stage and shows 𝑡𝑡 the future value functions at the end of the final stage𝑥𝑥 . The bolded equation at the top highlights the calculation for determining the future value function for time-period 1 and high storage, ( ). 𝑇𝑇 𝑇𝑇−1 ℎ The state discretization approach restricts the reservoir𝑡𝑡 − releases 𝐹𝐹evaluated𝑆𝑆 ; however, discretization of the state variables is required to utilize the dynamic programming procedure (Tejada-Guibert, Johnson and Stedinger 1995). Instead of discretizing end storage, the future value function, , in each stage is approximated by a continuous function, an example of which is shown in Figure𝐹𝐹𝑡𝑡 11. The set of water value functions are then used in a sequential simulation, where the reservoir decisions in each stage are found by solving the optimization problem in Equation 8.

Equation 8 ( ) = ( ( , ) + ( ))

𝑥𝑥𝑡𝑡 𝑆𝑆𝑡𝑡 𝑎𝑎𝑎𝑎𝑎𝑎: 𝑎𝑎𝑎𝑎𝑎𝑎 𝐵𝐵𝑡𝑡 𝑆𝑆𝑡𝑡 𝑆𝑆𝑡𝑡+1 𝑓𝑓𝑡𝑡+1 𝑆𝑆𝑡𝑡+1 𝑠𝑠𝑠𝑠𝑠𝑠=𝑠𝑠𝑠𝑠𝑠𝑠𝑠𝑠 𝑡𝑡𝑡𝑡 + 𝑡𝑡 𝑡𝑡+1 𝑡𝑡 𝑡𝑡 Where is the reservoir 𝑥𝑥release𝑆𝑆 decision,− 𝑆𝑆 𝑄𝑄 is the operator that finds the arguments that maximizes𝑥𝑥𝑡𝑡 a functions value, is the current𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 stage𝑎𝑎𝑎𝑎 transition value function, is the known 𝐵𝐵𝑡𝑡 𝑆𝑆𝑡𝑡 32

starting elevation, . is the unknown starting elevation for the next stage, is the

approximated future𝑆𝑆𝑡𝑡 +value1 function determined from the backwards dynamic programming𝑓𝑓𝑡𝑡+1 procedure, and are the reservoir inflows.

𝑄𝑄𝑡𝑡

𝑓𝑓𝑡𝑡

Figure 11 - Future water value function, as a function of storage. Solutions to the future water value function at the discretized state variable points, such as the low Sl and high Sh level storge points, can be used to approximate a continuous 𝑡𝑡 future water value function. Additional state 𝑓𝑓variable discretization points can be used to improve the approximation accuracy.

The results of the dynamic programming procedure, the optimization phase, produces a set of water value functions. The water value functions are used in a simulation phase to provide context of the future value of water, allowing the simulation models to make tradeoffs between current period and future period use of the resource.

Extensions to the multi-reservoir management problem can be accomplished by having the storage state variable replaced as a vector of storage state variables for each reservoir and including network flows for connected reservoirs. Using more than four state variables may make the problem intractable, but is not a concern for the Manitoba Hydro reservoir system, as there are only three major reservoirs, Lake Winnipeg, Cedar Lake and Southern Indian Lake, used for seasonal water management (Turgeon 2005) (Cote and Arsenault 2019). Extensions to the stochastic case use a probabilistic description of inflows occurring in each stage and is described further in the following section.

5.3. Sampling Stochastic Dynamic Problem Description The optimal decision to release water from a multi-reservoir system, based on Kelman et al (1990) that is discretized into discrete time stages , for a given starting storage state, , ,

and inflow scenario , where is the set of all inflow𝑡𝑡 ∈ 𝑇𝑇 scenarios, is shown in Equation 9. 𝑆𝑆𝑡𝑡

Equation 9 𝑖𝑖 ∈ 𝐼𝐼 𝐼𝐼

( , ) = ( , , S ) + ( , , ) | ( ) 𝐸𝐸 𝑥𝑥𝑡𝑡 𝑆𝑆𝑡𝑡 𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 �𝐵𝐵𝑡𝑡 𝑥𝑥𝑡𝑡 𝑖𝑖 t 𝛽𝛽 �𝑓𝑓 𝑡𝑡+1 𝑆𝑆𝑡𝑡+1 𝑗𝑗 𝑖𝑖 �� Where are the system operating decisions, 𝑗𝑗 𝑖𝑖is an operator that determines the

arguments𝑥𝑥𝑡𝑡 that maximizes the proceeding function,𝑎𝑎𝑎𝑎𝑎𝑎 𝑎𝑎𝑎𝑎𝑎𝑎 is the immediate benefit function for stage , is a depreciation factor to discount future values,𝐵𝐵𝑡𝑡 is the expectation operator, is the future𝑡𝑡 𝛽𝛽 stage inflow scenario, is the future value function𝐸𝐸 for stage , and is the end𝑗𝑗 storage vector for stage . The 𝑓𝑓𝑡𝑡 +vector1 can be calculated from the initial storage𝑡𝑡 𝑆𝑆 vector,𝑡𝑡+1 ( ), inflows to the reservoir 𝑡𝑡and the𝑆𝑆𝑡𝑡 +reservoir1 releases . Equation 9 assumes inflows in stage𝑆𝑆 𝑡𝑡t are𝑘𝑘 known with certainty and are associated with scenario𝑥𝑥𝑡𝑡 , that is the inflow volume in stage t is , . Equation 9 also assumes future inflows are not known𝑖𝑖 with certainty by considering the 𝑞𝑞expected𝑡𝑡 𝑖𝑖 value of the future value of water.

An optimization phase is first conducted using the SSDP approach to generate a set of water value functions in each stage, conditioned on the current stage inflow scenario . The optimization is

conducted over a subset of scenarios to reduce the size of the optimization𝑖𝑖 problem. Scenario selection aims to balance having𝐼𝐼𝑜𝑜 ⊂ an𝐼𝐼 accurate approximation of possible inflows and managing computational requirements. Equation 9 is then solved in a sequential simulation phase where the water value functions are used to approximate . The water value function

for the stage is selected from the set of water value functions based𝑓𝑓𝑡𝑡+ on1 the inflow scenario from the optimization subset, , that is most similar to the inflow scenario in the current stage, . The

problem is evaluated for𝐼𝐼 𝑜𝑜a full year to generate water value functions through all seasons𝑖𝑖 and months, allowing for continuous simulation of the system for multiple years.

The expectation operator, , in Equation 9 simplifies to the probability of future inflow scenarios occurring. By simplifying the𝐸𝐸 expectation operator considering the reduced set of optimization

34 𝐸𝐸

scenarios and the transition probabilities, Equation 9 develops into Equation 10. The corresponding function that is maximized in Equation 10 is shown in Equation 11.

Equation 10

( , ) = ( , , ) + 𝐼𝐼𝑜𝑜 ( | ) × ( )( , )

𝑥𝑥𝑡𝑡 𝑆𝑆𝑡𝑡 𝑖𝑖 𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎𝑎 �𝐵𝐵𝑡𝑡 𝑥𝑥𝑡𝑡 𝑖𝑖 𝑆𝑆𝑡𝑡 𝛽𝛽 � 𝑃𝑃 𝑗𝑗 𝑖𝑖 𝑓𝑓 𝑡𝑡+1 𝑆𝑆𝑡𝑡+1 𝑗𝑗 � 𝑗𝑗 Equation 11

( , ) = ( , , S ) + 𝐼𝐼𝑜𝑜 ( | ) × ( )( , )

𝑓𝑓𝑡𝑡 𝑆𝑆𝑡𝑡 𝑖𝑖 𝐵𝐵𝑡𝑡 𝑥𝑥𝑡𝑡 𝑖𝑖 t 𝛽𝛽 � 𝑃𝑃 𝑗𝑗 𝑖𝑖 𝑓𝑓 𝑡𝑡+1 𝑆𝑆𝑡𝑡+1 𝑗𝑗 𝑗𝑗 5.3.1. Considerations for Flow Travel Time in the Optimization Model The dynamic programming solution approach relies on the ability to separate the problem into distinct stages where the only connection between the stages are the state variables. However, the flow travel time in the network breaks this assumption depending on the length of the travel time lag between nodes in the network and the stage length. Two methods of accounting for the travel time lag were incorporated. Both approaches assume that the flow travel time or lag is equal to the duration of a single stage. Past stochastic optimization studies have utilized stage durations from one week up to a month, and even longer (Carpentier, Gendreau and Bastin 2012), (Faber and Stedinger 2001) (Kim, et al. 2007)). Internal studies by Manitoba Hydro have indicated the flow travel time is in the range of two to three weeks (Magura 2013). This estimate was corroborated with a time series analysis described in Appendix A1. The stage duration was set as semi-monthly, approximately equal to the travel time lag Manitoba Hydro has observed between Lake Winnipeg and Stephens Lake and is of a duration reflective of past approaches.

The first approach for incorporating travel time lag, denoted as the ‘with lag’ approach, incorporates the flow lag into the formulation by including a new, lagged inflow state variable, , that represents the sum of releases from upstream reservoirs from stage 1 that arrive at the𝑅𝑅𝑡𝑡 downstream location in stage . The flow lag is represented by a single state𝑡𝑡 − variable to limit the total number of state variables,𝑡𝑡 as past studies have found that using more than four state variables adds intractable computational burden (Turgeon 2005). Equation 11 is reformulated to

consider the lagged inflow state variable into Equation 12. Similar to , can be

determined from the reservoir releases that flow into the downstream location.𝑆𝑆𝑡𝑡+1 𝑅𝑅𝑡𝑡+1 𝑡𝑡 Equation 12 𝑥𝑥

( , , ) = ( , , , ) + 𝐼𝐼𝑜𝑜 ( | ) × ( , , )

𝑓𝑓𝑡𝑡 𝑆𝑆𝒕𝒕 𝑅𝑅𝑡𝑡 𝑖𝑖 𝐵𝐵𝑡𝑡 𝑥𝑥𝑡𝑡 𝑖𝑖 𝑆𝑆𝑡𝑡 𝑅𝑅𝑡𝑡 𝛽𝛽 � 𝑃𝑃 𝑗𝑗 𝑖𝑖 𝑓𝑓𝑡𝑡+1 𝑆𝑆𝑡𝑡+1 𝑅𝑅𝑡𝑡+1 𝑗𝑗 𝑗𝑗 The second method, denoted as the ‘no-lag’ approach, ignores the lag in the optimization phase by determining the water value functions with Equation 11.

All simulations incorporated the time lag using the ‘with lag’ approach of calculating using

results from the current stage to be used as inflows into the delayed inflow location𝑅𝑅𝑡𝑡+1 the subsequent stage.

5.3.2. State Variable Discretization Stage limits for the lakes reflect a combination of operational and license limits. Table 4 highlights the bounds and discretization limits for the three lakes in this research. Lakes are discretized evenly between the specified minimum and maximum points. Discretization of the storage variables starts above the lower bound for Lake Winnipeg and Cedar Lake to provide a reserve storage buffer so that the inflow scenarios can generate feasible solutions. Storage in the reserve storage section is given a benefit value to encourage storage use patterns that prevent infeasibilities in the sub problem that occur when there is insufficient storage and inflow to operate within the absolute stage limits. The specified minimums, maximums and the reserve storage max point are always included in the discretization. Lake Winnipeg does not have a maximum stage limit, but it has an if-then rule that requires the lake to operate at its maximum discharge when above 715 ft. A penalty value is applied to storage in the sub problem when at or above the maximum discretization stage to ensure adherence to the maximum discharge rule. The lagged inflow discretization limits are set from 50 kcfs to 175 kcfs and is also discretized evenly between the minimum and maximum bounds. The lower bound limit corresponds to the lowest monthly average inflow at Kettle. The upper bound limit corresponds to the maximum powerhouse flow for the largest station on the lower Nelson River. The impact of different

storage values and penalties, and the number of discretization points selected is explored in Chapter 6.

