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Authors: Sunil Kumar,Data Scientist, Eka Software Solutions

Vinay Mehendiratta, Director of Research, Eka Software Solutions

Business Case
It has been observed and widely accepted that demand of natural gas follows seasonal pattern with peak in winter and trough in summer. Production rate of natural gas is almost constant and generally not agile enough to meet peak demand in winter. It is a known business practice that gas is stored at storage facility to meet the demand in winter.

It has been observed in research literature that price of natural gas follows seasonal pattern with peak in winter and trough in summer. A trader exploits the price differential using spot price and forward market price information. The storage facility operator’s (owner or lessee) objective is to inject gas into storage facility at low price and withdraw gas from storage facility at high price, and benefit from price differential.


Storage facility

A storage facility has limited and known storage capacity. It requires threshold gas level to maintain pressure required. Quantity of gas that can be stored in a storage facility is constrained on both sides – minimum quantity to maintain the pressure and maximum quantity is limited by storage capacity agreed for the use. Storage facility has a limitation on injection rate and withdrawal rate. Generally, achievable injection and withdrawal rates are function of available gas in the storage facility. We assume that achievable injection and withdrawal rates are step function of inventory. Operation of injecting gas into facility or withdrawing gas from facility causes partial loss of gas. Rate of loss of gas during injection and withdrawal may vary with time.

Storage Contract
A storage contract between two parties is valid for a limited duration, generally for one year. A storage contract has following attributes:
• Contract start date
• Contract end date
• Rented storage capacity (this will be referred as storage capacity in this document and future discussions)
• Lowest gas level to be maintained in storage facility
• Available inventory at the start of contract
• Required inventory to be maintained at the end of contract.
• Per unit cost of injection and withdrawal (Cost of injection and withdrawal may vary with time)
• Reservoir ID and Location

Problem Statement
Determine the optimal schedule of injection and withdrawal of natural gas during the storage contract so that profit is maximized. The injection and withdrawal schedule depends on spot price, future prices of upcoming months in contract, fuel loss during injection and withdrawal, cost of injection and withdrawal, risk free interest rate, and operational constraints. Spot and future prices are of the location pricing group linked with the contract. The injection and withdraw schedule should consist of injection and withdrawal amount for remaining days of spot month (if user has selected this option), and upcoming months in contract.
Note - Schedule is defined in model output tables below.

Objective Function

To determine an injection and withdrawal schedule that maximizes total profit from storage contract by exploiting market price fluctuations while considering operational and contractual constraints.
Decision Variables
• Whether to inject or withdraw or to do nothing in a decision horizon, and
• Quantity of injection and withdrawal.
Business Constraints
• Quantity of gas stored in storage facility should not exceed rented storage capacity in a decision horizon.
• Quantity of gas left in storage should not be below threshold gas level in a decision horizon.
• Injection rate cannot be higher than the maximum achievable injection rate in a decision horizon. Achievable injection rate is step function of available inventory.
• Withdrawal rate cannot be higher than the maximum achievable withdrawal rate in a decision horizon. Achievable withdrawal rate is step function of available inventory.
• Injection and withdrawal should be such that storage facility has the agreed level of gas (and pressure) at the end of contract. Generally, Contractual agreement is such that it is required to bring the storage facility (at the end of contract) to the level where it was at the start of contract.
• One cannot withdraw more than available gas subtracted by minimum required gas in a decision horizon. This constraint does not allow negative inventory.
• User specified inventory requirements. If user wants certain inventory level to be maintained at the end of specific periods.
• User specifies certain months when injection or withdrawal can and cannot be conducted.

Cash Flow Discounting

Discounting of cash flow is performed to calculate time value of money. Cash flow discounting is important in presence of significant risk free interest rates. Suppose that cash receivable of USD 100 is expected after one year from today, and risk free annual interest rate is 5 %. Then current value of expected receivable is 100/(1 + 0.05) = USD 95.24 . It means that if we have USD 95.24 today and deposit it into bank at risk free interest rate then at the end of the year we will receive 95.24(1+0.05