Table 3 – Storage bounds and discretization limits

Lower Discretization Reserve Storage Discretization Upper Lake Unit Bound Minimum Maximum Maximum Bound

elevation (ft) 711.00 711.30 711.50 715.00 718.30 Lake Winnipeg storage (kcfs-d) 0.0 1000.0 1550.0 12160 22180.0

elevation (ft) 831.00 831.25 832.00 842.00 842.00 Cedar Lake storage (kcfs-d) 0.0 55.0 205.0 2945.0 2945.0

elevation (ft) 843.0 843.0 - 847.50 847.50 Southern Indian Lake storage (kcfs-d) 0.0 0.0 - 1390.0 1390.0

5.3.3. Subproblem Description Equation 11 for the ”no-lag” approach or Equation 12 for the ”with-lag” approach is approximated and solved as a linear programming problem, with the objective to maximize the current period net energy export revenue, generation costs, energy deficit cost, water management objective benefits or penalties, and the consideration of future period values. In its general form, the linear programming problem is formulated to maximize its objective function subject to a set of constraints shown in Equation 13.

Equation 13

max = + 𝐼𝐼𝑜𝑜 ( | ) × 𝑇𝑇 𝑧𝑧 𝑐𝑐 𝑥𝑥 𝛽𝛽 � 𝑃𝑃 𝑗𝑗 𝑖𝑖 𝑎𝑎𝑗𝑗 𝑗𝑗

𝐴𝐴𝐴𝐴 ≤ 𝑏𝑏 , × + , × × + , 𝑇𝑇 𝑙𝑙𝑙𝑙 0 𝑎𝑎𝑗𝑗 ≤ 𝜋𝜋𝑡𝑡+1 𝑗𝑗 𝑆𝑆𝑡𝑡+1 𝜋𝜋𝑡𝑡+1 𝑗𝑗 𝑅𝑅𝑡𝑡+1 𝑑𝑑𝑑𝑑 𝜋𝜋𝑡𝑡+1 𝑗𝑗 The water value function approximation, , has two terms for the ‘no lag’ approach, ending

storage for state variables, multiplied𝑎𝑎 𝑗𝑗by the storage cut value, , , and an intercept term, , , . The ‘with lag’𝑆𝑆 approach𝑡𝑡+1 includes an additional term for 𝜋𝜋the𝑡𝑡+1 delayed𝑗𝑗 𝑛𝑛 inflow state variable𝜋𝜋𝑡𝑡 +which1 𝑗𝑗 𝑛𝑛 is composed of the delayed inflow state variable, , multiplied by the stage 37 𝑅𝑅𝑡𝑡+1

duration, , to convert to a volumetric measurement, and the delayed inflow volume cut value

, . 𝑑𝑑𝑡𝑡 𝑙𝑙𝑙𝑙 𝜋𝜋𝑡𝑡+1 𝑗𝑗 5.3.3.1. Objective function Equation 14 shows the objective function in greater detail,

Equation 14

max = 𝐵𝐵 ( ) × 𝑆𝑆𝑆𝑆 , × ,

𝑍𝑍 � � 𝐸𝐸𝐸𝐸𝑝𝑝𝑏𝑏 − 𝐼𝐼𝐼𝐼𝑝𝑝𝑏𝑏 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑏𝑏 − ��𝐺𝐺𝐺𝐺𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 𝐶𝐶𝐶𝐶𝐶𝐶𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠� 𝑏𝑏 𝑠𝑠𝑠𝑠 × + + 𝐼𝐼𝑜𝑜 ( | ) ×

− 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑡𝑡𝑏𝑏 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑓𝑓𝑖𝑖𝑖𝑖𝑖𝑖𝑡𝑡𝑏𝑏� 𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 𝛽𝛽 � 𝑃𝑃 𝑗𝑗 𝑖𝑖 𝑎𝑎𝑗𝑗 𝑗𝑗 where is energy exported to the energy market in time block ( ), is the

energy 𝐸𝐸market𝐸𝐸𝑝𝑝𝑏𝑏 price, is energy imported from the energy 𝑏𝑏market𝑏𝑏 ∈ 𝐵𝐵, 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶, is𝑝𝑝 𝑏𝑏energy generated at generating𝐼𝐼𝐼𝐼𝑝𝑝 𝑏𝑏station ( ), , is the variable 𝐺𝐺generation𝐺𝐺𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 cost, is the total amount𝑠𝑠 of𝑠𝑠 undelivered𝑠𝑠𝑠𝑠 ∈ 𝑆𝑆𝑆𝑆 𝐶𝐶 electrical𝐶𝐶𝐶𝐶𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 demand in period , is the𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 value𝐸𝐸𝐸𝐸𝐸𝐸 of𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 unmet demand in period , is a representation of the satisfaction𝑏𝑏 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶 of water𝑡𝑡𝑏𝑏 management objectives, and (𝑏𝑏| 𝑊𝑊𝑊𝑊𝑊𝑊) × 𝑊𝑊𝑊𝑊 is the future period value. Energy market prices 𝐼𝐼𝑜𝑜 and energy demand are not static𝛽𝛽 ∑𝑗𝑗 for𝑃𝑃 the𝑗𝑗 𝑖𝑖 entire𝑎𝑎𝑗𝑗 stage, so the time block structure allows for the stage to be broken down into periods with similar characteristics. Here, the problem is broken down into two time periods, “onpeak” and “offpeak”. Onpeak represents weekday, daytime periods, and offpeak is all other periods. Onpeak typically has a larger demand and higher prices for power than offpeak periods.

Figure 12 showcases the semi-monthly demand and power prices from 2018 used in the analysis for the onpeak and offpeak periods. Energy from the Laurie River stations and the wind power stations were netted out from the Manitoba electric demand. The energy deficit cost, , is set at $5000 MW/h which is similar to values used in other jurisdictions (Carden,

𝐶𝐶Pfeifenberger𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑡𝑡𝑏𝑏 and Wintermantel 2011) (Newell, et al. 2014).

4500 45.00 4000 40.00 3500 35.00 3000 30.00 2500 25.00 2000 20.00 1500 15.00 Energy Price ($/MWh) Energy demand (MW) 1000 10.00 500 5.00 0 0.00 4/1/2018 5/1/2018 6/1/2018 7/1/2018 8/1/2018 9/1/2018 1/1/2019 2/1/2019 3/1/2019 4/1/2018 5/1/2018 6/1/2018 7/1/2018 8/1/2018 9/1/2018 1/1/2019 2/1/2019 3/1/2019 10/1/2018 11/1/2018 12/1/2018 10/1/2018 11/1/2018 12/1/2018

(a) Onpeak Offpeak (b) Onpeak Offpeak

Figure 12 – Manitoba electrical demand (a) and MISO Power Market prices (b)

5.3.3.2. Energy balance and hydropower constraints A central component of the objective is on energy interchange with an energy market, the cost to generate at the utility’s generating stations, and ensuring electrical supply matches demand (Equation 15).

Equation 15

𝑆𝑆𝑆𝑆 , + + =

� 𝐺𝐺𝐺𝐺𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 𝐼𝐼𝐼𝐼𝑝𝑝𝑏𝑏 − 𝐸𝐸𝐸𝐸𝑝𝑝𝑏𝑏 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑡𝑡𝑏𝑏 𝐿𝐿𝐿𝐿𝐿𝐿𝑑𝑑𝑏𝑏 𝑠𝑠𝑠𝑠 Where is the local energy demand or load for the stage and block. Import and export limits are determined𝐿𝐿𝐿𝐿𝐿𝐿𝑑𝑑𝑏𝑏 by the characteristics of the interconnection from the local system to the energy market, and is represented in Equation 16 and Equation 17 as simple limits on the export and import variables. Manitoba Hydro is interconnected to the MISO electricity market and can import up to 750 MW and export up to 2150 MW, which were used as limits on import and export volumes (Manitoba Hydro 2016).

Equation 16

𝑏𝑏 𝑏𝑏 39 ≤ 𝐼𝐼𝐼𝐼𝑝𝑝 ≤ 𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝑥𝑥

Equation 17

𝑏𝑏 𝑏𝑏 Where and translate≤ 𝐸𝐸𝐸𝐸𝑝𝑝 transmission≤ 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝑥𝑥 interconnection capacity into the amount 𝐼𝐼of𝐼𝐼𝐼𝐼𝐼𝐼𝐼𝐼 energy𝑥𝑥𝑏𝑏 that can𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 be imported𝑥𝑥𝑏𝑏 or exported into or from the utility system during the block.

Generation capability is represented in Equation 18 in a similar fashion to export or import variables.

Equation 18

, , ,

𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 ≤ 𝐺𝐺𝐺𝐺𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 ≤ 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 Where , and , respectively are the minimum and maximum generation volume 𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀for station𝑛𝑛𝑏𝑏 𝑠𝑠 𝑠𝑠 during𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀𝑀 block𝑛𝑛𝑏𝑏 𝑠𝑠.𝑠𝑠 𝑠𝑠𝑠𝑠 𝑏𝑏 The set of stations is composed of both hydro and thermal stations ( ( , )), where

is the set of thermal stations and is the set of hydro stations.𝑠𝑠𝑠𝑠 ∈ 𝑇𝑇𝑇𝑇 Hydro𝐻𝐻𝐻𝐻 station power𝑡𝑡ℎ ∈ 𝑇𝑇𝑇𝑇generation limits are also representedℎ by𝑦𝑦 power∈ 𝐻𝐻𝐻𝐻 curve constraints that represent the relationship between water flowing through the hydro station powerhouse and the energy generated. Following the current approach used in the Manitoba Hydro EMMA model (Manitoba Hydro 2020), the power curve is approximated by a piece-wise linear curve as in Equation 19, where the slope of each segment is monotonically decreasing to ensure convexity. The piece-wise linear approach allows the linear programming approach to approximate the non-linear relationship between power output station flow, including the impact outflow has on tailrace levels that contribute to head impacts. However, forebay level impacts on head and power are not included in this representation.

Equation 19

× , 𝑀𝑀 , , × , = 0

𝛿𝛿𝑏𝑏 𝐺𝐺𝐺𝐺𝑛𝑛𝑏𝑏 𝑠𝑠𝑠𝑠 − � 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑤𝑤𝑏𝑏 ℎ𝑦𝑦 𝑚𝑚 𝜇𝜇𝑠𝑠𝑠𝑠 𝑚𝑚 𝑚𝑚

Where is a rate to volume conversion between instantaneous generation (MW) and energy

generated𝛿𝛿𝑏𝑏 for the block, , , represents the flow rate through the hydro station that corresponds to segment 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆 of the𝑤𝑤 𝑏𝑏powercurve,ℎ𝑦𝑦 𝑚𝑚 and , is the slope of the powercurve for the

corresponding segment. The𝑚𝑚 , , variables𝜇𝜇 𝑠𝑠are𝑠𝑠 𝑚𝑚 bounded by the segments of the power curve as shown in Equation 20𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆. 𝑤𝑤𝑏𝑏 ℎ𝑦𝑦 𝑚𝑚

Equation 20

0 , , , , = 0

0 ≤ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆, , 𝑤𝑤𝑏𝑏 ℎ𝑦𝑦 𝑚𝑚 ≤, 𝑄𝑄𝑠𝑠𝑚𝑚 ℎ𝑦𝑦 , 𝐹𝐹𝐹𝐹𝐹𝐹 , 𝑚𝑚 > 0 ≤ 𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑤𝑤𝑏𝑏 ℎ𝑦𝑦 𝑚𝑚 ≤ 𝑄𝑄𝑠𝑠𝑚𝑚 ℎ𝑦𝑦 − 𝑄𝑄𝑠𝑠𝑚𝑚−1 ℎ𝑦𝑦 𝐹𝐹𝐹𝐹𝐹𝐹 𝑚𝑚 Where , is the station flow at the end of segment and hydro station .

𝑄𝑄𝑠𝑠𝑚𝑚 ℎ𝑦𝑦 𝑚𝑚 ℎ𝑦𝑦

5.3.3.3. Storage balance and outflow constraints Each hydro generating station operates on a lake . Total outflow from a hydro generating

station is equal to the total outflow from the upstream𝑙𝑙 ∈ lake𝐿𝐿 over the duration of the sub-problem. The relationship uses a time weighted average constraint as shown in Equation 21.

Equation 21

𝐵𝐵 𝑀𝑀 , , × = 0 , = 1, ( ) =

� ��𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑤𝑤𝑏𝑏 ℎ𝑦𝑦 𝑚𝑚� 𝛿𝛿𝑏𝑏� − 𝑄𝑄𝑐𝑐𝑐𝑐 𝐹𝐹𝐹𝐹𝐹𝐹 𝐴𝐴𝑐𝑐𝑙𝑙 𝑐𝑐 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 ℎ𝑦𝑦 𝑙𝑙 𝑏𝑏 𝑚𝑚 Where is the percent of time that block encompasses, is the outflow rate from controlled

outlet 𝛿𝛿𝑏𝑏, ( ), , is a routing𝑏𝑏 connectivity matrix𝑄𝑄𝑐𝑐𝑐𝑐 for controlled outlets, and 𝑐𝑐 𝑐𝑐( ∈ )𝐶𝐶= 𝐴𝐴 if𝑐𝑐 hydro𝑙𝑙 𝑐𝑐 station outflows from lake . 𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻𝐻 ℎ𝑦𝑦 𝑙𝑙 ℎ𝑦𝑦 𝑙𝑙 All lakes are subject to Equation 22, the hydraulic mass balance constraint that manages the stock and flows of water.

Equation 22

, + 𝐶𝐶 × , × + 𝑈𝑈 × , × = , + ×

𝑆𝑆𝑡𝑡+1 𝑙𝑙 ��𝑄𝑄𝑐𝑐𝑐𝑐 𝐴𝐴𝑐𝑐𝑙𝑙 𝑐𝑐� 𝑑𝑑𝑑𝑑 ��𝑄𝑄𝑢𝑢𝑢𝑢 𝐴𝐴𝑢𝑢𝑙𝑙 𝑢𝑢� 𝑑𝑑𝑑𝑑 𝑆𝑆𝑡𝑡 𝑙𝑙 𝑄𝑄𝑙𝑙𝑙𝑙 𝑑𝑑𝑑𝑑 𝑐𝑐 𝑢𝑢 41

Where, , is the volume of water stored, in lake at the beginning of sub-problem , ,

is storage𝑆𝑆𝑡𝑡 in𝑙𝑙 the lake at the end of the sub-problem,𝑙𝑙 is the duration of the sub-problem𝑡𝑡 ∈ 𝑇𝑇, 𝑆𝑆𝑡𝑡+1is𝑙𝑙 the outflow from uncontrolled outlet , 𝑑𝑑𝑑𝑑 , is a routing connectivity matrix𝑄𝑄 𝑢𝑢for𝑢𝑢 uncontrolled outlets, and is local inflow𝑢𝑢 into∈ 𝑈𝑈 lake𝐴𝐴𝑢𝑢 𝑙𝑙. 𝑢𝑢The connectivity matrix takes on a value of 1 when outlet or is an𝑄𝑄𝑙𝑙 𝑙𝑙outlet of lake , a value of𝑙𝑙 -1 when or is an inlet into lake , and 0 in all other cases.𝑐𝑐 When𝑢𝑢 the flow travel time𝑙𝑙 lag is not ignored,𝑐𝑐 as in𝑢𝑢 the “with lag” formulation𝑙𝑙 and in the simulation phase, Equation 22 is modified into Equation 23 for the lagged inflow lake, , to ignore inflows from upstream network flows. is instead used as the inflow in the current

𝑙𝑙𝑙𝑙stage and is calculated from the ignored inflows in the𝑅𝑅𝑡𝑡 previous stage, shown in Equation 24.

Equation 23

+ 𝐶𝐶𝐶𝐶 × , × + 𝑈𝑈𝑈𝑈 × , × = + + × 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑆𝑆𝑡𝑡+1 ��𝑄𝑄𝑐𝑐𝑐𝑐 𝐴𝐴𝑐𝑐𝑙𝑙 𝑐𝑐� 𝑑𝑑𝑑𝑑 ��𝑄𝑄𝑢𝑢𝑢𝑢 𝐴𝐴𝑢𝑢𝑙𝑙 𝑢𝑢� 𝑑𝑑𝑑𝑑 𝑆𝑆𝑡𝑡 𝑅𝑅𝑡𝑡 𝑄𝑄𝑙𝑙𝑙𝑙 𝑑𝑑𝑑𝑑 𝑐𝑐 𝑢𝑢

where , 0

𝐶𝐶𝐶𝐶 ⊂ 𝐶𝐶 𝐴𝐴𝑐𝑐𝑙𝑙𝑙𝑙 𝑐𝑐 ≥ 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑐𝑐 where , 0

𝑙𝑙𝑙𝑙 𝑢𝑢 Equation 24 𝐶𝐶𝐶𝐶 ⊂ 𝐶𝐶 𝐴𝐴𝑢𝑢 ≥ 𝑓𝑓𝑓𝑓𝑓𝑓 𝑎𝑎𝑎𝑎𝑎𝑎 𝑢𝑢

= 𝐶𝐶 × , 𝑈𝑈 × ,

𝑅𝑅𝑡𝑡+1 − ��𝑄𝑄𝑐𝑐𝑐𝑐 𝐴𝐴𝑐𝑐𝑙𝑙 𝑐𝑐� − ��𝑄𝑄𝑢𝑢𝑢𝑢 𝐴𝐴𝑢𝑢𝑙𝑙 𝑢𝑢� 𝑐𝑐 = , , = 1, 𝑢𝑢 , = 1

𝑓𝑓𝑓𝑓𝑓𝑓 𝑙𝑙 𝑙𝑙𝑑𝑑 𝐴𝐴𝑐𝑐𝑙𝑙 𝑐𝑐 − 𝐴𝐴𝑢𝑢𝑙𝑙 𝑢𝑢 − Lakes may also have multiple storage segments to allow for benefit and penalty values on different storage segments of the lake. Equation 25 shows storage bounds for each storage segment. These benefit or penalty values provide a way of incorporating water management objectives into the objective function.

Equation 25

, , , , , ,

𝑆𝑆𝑚𝑚𝑚𝑚𝑛𝑛𝑡𝑡+1 𝑙𝑙 𝑛𝑛 ≤ 𝑆𝑆𝑡𝑡+1 𝑙𝑙 𝑛𝑛 ≤ 𝑆𝑆𝑆𝑆𝑆𝑆𝑥𝑥𝑡𝑡+1 𝑙𝑙 𝑛𝑛

Where , , and , , are the minimum and maximum storage volumes for lake at

the start𝑆𝑆𝑆𝑆𝑆𝑆 of 𝑛𝑛time𝑡𝑡+1 𝑙𝑙 period𝑛𝑛 𝑆𝑆 𝑆𝑆 and𝑆𝑆𝑥𝑥𝑡𝑡 for+1 𝑙𝑙 the𝑛𝑛 storage segment . Whereas , , is the amount𝑙𝑙 of storage in segment 𝑡𝑡for the given lake and time period.𝑛𝑛 ∈ 𝑁𝑁𝑙𝑙 𝑆𝑆𝑡𝑡+1 𝑙𝑙 𝑛𝑛 𝑛𝑛 The water management objective in Equation 14 is entirely based on the storage segment values, as shown in Equation 26. Storage segment values must be configured as monotonically

decreasing functions of storage to ensure the objective function is convex. , is the water value for lake , storage segment . 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑟𝑟𝑙𝑙 𝑛𝑛

Equation 26 𝑙𝑙 𝑛𝑛

= 𝐿𝐿 𝑁𝑁𝑙𝑙 , , × ,

𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊𝑊 � � 𝑆𝑆𝑡𝑡+1 𝑙𝑙 𝑛𝑛 𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝐶𝑟𝑟𝑙𝑙 𝑛𝑛 𝑙𝑙 𝑛𝑛 Controlled flow variable limits correspond with the outlet outflow capability, shown in Equation 26.

Equation 27

𝑐𝑐 𝑐𝑐 𝑐𝑐 Where and represent𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑛𝑛 the≤ 𝑄𝑄 minimum𝑐𝑐 ≤ 𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 and𝑥𝑥 maximum outflow capability of

controlled𝑄𝑄𝑄𝑄𝑄𝑄𝑄𝑄 outlet𝑛𝑛𝑐𝑐 . All𝑄𝑄 h𝑄𝑄𝑄𝑄𝑄𝑄ydro𝑥𝑥 𝑐𝑐stations, the Notigi and Missi Falls control structure, and Nelson West Channel are𝑐𝑐 configured as controlled outlets. The Nelson West Channel outlet maximum outflow capability is also a function of upstream storage and has an additional constraint, Equation 28, to reflect the maximum outflow rating curve.

Equation 28

, × + 2 , , 𝑆𝑆𝑡𝑡 𝑙𝑙 𝑄𝑄𝑐𝑐𝑐𝑐 ≤ 𝜙𝜙𝑐𝑐 max 𝜃𝜃𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚 Where , and , are respectively the slope and intercept values of a linearly

approximated𝜙𝜙𝑐𝑐 𝑚𝑚𝑚𝑚𝑚𝑚 outflow𝜃𝜃 𝑐𝑐function𝑚𝑚𝑚𝑚𝑚𝑚 or rating curve.

Uncontrolled flow, , represents outflow from lakes through outlets that do not have a control

structure. Nelson East𝑄𝑄𝑢𝑢 𝑢𝑢channel is represented by an uncontrolled outlet. Equation 24 shows the outflow as a function of storage.

Equation 29

, = × + 2 𝑆𝑆𝑡𝑡 𝑙𝑙 𝑄𝑄𝑢𝑢𝑢𝑢 𝜙𝜙𝑢𝑢 𝜃𝜃𝑢𝑢 Where and are respectively the slope and intercept values of a linearly approximated rating

curve. 𝜙𝜙𝑢𝑢 𝜃𝜃𝑢𝑢

Historical minimums and maximums outflows for the 1978/79 to 2015/16 period and modeled outflow limits for the modeled lakes are shown in Table 6. Linear approximations for the Nelson West Channel and Nelson East Channel rating curves were provided by Manitoba Hydro. The Nelson West Channel minimum outflow of 0 kcfs represents the ability for Manitoba Hydro to reduce the outflow from Lake Winnipeg by raising the elevation of the Jenpeg forebay, reducing the gradient between the two lakes. The minimum of 0 kcfs also helps prevent the optimization model from violating the minimum storage bound during negative inflow periods. The Grand Rapids hydro station does not have operating limits on its outflow (Manitoba Hydro 2020). The Missi Falls minimum outflows and Notigi maximum outflows reflect current operating limits for these outlets, whereas the historical values were observed in periods prior to the current mode of operation (Manitoba Hydro 2020).

Table 4 - Outflow limits on modeled lake outlets

Lake Outlet Modeled Minimum Flow Modeled Max. Flow Historic Min Historic Max (kcfs) (kcfs) (kcfs) (kcfs)

Nelson East Channel As per rating curve As per rating curve 3.61 25.4 Lake Winnipeg Nelson West Channel 0 As per rating curve 12.35 152.7

Cedar Lake Grand Rapids 0.00 200.00 0.1 79.88

0.75 May 1 to Oct 31 4.25 Nov 1 to Nov 30 3.85 Dec 1 to Dec 15 Missi 100.00 0.48 79.55 Southern 3.25 Dec 16 to Jan 15 Indian Lake 3.125 Jan 16 to Jan 31 2.25 Feb 1 to Feb 28 1.75 Mar 1 to Apr 30

34.9 May 16 to Oct 31 Notigi 10 15.1 39.65 33.9 Nov 1 to Apr 30

5.3.4. Water Value Function Approximation The future water value is approximated by , a linear approximation of the future value

based on the state variables𝑓𝑓𝑡𝑡+1 at the end of the current𝛼𝛼 stage , and , and a set of water value function cuts. Equation 30 shows the approximation for𝑡𝑡 a𝑆𝑆 𝑡𝑡single+1 cut𝑅𝑅𝑡𝑡 +1 and inflow scenario . 𝑛𝑛

Equation𝑖𝑖 30

, 𝐿𝐿𝐿𝐿 × + × × 𝑙𝑙 𝑙𝑙 𝑙𝑙𝑙𝑙 𝛼𝛼𝑖𝑖 𝑛𝑛 ≤ ��𝜋𝜋𝑡𝑡+1 𝑆𝑆𝑡𝑡+1� 𝜋𝜋𝑡𝑡+1 𝑅𝑅𝑡𝑡+1 𝑑𝑑𝑑𝑑 𝑙𝑙 Where for lakes with storage state variables, is the water value function cut, and is

the duration𝐿𝐿𝐿𝐿 ⊂ of𝐿𝐿 the stage. 𝜋𝜋 𝑑𝑑𝑑𝑑

Water value function cuts are computed as shown in equations Equation 31 and Equation 32from the linear programming solution of Equation 14, using the dual values, , from all constraints that the state variables participated in. In this formulation, only Equation 22𝜆𝜆 includes the storage state variables and Equation 23 is also relevant for the “with-lag” approach.

Equation 31

= + 𝐶𝐶 × , × + 𝑈𝑈 × , × , 𝑙𝑙 𝑙𝑙 𝑆𝑆𝑡𝑡+1 𝑆𝑆𝑡𝑡 ��𝑄𝑄𝑐𝑐𝑐𝑐 𝐴𝐴𝑐𝑐𝑙𝑙 𝑐𝑐� 𝑑𝑑𝑑𝑑 ��𝑄𝑄𝑢𝑢𝑢𝑢 𝐴𝐴𝑢𝑢𝑙𝑙 𝑢𝑢� 𝑑𝑑𝑑𝑑 �𝜆𝜆1 𝑙𝑙� 𝑙𝑙 ∈ 𝐿𝐿𝐿𝐿 𝑐𝑐 𝑢𝑢

= + 𝐶𝐶 × , × + 𝑈𝑈 × , × , 𝑙𝑙𝑙𝑙 𝑙𝑙𝑙𝑙 𝑆𝑆𝑡𝑡+1 𝑆𝑆𝑡𝑡 ��𝑄𝑄𝑐𝑐𝑐𝑐 𝐴𝐴𝑐𝑐𝑙𝑙𝑙𝑙 𝑐𝑐� 𝑑𝑑𝑑𝑑 ��𝑄𝑄𝑢𝑢𝑢𝑢 𝐴𝐴𝑢𝑢𝑙𝑙𝑙𝑙 𝑢𝑢� 𝑑𝑑𝑡𝑡 �𝜆𝜆2 𝑙𝑙𝑙𝑙� 𝑐𝑐 𝑢𝑢

Equation 32

= , 𝑙𝑙 𝜋𝜋𝑡𝑡 𝜆𝜆1 𝑙𝑙

= , 𝑙𝑙𝑙𝑙 𝜋𝜋𝑡𝑡 𝜆𝜆2 𝑙𝑙𝑙𝑙

= 𝐿𝐿𝐿𝐿 × × × 0 𝑙𝑙 𝑙𝑙 𝑙𝑙𝑙𝑙 𝜋𝜋𝑡𝑡 𝑣𝑣𝑡𝑡 − � 𝜋𝜋𝑡𝑡 𝑆𝑆𝑡𝑡 − 𝜋𝜋𝑡𝑡 𝑅𝑅𝑡𝑡 𝑑𝑑𝑑𝑑 𝑙𝑙 Equation 32 shows how the dual values from the constraints are used, in conjunction with the objective function value, and the state variable discretization for storage, , and delayed 𝑙𝑙 inflow, , to determine the𝑣𝑣𝑡𝑡 water value function cut intercept value, . 𝑆𝑆𝑡𝑡 0 𝑅𝑅𝑡𝑡 𝜋𝜋𝑡𝑡 Water value function cuts are calculated following the backwards dynamic programming procedure. Cut values for starting storage in period are determined for all discretized values of

, , and starting in period = , proceeding backwards𝑡𝑡 until the first period = 1. Water 𝑆𝑆value𝑡𝑡 𝑅𝑅𝑡𝑡 function𝑖𝑖 cuts calculated 𝑡𝑡in 𝑇𝑇+ 1 are used as the ending period values in period𝑡𝑡 . The procedure is repeated by setting the𝑡𝑡 water value function cuts at the end of period to the 𝑡𝑡water value function cuts at the beginning of period = 1 for a set number of iterations.𝑇𝑇 𝑡𝑡 Only a subset of the water value function cuts is utilized in the linear programming problem to reduce the problem size following an approach similar to (Cote and Arsenault 2019). Cuts are added up to a set maximum number of cuts or until a maximum tolerable error in the water value function approximation has been reached. The error in the water value function approximation

is calculated at all the discretization points after each cut is added. The next cut added is at the point where the maximum error was calculated. Figure 13 shows an example of the water value function cut selection process.

( , )

𝑓𝑓𝑡𝑡 𝑆𝑆𝑡𝑡 𝑗𝑗

𝑆𝑆𝑡𝑡

Figure 13 - Example of water value function cut selection, shown for a single reservoir. The 'true' water value function is shown in blue. Measurements of the water value are calculated at the discretized storage points, shown as dashed lines. Water value function cuts are the tangential lines. The maximum error of the current approximation is shown as the red double-sided arrow. The next water value function cut to be added to the approximation is at the discretization with the highest water value function error, which in this case, is at the red double-sided arrow.

5.3.5. Inflow Scenario Selection The inflow scenarios for the analysis include 38 historical scenarios from 1978 to 2015 from the LTFD dataset (Manitoba Hydro 2017). The start of the period (1978) corresponds to the first full year of available historical data after the completion of the Lake Winnipeg Regulation and Churchill River Diversion projects.

Figure 14 expands on the LTFD dataset presented in Figure 5 by showing the historical inflow traces for all inflow locations in Figure 4, the network studied. It is worth noting that inflows local to a reservoir are calculated based on observed inflows, observed changes in reservoir storage, and observed outflows, and will incorporate any errors in measurement as well as other hydrological processes, such as evaporation, ground water exchange. Several traces have net negative inflows for Cedar Lake, the ‘SRIAO’ inflow location, and Lake Winnipeg, the ‘LWPIAO, NR0’ inflow location.

450 120 400 Total Inflow SRIAO 100 350 300 80 250 60 200 150 40 Inflow (kcfs)

100 Inflow (kcfs) 20 50 0 0 -50 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 10/1 11/1 12/1

-20 10/1 11/1 12/1 Month/Day Month/Day

200 120 LWPIAO, NR0 SF 150 100 80 100 60 50

Inflow (kcfs) 40

Inflow (kcfs) 0 20

-50 0 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 10/1 11/1 12/1 -100 10/1 11/1 12/1 Month/Day Month/Day

120 30 CRIAO, BR1 BR3,BR4,NR3,NR4 100 25 80 20 60 15 Inflow (kcfs) Inflow (kcfs) 40 10 20 5 0 0 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 10/1 11/1 12/1 10/1 11/1 12/1 Month/Day Month/Day

16 8 NR1, NR2 BR2 14 7 12 6 10 5 8 4 6 3 Inflow (kcfs) Inflow (kcfs) 4 2 2 1 0 0 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 10/1 11/1 12/1 10/1 11/1 12/1 Month/Day Month/Day

Figure 14 - Historical inflow traces and average as a black line from 1978/79 to 2015/16 by location and total of all locations.

Selection of the scenarios for the water value function is based on the quantiles of the total system inflow for a year that corresponds to the number of scenarios selected for optimization. Figure 15 shows an example of the percentile ranking selection when 8 scenarios and 4 scenarios are selected.

80000 80000 70000 70000 d) d) - 60000 - 60000 50000 50000 40000 40000 30000 30000 20000 20000 Inflow volume (kcfs Inflow volume (kcfs 10000 10000 0 0 0% 50% 100% 0% 50% 100% (a) Percentile (b) Percentile

Figure 15 – Scenario total system inflow volume, highlighting the quantile inflow selection method for 8 scenarios (a) and 4 scenarios(b)

The stability of the solution with respect to the scenario selection process is tested by comparing the objective function values from solutions with varying number of scenarios. Adding more scenarios brings the approximation closer to the true distribution of possible inflows and generally improves stability (Kaut and Wallace 2007).

5.3.6. Transition Probability Calculation Transition probabilities are used to represent the probability of an inflow scenario, , occurring

in a stage + 1 given scenario occurred in stage . The method uses a linear least𝑘𝑘 -square procedure and𝑡𝑡 Bayes Theorem from𝑗𝑗 Faber and Stedinger𝑡𝑡 (2001). The Bayes theorem can be used to represent the conditional probability of a future inflow scenario occurring in period +

1 given an inflow volume in period as in Equation 32. 𝑘𝑘 𝑡𝑡

Equation 33 𝑡𝑡

Pr , × Pr[ ] Pr[ | ] = Pr × Pr [ ] �𝑞𝑞𝑡𝑡�𝑞𝑞𝑡𝑡+1 𝑘𝑘� , 𝑘𝑘 𝑞𝑞𝑡𝑡+1𝑘𝑘 𝑞𝑞𝑡𝑡 𝐼𝐼𝑜𝑜 ∑𝑗𝑗=1 �𝑞𝑞𝑡𝑡�𝑞𝑞𝑡𝑡+1 𝑗𝑗� 𝑗𝑗

Without any additional information, any given scenario is assumed equally likely to occur, i.e.

Pr[ ] = . To determine Pr , , the volume of flow in period for scenario , , , is first 1 𝑜𝑜 𝑡𝑡 𝑡𝑡+1 𝑘𝑘 𝑗𝑗 𝑡𝑡 related𝑗𝑗 to𝐼𝐼 the volume of flow�𝑞𝑞 in� 𝑞𝑞period� + 1 for scenario , , in a 𝑡𝑡linear model as𝑗𝑗 in𝑞𝑞 Equation

34. 𝑡𝑡 𝑗𝑗 𝑞𝑞𝑡𝑡+1 𝑗𝑗

Equation 34

, = , , + , +

𝑞𝑞𝑗𝑗 𝑡𝑡 𝜙𝜙1 𝑡𝑡𝑞𝑞𝑡𝑡+1 𝑗𝑗 𝜙𝜙𝑜𝑜 𝑡𝑡 𝜖𝜖𝑡𝑡 The probability of the volume of flow in period for scenario occurring given the volume of

flow in period + 1 and scenario , Pr [ , |𝑡𝑡 , ], can be𝑘𝑘 determined from a Normal distribution, given𝑡𝑡 that the residuals 𝑗𝑗of the𝑞𝑞 least𝑘𝑘 𝑡𝑡 𝑞𝑞𝑡𝑡-+square1 𝑗𝑗 fitting of Equation 34 are Normally distributed (Faber and Stedinger 2001). Figure 16 shows an example of the regression and the

division of the probability space for determining Pr [ , | , ], which is used in Equation 34 to

determine the transition probability for each inflow 𝑞𝑞scenario,𝑘𝑘 𝑡𝑡 𝑞𝑞𝑡𝑡+1 𝑗𝑗 , ,.in the optimization scenario set, . 𝑞𝑞𝑡𝑡 𝑖𝑖 𝐼𝐼𝑜𝑜 3500

3000 days)

- 2500

2000

1500

flow volume (kcfs 1000 t+1 q 500

0 500 1000 1500 2000 2500 3000

qt, flow volume (kcfs-days)

Figure 16 – Example of the linear regression approach for determining [ , | , ]. (a) shows the linear regression between and . Here, four scenarios ( = 4) are included, represented by the highlighted circles. (b) shows the normal distribution 𝑃𝑃𝑃𝑃 𝑞𝑞𝑘𝑘 𝑡𝑡 𝑞𝑞𝑡𝑡+1 𝑗𝑗 of the residuals for the linear regression of the third scenario, , , with mean , , + , . The distribution is divided into 𝑡𝑡 𝑡𝑡+1 𝑜𝑜 𝑞𝑞segments𝑞𝑞 at the scenario values from𝐼𝐼 (a). 𝑞𝑞𝑡𝑡 3 𝜙𝜙1 𝑡𝑡𝑞𝑞𝑡𝑡+1 3 𝜙𝜙0 𝑡𝑡 5.4. Analysis Setup The analysis was conducted for the two lag approaches, “with-lag” and “no-lag”, semi-monthly for one year starting on and ending on April 1st. Optimization was conducted over the selected

subset of inflow scenarios, and the simulation, the evaluation of the optimal solution, was conducted over all 38 inflow scenarios. The sensitivity of the economic value of the solution, total energy deficits, Lake Winnipeg maximum outflow license violations, and computation time relative to the number of optimization sequences, state variable discretization, and storage benefit and penalty values was evaluated. The economic objective is the sum of the objective function value for all simulated scenarios with the water management objective values removed. The energy deficit is the sum of all energy deficits from the simulation. Lake Winnipeg license violations occur when Lake Winnipeg is at or above 715.00 ft and outflow the lake outflow is not at maximum. Good solutions are taken to mean having non-negative economic value, no energy deficits and no license violations. Historic operating decisions were simulated in the model by forcing historic reservoir releases at Cedar Lake, Lake Winnipeg and Southern Indian Lake and then compared against the results from the SSDP model. An overview of the process for each method and the historic reservoir releases is shown in Figure 16.

Ignored Lag With Lag Historic

No Lag Formulation With Lag Formulation 0 for all …

, , State Variables: State Variables:

1 1 N/A - Lake Winnipeg Storage Lake Winnipeg Storage T , Cedar Lake Storage Cedar Lake Storage T

= Southern Indian Lake Storage

t Southern Indian Lake Storage Stephens Lake Lagged Inflows scenarios For all optimization Optimization Phase Optimization

Ignored Lag With Lag Historic Water Value Water Value Releases Functions Functions

, , No Lag Reservoir releases With Lag With Lag Sub- Sub- Sub- … T Problem Problem Problem 1 1 ,

0 With Delay = t+1, t+1, t+1, Stephens Lake Sub- Ending storage to Ending storage to lagged inflow Problem starting storage, starting storage, updated Stephens Lake Stephens Lake for all t For all scenarios Simulation Phase Simulation t+1, lagged inflow lagged inflow Ending storage to updated updated starting storage

Results Results Results

Figure 17 – Overview of the optimization process and simulation process for the “no-lag”, “with-lag” and Historic cases

The SSDP modeling platform provided by Manitoba Hydro, modified to include the delayed inflow state variable was implemented with a combination of the Python, Cython, and C++ programing languages, as well as the CLP LP optimizer from COIN-OR and the Open MPI package for parallel processing. Cython facilitates direct access to the CLP C++ class interface from the main python program as shown in Figure 18, which allows the lower level mathematical operations in the LP solver to run more efficiently (Behel, et al. 2011). Open MPI facilitates splitting the task of solving the LP for each state in a given stage across the available processors, aggregating the results for approximating the water value function, and then distributing the water value function cuts back to each processor for the next stage or iteration as shown in Figure 19. Open MPI is accessed with the MPI for Python module (Dalcin, et al. 2011). CLP is an open source LP solver available

from COIN-OR Foundation, see (Lougee-Heimer 2003). The analysis was completed on a RHEL 7

Linux server with 32 GB of RAM and utilizing 30 Intel(R) Xeon(R) Platinum 8180 CPU @ 2.50GHz processors.

MBHydroModel -lpmod : lpModel

* 0..* -lpmod lpModel -lpsolver : CyCLP ...

* 0..1 -lpsolver

CyCLP -clpSolver* : clpSolver ...

* 0..1 -clpSolver*

clpSolver ...

Figure 18 - Conceptual UML Class Diagram describing the association between the MBHydroModel class through to the CLP solver. The CyCLP class is compiled C++ and python code and directly communicates with both the python based lpModel class and the clpSolver class that executes CLP.

The LP in each stage has 177 variables and 413 constraints prior to the water value function cuts being added. Up to 2000 water value function cuts per optimization sequence were added or until the water value function cuts error tolerance of 200 was satisfied.

Optimization phase with parralel processing

Process 0 Process 1 ... Process N

Build all LPs Build all LPs Build all LPs

Populate LP Populate LP Populate LP with assigned with assigned with assigned States States States

Solve Solve Solve Assigned LPs Assigned LPs Assigned LPs

Send/Gather Send Dual Send Dual Repeat for all Dual Values Values Values stages and all iterations Determine Water Value Wait Wait Function Cuts

Broadcast/Receive Receive Water Receive Water Water Value Value Value Function Cuts Function Cuts Function Cuts

Update LP Update LP Update LP Water Value Water Value Water Value Function Cuts Function Cuts Function Cuts

Figure 19 - Description of the optimization phase with parallel processing. Process 0 is the master process that completes the non-parallel tasks. The gather and broadcast are MPI calls that allow the multiple processes to collaborate by gathering the dual values from the individual processes and then broadcasting the water value function cuts for the next stage or iteration.

6. Results and Discussion 6.1. Historic Results Although applying historical operating decisions to the current system is not ideal, it provides a baseline for comparing the success of the SSDP derived operating policies. Simulating the historic operating decisions resulted in an economic objective of $M -5,835.84 and 1296.17 GWh of energy deficits. Each MWh of energy deficit has a cost of $5,000. Therefore, the 1296.17 GWh energy deficit had a -$M 6,480.85 impact on the economic objective.

Energy deficits occurred in four of the simulated inflow years, highlighted in Table 6. All energy deficits occur between October 16th and April 1st. Figure 20 highlights the elevation traces and Stephens Lake inflows for the energy deficit years as well as the average of all the simulated years from 1978/79 to 2015/16. Lake elevations generally rise in the spring and summer and discharge in the fall and winter. The 2003/04 year is an exception, where Lake Winnipeg does not increase in elevation through the spring and summer, and Cedar Lake does not discharge in the fall and winter. The Stephens Lake inflows in the fall and winter of the simulated 2003/04 flow year are the lowest of the energy deficit years. Low inflows into Stephens Lake greatly reduce the energy generation capability of the Manitoba Hydro system, leading to high energy deficits.

The 2003/04 operating decision highlights the challenge of using historic operating decisions to simulate operations for future systems. Energy demand in the province totaled 21,890 GWh in 2003/04, whereas the simulated system has an annual energy demand of 26,388 GWh (Manitoba Hydro 2011). The growth in energy demand between 2003/04 and the simulated year more than account for the energy deficits.

Appendix A2 contains complete hydrological and energy system results for the entire simulation period.

Table 5 - Energy deficits from simulated historic operating decisions by flow year

Flow Year Energy Deficit (GWh) Percent of total (%)

1987/88 29.73 2% 1988/89 185.62 14% 1990/91 8.23 1% 2003/04 1072.59 83% Total 1296.17

717 842

716 840

715 838

714 836 1987/88 713 834 Lake Elevation (ft) Lake Elevation (ft) 1988/89 1990/91 712 832 2003/04 Average, 1978/79 to 2015/16 711 830 (a) 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 (b) 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 10/1 11/1 12/1 10/1 11/1 12/1

848 160

140 847 120 846 100

845 80 Monthly Inflow (kcfs) 60 844 Lake Elevation (ft) 40 843

Average Semi - 20

842 0

(c) 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 (d) 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 10/1 11/1 12/1 10/1 11/1 12/1 Figure 20 - Lake elevation traces from the simulated historic operating decisions for (a) Lake Winnipeg, (b) Cedar Lake and (c) Southern Indian Lake and (d) average inflows for Stephens Lake. Traces are shown for the flow years with energy deficits and the average of all simulated flow years.

6.2. “No-Lag” Approach Results The “no-lag” approach results are from the model that did not include a representation of the travel time lag in the optimization phase but did include the travel time lag in the simulation

phase. Table 7 highlights solutions from the “No-lag” approach with varying storage discretization and water management objective storage values. Eight iterations and eight inflow sequences, = 8, were used unless otherwise noted. Solutions marked ‘does not solve’ either failed to

𝐼𝐼generate𝑜𝑜 a suitable water value function set or had an infeasible solution during simulation. Infeasible solutions resulted when there was a combination of inflows and minimum outflows that would result in storage going below the minimum stage. This was offset in some cases by having using higher reserve storage benefit values to generate water value functions that avoid operating the reservoirs at low storage during periods of low or negative inflows. Increasing the reserve storage buffer or allowing storage to go below the minimum stage limit are other approaches that could be used to alleviate infeasibilities.

The approach was able to produce solutions with non-negative economic objective values, an improvement over the historic operating decisions by over $M 6,000 as outlined in Table 7. Good solutions, taken here to mean there is a non-negative economic value, no energy deficits and no license violations, were only found when the minimum number of Lake Winnipeg discretization points were used and over five discretization points for Southern Indian Lake and Cedar Lake were used. The solutions that performed better than the historic operating decisions solved in a suitable duration of time for operational decision making.

The effect of changing storage values on the results is explored further in Table 8, using a 7-3-7 discretization points pattern for the Cedar Lake, Lake Winnipeg and Southern Indian Lake state variables. Setting the Cedar Lake reserve storage value in the order of magnitude of 1.0E+10 $/kcfs-d produced good solutions. The Lake Winnipeg reserve storage value produced good solutions when at or above 3.0E+7 $/kcfs-d, and the Lake Winnipeg maximum outflow penalty had good solutions when at the 1.5E+04 $/kcfs-d order of magnitude. The solution with the highest economic objective value was found with the Cedar Lake reserve storage value of 1.0E+10 $/kcfs-d, Lake Winnipeg reserve storage value of 5.0E+07 $/kcfs-d and the Lake Winnipeg outflow penalty of 1.5E+04 $/kcfs-d. This solution had an economic objective function value of $M 1,018.43 and is used as a further comparison against the “with-lag” scenarios. Additional hydrological and energy results for this scenario are presented in Appendix A2.

Table 6 – “No-lag” approach with varying number of discretization points for the state variables and storage values. The Lake Winnipeg Max Outflow Penalty value was 1.5E+04 for all cases

Reserve Storage Number of Discretization Points ($/kcfs-d) Economic Energy Number of Cedar Lake Southern Cedar Lake Objective Deficit License Computation Lake Winnipeg Indian Lake Lake Winnipeg ($M) (GWh) Violations Time (s)

3 3 2 1.E+07 5.E+04 does not solve 3 3 2 1.E+06 1.E+05 does not solve 3 3 2 1.E+07 1.E+05 -3,517.02 901.14 138 31 3 3 2 1.E+07 1.E+05 -4,890.71 1081.8 0 31

4 3 2 1.E+07 1.E+05 -4,550.81 1012.9 0 31 5 3 2 1.E+07 1.E+05 -3,529.43 396.09 0 33 6 3 2 1.E+07 1.E+05 -3,164.71 745.02 0 34 6 3 2 1.E+07 1.E+04 -1,496.30 432.44 0 34 6 3 2 1.E+07 5.E+4 does not solve 7 3 2 1.E+07 1.E+04 -1,246.52 388.52 0 34

4 4 3 1.E+07 5.E+04 does not solve 4 4 4 1.E+07 5.E+04 does not solve 5 5 5 1.E+07 5.E+04 -3,656.53 972.62 0 58 6 6 6 1.E+07 5.E+04 -2,674.55 779.36 0 221 7 7 7 1.E+07 5.E+04 -3,096.52 866.29 0 432 7 7 7 1.E+07 1.E+04 -2,896.34 830.26 0 429

4 3 2 1.E+07 1.E+05 -4,550.81 1012.9 0 31 4 4 5 1.E+07 5.E+04 -2,896.34 830.26 0 48 5 5 5 1.E+07 5.E+04 -3,656.53 972.62 0 67

3 3 3 1.E+07 5.E+04 does not solve 4 3 4 1.E+07 5.E+04 does not solve 5 3 5 1.E+07 5.E+04 935.33 0 0 47 6 3 6 1.E+07 5.E+04 974.65 0 0 65 7 3 7 1.E+07 5.E+04 1,018.43 0 0 136 8 3 8 1.E+07 5.E+04 845.14 0 0 209

Table 7 – “No-lag” approach sensitivity to storage values for the 7/3/7 discretization

Reserve Storage Value ($/kcfs-d) Lake Winnipeg Maximum Energy Number of Outflow Penalty Economic Deficit License Cedar Lake Lake Winnipeg ($/kcfs-d) Objective ($M) (GWh) Violations

1.0E+09 5.0E+07 1.5E+04 does not solve 1.0E+10 5.0E+07 1.5E+04 1,018.43 0 0

1.0E+11 5.0E+07 1.5E+04 776.58 44.29 0 1.0E+12 5.0E+07 1.5E+04 does not solve

1.0E+10 5.0E+07 1.5E+03 does not solve 1.0E+10 5.0E+07 1.5E+04 1,018.43 0 0 1.0E+10 5.0E+07 1.5E+05 -55.57 103.52 0

1.0E+10 1.0E+07 1.5E+04 does not solve 1.0E+10 1.0E+08 1.5E+04 773.14 41.58 0 1.0E+10 1.0E+09 1.5E+04 709.20 56.93 0 1.0E+10 1.0E+10 1.5E+04 6.45 193.32 0

1.0E+10 2.0E+07 1.5E+04 693.63 56.88 0 1.0E+10 3.0E+07 1.5E+04 997.94 0 0 1.0E+10 4.0E+07 1.5E+04 1,006.51 0 0 1.0E+10 5.0E+07 1.5E+04 1,018.43 0 0 1.0E+10 6.0E+07 1.5E+04 587.98 76.59 0

The sensitivity of the results with respect to the number of inflow sequences is shown in Table 9, with other parameters the same as in Table 7. Selecting the number of inflow sequences between 2 and 10 was found to vary the economic objective function from $M (560.23) to $M 1,034.52, and the energy deficit varied from 0 to 338.8 GWh, all superior to the historic operating decision results. The number of inflow sequences and the storage values also have a combinatory impact. Comparing the 8 inflow sequences and 9 inflow sequences using the storage values of 1.0E+07 $/kcfs-d, 1.0E+4 $/kcfs-d and 1.5E+04 $/kcfs-d for Cedar Lake, Lake Winnipeg and the Lake Winnipeg maximum outflow penalty respectively results in an economic value of $M 1,034.52 for 9 inflow sequences but fails to solve for the 8 inflow sequence case. Using storage values of 1.0E+07 $/kcfs-d, 5.0E+4 $/kcfs-d and 1.5E+04 $/kcfs-d for Cedar Lake, Lake Winnipeg and the Lake Winnipeg maximum outflow penalty respectively results in an economic objective value of

$M 201.14 for the 9 inflow sequences case and 129.16 GWh of energy deficits, whereas the 8 inflow sequence case has an economic value of $M 1018.43 and no energy deficits.

Increasing the number of scenarios did not improve the stability of the economic objective function results, contrary to what was expected. The scenario selection method can partially account for this. Scenarios were selected based on the quantile distribution of only a single metric, the total volume of inflows into the system over the year. However, there are other factors that could be considered when selecting a scenario, such as the spatial variability of inflows across the system and the temporal variability of inflows within the year. Selecting scenarios based only on the total inflow volume can result in the scenario sets including or excluding variability across these other factors in a non-systematic way. This can result in a scenario sets with increasing representation of the total inflow volume as the number of scenarios increase, but highly variable representation of the other factors, which may be leading to instability in the solution as scenarios are increased.

Table 8 – Impact on the “No-lag” approach results due to the number of inflow sequences selected for optimization. The Lake Winnipeg Max Outflow Penalty value was 1.5E+04 in all cases.

Reserve Storage Value ($/kcfs-d) Energy Number of Economic Deficit License Computation Cedar Lake Lake Winnipeg Objective ($M) (GWh) Violations Time (s)

𝑜𝑜 1 𝐼𝐼 1.0E+07 1.0E+04 does not solve 1 1.0E+07 5.0E+04 does not solve 1 1.0E+07 1.0E+05 does not solve 2 1.0E+07 1.0E+04 178.19 0 0 37 3 1.0E+07 1.0E+04 -560.23 338.8 0 52 4 1.0E+07 1.0E+04 260.70 61.01 0 68 5 1.0E+07 1.0E+04 575.17 38.69 0 82 6 1.0E+07 1.0E+04 666.80 10.94 0 99 7 1.0E+07 1.0E+04 -422.54 254.89 0 113 8 1.0E+07 1.0E+04 does not solve 8 1.0E+07 2.0E+04 693.63 56.88 0 141 8 1.0E+07 2.5E+04 756.77 14.91 0 137 8 1.0E+07 5.0E+04 1,018.43 0 0 136 9 1.0E+07 1.0E+04 1,034.52 0 0 148 9 1.0E+07 5.0E+04 201.14 129.16 0 161 10 1.0E+07 1.0E+04 360.03 29.2 0 169

Figure 21 show the economic metric varying based on the number of iterations. Discretization points followed the 7-3-7 Cedar Lake, Lake Winnipeg, Southern Indian Lake pattern, with reserve storage values of 1.0E+07 $/kcfs-d and 5.0E+04 $/kcfs-d for Cedar Lake and Lake Winnipeg, and an outflow penalty value of 1.5E+04 $/kcfs-d. The results converged quickly after 3 iterations, but continued to vary from iteration to iteration, with a standard deviation of $M 162 for the economic metric from iteration three through to iteration 10. The results do not converge as additional iterations are added. If the impact of the energy deficits is removed, the standard deviation for the economic metric modified to exclude energy deficits becomes $M 17.31, suggesting the energy deficits have a large impact on the variability of the economic objective function value. Figure 22 and Figure 23 highlight the elevation and Stephens Lake inflow traces from cases with 8 and 10 iterations on average from all inflow scenarios and in the 2003/04 simulated inflow year. The 10 iterations case had 49.24 GWh of energy deficits total, all of which occurred in the 2003/04 flow year. The energy deficits were split between the first half of November and the first half of December. The average traces for the 8 iteration case and the 10 iteration case in Figure 22 are nearly indistinguishable from each other, whereas the traces in Figure 23 for the 2003/04 simulated inflow year show the 8 iteration case utilizing more Lake Winnipeg storage in the late fall, conserving more Cedar Lake storage that is eventually utilized in the late fall and early Winter period, and resulting in no energy deficits.

2,000

1,000

-1,000

-2,000

-3,000 Economicobjective value ($M)

-4,000

-5,000 0 5 10 15 20 25 30 35 Number of iterations

Figure 21 – Impact of number of iterations on the economic objective for the ”no-lag” approach

717 844

716 842 840 715 838 714 836 10 Iterations 713 834

Lake Elevation (ft) Lake Elevation (ft) 8 Iterations 712 832 711 830 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 10/1 11/1 12/1 (a) 10/1 11/1 12/1 (b) 848 160 140 847 120 846 100

845 Monthly Inflow 80 - (kcfs) 60 844

Lake Elevation (ft) 40 843 20

842 Average Semi 0 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 (c) 10/1 11/1 12/1 (d) 10/1 11/1 12/1 Figure 22 - Lake elevation traces from the 8 and 10 iteration results for (a) Lake Winnipeg, (b) Cedar Lake and (c) Southern Indian Lake and (d) average inflows for Stephens Lake. Traces are shown for the average of all flow years.

717 842

716 840

715 838

714 836 8iter 713 834 10iter Lake Elevation (ft) Lake Elevation (ft) 712 832

711 830 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 (a) 10/1 11/1 12/1 (b) 10/1 11/1 12/1 848 120

847 100

846 80

845 Monthly Inflow 60 - (kcfs) 844 40 Lake Elevation (ft) 843 20

842 Average Semi 0 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 (c) 10/1 11/1 12/1 (d) 10/1 11/1 12/1 Figure 23 - Lake elevation traces from the 8 and 10 iteration results for (a) Lake Winnipeg, (b) Cedar Lake and (c) Southern Indian Lake and (d) average inflows for Stephens Lake. Traces are from the 2003/04 inflow year.

The discrepancy for the “no-lag” approach between the optimization phase, where the lag is ignored and the simulation phase where the lag is incorporated also contributes to the energy deficits in the 10-iteration case. Figure 24 highlights the lagged outflows, Stephens Lake inflows, and the energy deficits during the fall and early winter of the simulated 2003/04 flow year. Energy deficits occur in the first half of November and first half of December in the 10-iteration case in the 2003/04 simulation year, when there is insufficient hydro and thermal generations, and import is required to meet electrical demand. The outflows at the lagged flow locations, Kelsey and Wuskwatim, were increased during the first half of November and the first half of December; however, the lagged outflows do not arrive at Stephens Lake until the following time step. The reservoir operating decision for Cedar Lake is also made assuming no lag and does not release enough outflow to account for the shortfall in supply.

Energy deficits Stephens Lake Network inflow Wuskwatim plus Kelsey outflows 120 35

100 30

25 80 20 60 15 Flow (kcfs) 40 10 Energy deficit (GWh)

20 5

0 0 1/1 9/1 9/16 10/1 11/1 12/1 10/16 11/16 12/16

Figure 24 - Outflows and lagged inflows during the 2003/04 simulated inflow year for the 10-iteration case. Energy deficits occur in the timestep immediately prior to inflows at Stephens Lake increasing from lagged outflows.

6.3. ”With-Lag” Case Results The “with-lag” approach results used the model that incorporated the travel time lag as a lagged inflow state variable in the optimization phase. Table 10 highlights results where the lagged inflow representation was used with varying storage discretization and water management objective storage values. The lagged inflow column in Table 10 showcases the number of lagged inflow discretization points used. Solutions that use only three Lake Winnipeg segments and more than four Cedar Lake and Southern Indian lake segments generated good solutions with non-negative economic values, no energy deficits and no license violations. The time to solve was significantly longer than the ”no-lag” approach, with over an hour of computation time for the cases with the most state variable discretization points (see last columns of Table 10). However, the case that had the highest economic objective value took less than 15 minutes to solve for the 7-3-7-3 discretization pattern for Cedar Lake, Lake Winnipeg, Southern Indian Lake, and lagged inflow.

The effect of different storage values to the “with-lag” approach results are show in Table 11. Good solutions were found when the reserve storage values were 1.0E+06 $/kcfs-d for Cedar

Lake, the reserve storage values were between 1.0E+01 $/kcfs-d and less 1.0E+05 $/kcfs-d for Lake Winnipeg and a maximum outflow penalty value of 1.0E+04 $/kcfs-d or greater for Lake Winnipeg were used. The “with-lag” approach with the largest economic objective had a Cedar Lake reserve storage value of 1.0E+06 $/kcfs-d, a Lake Winnipeg reserve storage value of 1.0E+03 $/kcfs-d and a Lake Winnipeg maximum outflow penalty value of 1.5E+04 $/kcfs-d and produced an economic objective vale of $M 1,088.91. Additional hydrological and energy system results for the entire period for this scenario are presented in Appendix A2.

Table 9 -“With-lag” approach with varying number of discretization points for the state variables and storage values. The Lake Winnipeg Ma Outflow Penalty value was 1.5E+04 for all cases.

Reserve Storage Number of Discretization Points ($/kcfs-d)

Lake Souther Lake Energy Number of Cedar Lagged Winnipe n Indian Cedar Winnipe Economic Deficit License Computation Lake Inflow g Lake Lake g Objective ($M) (GWh) Violations Time (s)

3 3 2 3 1.0E+0 1.0E+04 does not solve 6 3 3 2 3 1.0E+0 5.0E+04 does not solve 6 3 3 2 3 1.0E+0 1.0E+05 does not solve 6 3 3 2 3 1.0E+0 1.0E+05 does not solve 7

4 4 3 3 1.0E+0 1.0E+04 does not solve 6 4 4 3 3 1.0E+0 1.0E+04 -10,152.81 2240.65 0 106 7

4 3 4 4 1.0E+0 1.0E+04 does not solve 5 4 3 4 4 1.0E+0 1.0E+04 794.18 13.76 0 182 6

5 3 5 3 1.0E+0 1.0E+04 683.67 49.93 0 236 6 5 3 5 4 1.0E+0 1.0E+04 903.50 0 0 427 6 5 3 5 5 1.0E+0 1.0E+04 870.30 5.72 0 570 6 5 3 5 6 1.0E+0 1.0E+04 880.57 0 0 758 6

5 4 5 6 1.0E+0 1.0E+04 -2,925.52 803.07 0 1002 6

6 3 6 3 1.0E+0 1.0E+04 646.42 57.36 0 440 6 6 3 6 4 1.0E+0 1.0E+04 892.56 2.52 0 827 6 6 3 6 5 1.0E+0 1.0E+04 889.04 5.32 0 1252 6 6 3 6 6 1.0E+0 1.0E+04 918.62 0 0 1795 6

7 3 7 3 1.0E+0 1.0E+04 942.87 0 0 824 6 7 3 7 4 1.0E+0 1.0E+04 912.36 0 0 1588 6 7 3 7 5 1.0E+0 1.0E+04 508.12 83.39 0 2414 6 7 3 7 6 1.0E+0 1.0E+04 921.24 0 0 3670 6

Table 10 – “With-lag” approach sensitivity to storage values

Reserve Storage Number of Discretization Points ($/kcfs-d)

Southern Lake Winnipeg Max. Economic Energy Number of Cedar Lake Indian Lagged Cedar Lake Outflow Penalty Objective Deficit License Computation Lake Winnipeg Lake Inflow Lake Winnipeg ($/kcfs-d) ($M) (GWh) Violations Time (s)

7 3 7 3 1.0E+09 1.0E+04 1.5E+04 does not solve 7 3 7 3 1.0E+08 1.0E+04 1.5E+04 475.91 92.61 0 839 7 3 7 3 1.0E+07 1.0E+04 1.5E+04 461.83 96.68 0 861 7 3 7 3 1.0E+06 1.0E+04 1.5E+04 942.87 0 0 824 7 3 7 3 1.0E+05 1.0E+04 1.5E+04 does not solve

7 3 7 3 1.0E+06 1.0E+01 1.5E+04 does not solve 7 3 7 3 1.0E+06 1.0E+02 1.5E+04 618.95 0 0 827 7 3 7 3 1.0E+06 1.0E+03 1.5E+04 1088.91 0 0 833 7 3 7 3 1.0E+06 1.0E+04 1.5E+04 942.87 0 0 824 7 3 7 3 1.0E+06 5.0E+04 1.5E+04 955.26 0 0 841 7 3 7 3 1.0E+06 1.0E+05 1.5E+04 943.90 0 0 833 7 3 7 3 1.0E+06 1.0E+06 1.5E+04 962.36 0.24 0 852 7 3 7 3 1.0E+06 1.0E+07 1.5E+04 424.11 43.69 0 845

7 3 7 3 1.0E+06 1.0E+04 5.0E+03 334.62 166.21 35 854 7 3 7 3 1.0E+06 1.0E+04 1.0E+04 426.41 116.27 0 860 7 3 7 3 1.0E+06 1.0E+04 1.5E+04 942.87 0 0 824 7 3 7 3 1.0E+06 1.0E+04 1.5E+05 424.11 43.69 0 836

Figure 25 shows the results of the “with-lag” approach with increasing number of iterations. The discretization pattern and storage values were the same as the case with the largest economic objective from Table 11. The model appears to converge rapidly after six iterations, with a standard deviation of the economic objective value from iteration six through 10 of $M 15.63. The results are more stable than the ”no-lag” approach which had a standard deviation of $M 162, a 90% reduction. The inclusion of the lag as a lagged inflow state variable within the optimization phase allows for the same representation of the system in both the optimization and simulation phase. This contrasts with the “no-lag” case, which has a discrepancy in system representation between the optimization and simulation phases.

5,000

-5,000

-10,000

-15,000

-20,000 Economicobjective value ($M) -25,000

-30,000 0 2 4 6 8 10 12 14 16 Number of iterations

Figure 25 - Impact of the number of iterations on the “With-lag” results

The “with-lag” approach result with the largest economic objective value of $M 1,088.91 was larger than the largest economic objective from with the ”no-lag” approach results of $M 1,018.43, a 7% improvement. Computation time for the “with-lag” results was 833 seconds compared to 136 seconds for the ”no-lag”, a 513% increase. A more thorough search of the various storage values could result in larger economic objective values for either travel time lag method. However, the ”with-lag” approach results proved more stable than the ”no-lag” approach results with respect to the number of iterations of the model. The purpose of the water storage values was for ensuring feasible solutions, which has been achieved, but they also have a significant impact on the operating policy produced and the resulting economic objective and energy deficit values that is not intended. Approaches that do not use storage values would simplify the process of configuring the model parameters and allow for a clearer comparison of the ”with-lag” and ”no-lag” approaches.

6.4. Hydrological Results Hydrological results from the simulated history, “no-lag”, and “with-lag” scenarios are presented and compared. The “no-lag” approach results are the 7-3-7 Cedar Lake, Lake Winnipeg, Southern Indian storage discretization configuration from Table 7. The “with-lag” approach results are from

the 7-3-7-3 Cedar Lake, Lake Winnipeg, Southern Indian Lake, lagged inflow state variable discretization configuration from Table 10.

715.00 844.00

842.00 714.50 840.00

714.00 838.00 836.00 Simulated History 713.50 Simulated History 834.00

Lake Elevation (ft) No-Lag Lake Elevation (ft) No-Lag 832.00 713.00 With-Lag With-Lag 830.00

712.50 828.00 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 (a) 10/1 11/1 12/1 (b) 10/1 11/1 12/1

847.50 180.00 847.00 160.00 846.50 140.00 846.00 120.00 845.50 845.00 100.00

844.50 80.00 Simulated History 844.00 Simulated History 60.00 Lake Elevation (ft) Lake Elevation (ft) No-Lag 843.50 No-Lag 40.00 With-Lag 843.00 With-Lag 842.50 20.00 842.00 0.00 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 1/1 2/1 3/1 (c) 10/1 11/1 12/1 (d) 10/1 11/1 12/1 Figure 26 - Average results from the simulated history, "no-lag", and "with-lag approach from all simulated flow years for (a) Lake Winnipeg elevation, (b) Cedar Lake elevation (c) Southern Indian Lake elevation, and (d) Stephen's Lake network inflow.

Figure 26 shows the semi-monthly traces for Lake Winnipeg elevations, Cedar Lake elevations, Sothern Indian Lake elevations, and Stephen’s Lake network inflow, averaged over all the historic inflow scenarios. Generally, lake elevations increase in the Spring and Summer and decrease in Fall and Winter for all results, with differences in timing and magnitude. Lake Winnipeg elevation peaks in July for all cases. The Lake Winnipeg peak for the “no-lag” and “with-lag” cases is less than the simulated history, and have a smaller, second rise peaking in January. The Cedar Lake elevation trace is similar for all scenarios, except the “no-lag” case rises higher through to November, after which both the “no-lag” and “with-lag” results have a steeper decline in elevation, lower bottom, and earlier rise starting February. The “no-lag” and “with-lag” Southern 69

Indian Lake have a steeper rise in the Spring and a later and steeper decline than the simulated history. The Stephens Lake network inflow varies fluctuates for all cases, with less distinctive trends than the elevation traces.

Figure 27 shows the elevation traces for Lake Winnipeg, Cedar Lake and Southern Indian Lake and the Stephens Lake network inflows for a subset of the inflow cases, 2003/04 through to 2015/16. The remaining cases show a similar pattern of results and are available in Appendix A2. The “no-lag” and “with-lag” traces generally follow similar traces that are different from the simulated history trace.

There are several causes for the differences in the simulated history and the “no-lag” and “with- lag” modeled results.

• The simulated history results are based on operating decisions made under a different electrical system configuration and demand. Section 6.1 elaborates on how the historical operating decisions made in the 2003/04 would lead to over 1000 GWh of energy deficits and $6,000 M in economic costs if repeated for the simulated electrical system. • The simulated history results are operational results aggregated from hour to hour and day to day operational decisions. Considerations for short-term operations, such as the management of short-term fluctuations in reservoir inflow will be imbedded in the simulated history results but may not be accurately captured in the “no-lag” or “with-lag” modeled results. • The “no-lag” and “with-lag” results are based on water value functions derived from a subset of the inflow scenarios. The subset did not select the lowest or highest inflow scenarios and may not have an accurate reflection of the water value in those periods. • Lake Winnipeg max outflow results for the “no-lag” and “with-lag” modeled results are based on an average maximum outflow rating curve for the period, whereas the simulated history used observed values from the historic record, which can be higher or lower than the average rating curve.

Lake Winnipeg - 2003/04-2015/16

718.00

717.00

716.00

715.00

714.00

713.00 Elevation (ft) 712.00

711.00 Simulated History No Lag With Lag 710.00 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Cedar Lake - 2003/04-2015/16

844.00

842.00

840.00

838.00

836.00

Elevation (ft) 834.00

832.00 Simulated History No Lag With Lag 830.00 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Southern Indian Lake - 2003/04-2015/16

848.00

847.00

846.00

845.00

Elevation (ft) 844.00

843.00 Simulated History No Lag With Lag 842.00 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Stephens Lake Network Inflow - 2003/04-2015/16

300.00

250.00

200.00

150.00

Inflow (kcfs) 100.00

50.00 Simulated History No Lag With Lag - 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Figure 27 – Lake Winnipeg, Cedar Lake, and Sothern Indian Lake elevation traces the Stephens Lake network inflow results for the 2003/04 flow year through to the 2015/16 flow year for the simulated history, “no-lag” and “with-lag” results.

7. Conclusions and Recommendations for Future Work 7.1. Summary of Findings This thesis successfully built and configured a Sampling Stochastic Dynamic Programming (SSDP) model to optimize the multi-reservoir operation problem under uncertainty for the Manitoba Hydro hydroelectric reservoir system. The approach generates water value functions that are then used to simulate operation of the hydroelectric reservoir system for historical inflows from 1978/79 to 2015/16. Consideration for the flow travel time lag between the Lake Winnipeg and Southern Indian Lake reservoirs is included in the simulation model. Two different approaches for incorporating the delay in the generation of the water value functions were used, a “no-lag” approach where the travel time lag is ignored and a “with-lag” approach where the lag is incorporated as a lagged inflow state variable. Comparisons were also made to historical reservoir release decisions.

Results demonstrated that the “with lag” approach that incorporated the flow travel time in the optimization phase as a lagged state variable generated slightly higher (7%) economic objective values with 90% less variability with respect the number of iterations, compared to the method that ignored the flow travel time. This is expected, as the ignored lag method has a discrepancy between the system representation when optimizing the water value functions and simulating the system. Computation time when the flow travel time was considered was 513% more than the problem with no lag, largely due to the incorporation of an additional state variable, the lagged inflow state variable. However, the total computational cost of the optimization problem that incorporated the flow travel time was 833 seconds on modern computer hardware, a comparable computational budget to HERMES, the existing decision model for the reservoir system.

Results also demonstrated that using historical reservoir release decisions for evaluating hydroelectric reservoir operations under current electrical system demands is problematic. Simulated results using historical reservoir releases resulted in energy supply deficits during lower inflow volume years even when storage was available.

The thesis also found that, the use of storage benefit and penalty values to reflect operating rules and limits to be effective in ensuring adherence to operating rules and limits, but laborious to calibrate. Results demonstrated that the economic objective function values and the amount of energy deficits was sensitive to the selection of the storage benefit and penalty values.

7.2. Limitations A major limitation of the study was the selection of the timestep and stage of the model to match the flow travel time (lag) between Lake Winnipeg and Stephens Lake. The same time lag was used for releases from Lake Winnipeg and Southern Indian Lake so that only a single additional state variable would be required to represent the lagged inflow into Stephens Lake, reducing the expected computation time. This assumption has several consequences on the accuracy and validity of the model results, outlined as follows.

The time lag is a fixed lag that does not change in time or in response to system conditions. Factors that might impact the duration of the lag are not included. The sensitivity of the time lag to factors such as the flow rate, intermediate lake levels and channel and outlet ice impacts were not investigated. This simplification can ultimately impact the accuracy of the water value functions generated and used in determining optimal reservoir releases.

The time lag for Southern Indian Lake releases use the Lake Winnipeg lag time. The use of a less accurate lag time again introduces inaccuracies into the representation of the hydroelectric reservoir system, which could impact the validity of the water value functions generated and used in determining optimal reservoir releases.

The timestep size and block resolution is not conducive to evaluate operations at finer resolutions. The semi-monthly timestep of the model and simulation produces reservoir releases at semi-monthly intervals. Disaggregation or other downscaling techniques need to be applied before the hydrological results can be used for applications at finer timescales. The time block representation of the electrical system helps to alleviate this issue for electric system operations; however, only two blocks representing onpeak and offpeak periods are included.

Another major limitation of this work is that the optimization approach assumes operating rules and restrictions are reflective of non-economic water management objectives. The model constraints include representations of the hydrological system operating rules, which reflect operating licenses, as well as electric system balancing. The objective function of the model focuses on economic objectives resulting from the electric system operations, with considerations for the Lake Winnipeg maximum outflow rule adherence, and storage benefit values for Lake Winnipeg and Cedar Lake to keep the storage level above minimum levels. Other water management objectives are assumed to be reflected in the hydrological system operating rules. However, Manitoba Hydro does incorporate other stakeholder information and feedback in determining reservoir operations, which may not be incorporated into the optimization approach.

7.3. Applications The model could be used to provide guidance to reservoir operators. The SSDP decision model determines optimal reservoir release decisions, based on the representation and approximation of the Manitoba Hydro hydroelectric reservoir system. Reservoir operators can use the results of the SSDP model as an additional tool in their decision support systems. Operators are cautioned to review and understand the model scope, limitations and accuracy when considering using the reservoir release decisions from the SSDP model.

The model could also be coupled with hydrological models to improve representation of reservoir release decisions in other simulation studies. Ensemble streamflow prediction traces could replace the use of historic inflow scenarios to provide the optimization more accurate inflow information. Reservoir release decisions from the SSDP model could be passed back to the river network simulations models. Combining the reservoir release decisions from the SSDP model with hydrological simulation models that do not incorporate the hydroelectric system into the reservoir release decision methods can help to bridge and balance the operating priorities of the hydrological and electrical systems. This could be especially helpful when simulating future scenarios, where the hydro-climatic system has changed, and historic operating decisions may no longer be applicable or relevant for the system under evaluation.

7.4. Recommendations for Future Work The following recommendations are natural extensions that can be built on this research work.

Investigate incorporating ensemble streamflow prediction. Other studies have demonstrated improved model performance by using ensemble streamflow prediction forecasts to represent both the current conditions and future variability and magnitude of streamflow volumes (Faber and Stedinger 2001) (Kim, et al. 2007) (Cote and Leconte 2015). The SSDP model benefits from incorporating the available state-of-the-system information used in producing the ensemble forecast in representing future possible streamflow traces. Additionally, the ensemble forecast includes the spatial and temporal correlations, similar to using historical streamflows.

Investigate approaches that do not require storage benefit and penalty values. Configuring the storage benefit and penalty values require a laborious trial and error approach. The purpose of the water storage values was for ensuring feasible solutions, which has been achieved, but they also have a significant impact on the operating policy produced and the resulting economic objective and energy deficit values that is not intended. Approaches that eliminate the need of these values could improve model usability through a less laborious and sensitive configuration process. This could include changes to the streamflow input data that eliminates negative inflows by correcting or adjusting the inflow history or using ensemble streamflow prediction would eliminate the need for the Cedar Lake and Lake Winnipeg benefit values. Alternative sub-problem optimization approaches that can represent if-then constraints such as mixed integer linear programming could be used to represent the if-then maximum outflow constraint, eliminating the need for the Lake Winnipeg maximum outflow goal.

Investigate alternative hydrological variables. This thesis looked at only a single but common hydrological variable as a predictor of future period inflows, the current period total system inflows. However, studies in other systems that utilize the SSDP method have shown improvements in model results by using other hydrological variables (Tejada-Guibert, Johnson and Stedinger 1995) (Cote, Haguma, et al. 2011).

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Appendix A1: Estimation of the travel time delay in the Manitoba Hydro hydroelectric reservoir system Manitoba Hydro has estimated that flows takes 15 days to arrive at Kettle, the first station in the Lower Nelson cascade, from Lake Winnipeg and 9 days for from Southern Indian Lake to arrive at Kettle; however, details of the method employed and results are not published (Magura 2013). To corroborate the Manitoba Hydro estimate, a time series analysis on the weekly aggregated flow volumes using the 1978 to 2015 LTFD dataset to estimate the lag times was conducted. The Nash-Sutcliffe efficiency is a common method for evaluating the goodness of fit of hydrological models (McCuen, Knight and Cutter 2006). A model with a zero-estimation error would have a NSE value 1.0, whereas a NSE value of 0 indicates the model error variance is equal to the variance in the observed timeseries. The lagged flows, Notigi outflows plus Lake Winnipeg outflows plus local inflows on the way to Stephens Lake, were compared to the observed Stephens Lake inflows to calculate the Nash-Sutcliffe efficiency (NSE) index, as shown in Equation 35, modified such that the timeseries are shifted by the time lag.

Equation 35

, = 1 𝑇𝑇 𝐼𝐼 𝑜𝑜 2 ∑𝑡𝑡 �∑𝑖𝑖=0�𝑄𝑄𝑖𝑖 𝑡𝑡−𝑘𝑘𝑖𝑖� − 𝑄𝑄𝑡𝑡 � 𝑁𝑁𝑁𝑁𝑁𝑁 − 𝑇𝑇 𝑜𝑜 𝑜𝑜 2 ∑𝑡𝑡 �𝑄𝑄𝑡𝑡 − 𝑄𝑄��𝑡𝑡�� is the set of timeseries considered for the lag, is the timeseries for location , is the time

𝐼𝐼lag for location , is the target timeseries for𝑄𝑄 𝑖𝑖comparison, observed Stephen’s𝑖𝑖 𝑘𝑘 𝑖𝑖Lake inflow 𝑜𝑜 volumes for timestep𝑖𝑖 𝑄𝑄𝑡𝑡 , and is the mean of . The timestep of the timeseries was weekly, 𝑜𝑜 𝑜𝑜 which limited the search𝑡𝑡 of the𝑄𝑄��𝑡𝑡� time lag, , to multiples𝑄𝑄𝑡𝑡 of a week. A search of the time lag for each location, , was conducted to find those𝑘𝑘𝑖𝑖 that produced the highest NSE values was using the Evolutionary𝑘𝑘𝑖𝑖 Algorithm solver from the Microsoft Excel Solver. Results are shown in Table 12. The search found that the lag between Lake Winnipeg and Stephen’s Lake was best represented by a 2-week time shift, and that the lag from Notigi to Stephens Lake was best represented as a 3-week lag with a NSE of 0.9301 compared to a NSE of 0.8627 when no time lags were used. This compares well to the Manitoba Hydro estimate of 15 days from Jenpeg to Stephen’s Lake, but disagrees with the estimate of 9 days from Notigi to Stephen’s Lake. However, the lagged inflow

state variable requires the time lags to be equal. Therefore, in consideration of past evidence and the constraints𝑅𝑅𝑡𝑡 of the model formulation, the Lake Winnipeg and Notigi time shifts were taken to be semi-monthly or approximately 2 weeks, which decreased the NSE index to 0.9252.

Table 11 - Comparison of different time lags and the Nash Sutcliffe efficiency index for the time lag estimation of Stephens Lake inflows

Case NSE

No time lags 0.8627

2-week Lake Winnipeg lag, 3-week Notigi lag 0.9301

2-week Lake Winnipeg lag, 2-week Notigi lag 0.9252

Appendix A2: Hydrologic and energy system results from the simulation of historical releases Hydrological results show elevation traces for storage state variables, Cedar Lake, Lake Winnipeg and Southern Indian Lake, and network inflow trace for the delayed inflow location, Stephens Lake from the simulated history, “no-lag”, and “with-lag” approaches. Energy supply and energy demand charts show the total supply or demand by type for each period.

Lake Winnipeg - 1978/79-1989/90

716.00

715.00

714.00

713.00

Elevation (ft) 712.00

711.00 Simulated History No Lag With Lag 710.00 4/1/1978 4/1/1979 4/1/1980 4/1/1981 4/1/1982 4/1/1983 4/1/1984 4/1/1985 4/1/1986 4/1/1987 4/1/1988 4/1/1989

Cedar Lake - 1978/79-1989/90

844.00 842.00 840.00 838.00 836.00

Elevation (ft) 834.00 832.00 Simulated History No Lag With Lag 830.00 4/1/1978 4/1/1979 4/1/1980 4/1/1981 4/1/1982 4/1/1983 4/1/1984 4/1/1985 4/1/1986 4/1/1987 4/1/1988 4/1/1989

Southern Indian Lake - 1978/79-1989/90

849.00 848.00 847.00 846.00 845.00

Elevation (ft) 844.00 843.00 Simulated History No Lag With Lag 842.00 4/1/1978 4/1/1979 4/1/1980 4/1/1981 4/1/1982 4/1/1983 4/1/1984 4/1/1985 4/1/1986 4/1/1987 4/1/1988 4/1/1989

Stephens Lake Network Inflow - 1978/79-1989/90

250

200

150

100 Elevation (ft) 50 Simulated History No Lag With Lag 0 4/1/1978 4/1/1979 4/1/1980 4/1/1981 4/1/1982 4/1/1983 4/1/1984 4/1/1985 4/1/1986 4/1/1987 4/1/1988 4/1/1989

Figure A 1 – Elevation and network inflow traces for historic inflow years 1978/79 to 1989/90

Lake Winnipeg - 1990/91-2002/03

717.00 716.00 715.00 714.00 713.00

Elevation (ft) 712.00 711.00 Simulated History No Lag With Lag 710.00 4/1/1990 4/1/1991 4/1/1992 4/1/1993 4/1/1994 4/1/1995 4/1/1996 4/1/1997 4/1/1998 4/1/1999 4/1/2000 4/1/2001 4/1/2002 4/1/2003

Cedar Lake - 1990/91-2002/03

844.00 842.00 840.00 838.00 836.00

Elevation (ft) 834.00 832.00 Simulated History No Lag With Lag 830.00 4/1/1990 4/1/1991 4/1/1992 4/1/1993 4/1/1994 4/1/1995 4/1/1996 4/1/1997 4/1/1998 4/1/1999 4/1/2000 4/1/2001 4/1/2002 4/1/2003

Southern Indian Lake - 1990/91-2002/03

849.00 848.00 847.00 846.00 845.00

Elevation (ft) 844.00 843.00 Simulated History No Lag With Lag 842.00 4/1/1990 4/1/1991 4/1/1992 4/1/1993 4/1/1994 4/1/1995 4/1/1996 4/1/1997 4/1/1998 4/1/1999 4/1/2000 4/1/2001 4/1/2002 4/1/2003

Stephens Lake Network Inflow - 1990/91-2002/03

250.00

200.00

150.00

100.00 Elevation (ft) 50.00 Simulated History No Lag With Lag - 4/1/1990 4/1/1991 4/1/1992 4/1/1993 4/1/1994 4/1/1995 4/1/1996 4/1/1997 4/1/1998 4/1/1999 4/1/2000 4/1/2001 4/1/2002 4/1/2003

Figure A 2 – Elevation and network inflow traces for historic inflow years 1990/91 to 2002/03

Lake Winnipeg - 2003/04-2015/16

718.00 717.00 716.00 715.00 714.00 713.00 Elevation (ft) 712.00 711.00 Simulated History No Lag With Lag 710.00 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Cedar Lake - 2003/04-2015/16

844.00 842.00 840.00 838.00 836.00

Elevation (ft) 834.00 832.00 Simulated History No Lag With Lag 830.00 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Southern Indian Lake - 2003/04-2015/16

848.00

847.00

846.00

845.00

Elevation (ft) 844.00

843.00 Simulated History No Lag With Lag 842.00 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Stephens Lake Network Inflow - 2003/04-2015/16

300.00

250.00

200.00

150.00

Elevation (ft) 100.00

50.00 Simulated History No Lag With Lag - 4/1/2003 4/1/2004 4/1/2005 4/1/2006 4/1/2007 4/1/2008 4/1/2009 4/1/2010 4/1/2011 4/1/2012 4/1/2013 4/1/2014 4/1/2015

Figure A 3 - Elevation and network inflow traces for historic inflow years 2003/04 – 2015/16

Energy Supply - Simulated History 1978/89-1989/90 Total Hydro Total Import Total Thermal Supply Deficit 1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Supply - Simulated History 1990/91-2002/03

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Supply - Simulated History 2003/04-2015/16

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Figure A 4 – Energy supply volumes for the simulated history

Energy Demand - Simulated History 1978/89-1989/90 Total Manitoba Demand Total Export 1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Demand - Simulated History 1990/91-2002/03

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Demand - Simulated History 2003/04-2015/16

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Figure A 5 – Energy demand for the simulated history

Energy Supply -"No-lag" approach 1978/89-1989/90 Total Hydro Total Import Total Thermal Supply Deficit 1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Supply -"No-lag" approach 1990/91-2002/03

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Supply -"No-lag" approach 2003/04-2015/16

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Figure A 6 – Energy supply for the ‘no-lag’ approach

Energy Demand -"No-lag" approach 1978/89-1989/90 Total Manitoba Demand Total Export 1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Demand -"No-lag" approach 1990/91-2002/03

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Demand -"No-lag" approach 2003/04-2015/16

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Figure A 7 – Energy demand for the ‘no-lag’ approach

Energy Supply -"With-lag" approach 1978/89-1989/90 Total Hydro Total Import Total Thermal Supply Deficit 1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Supply -"With-lag" approach 1990/91-2002/03

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Supply -"With-lag" approach 2003/04-2015/16

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Figure A 8 – Energy supply for the ‘with-lag’ approach

Energy Demand -"With-lag" approach 1978/89-1989/90 Total Manitoba Demand Total Export 1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Demand -"With-lag" approach 1990/90-2002/03

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Energy Demand -"With-lag" approach 2003/04-2015/16

1,900 1,700 1,500 1,300 1,100

Energy (GWh) 900 700 500

Figure A 9 – Energy demand for the ‘with-lag’ approach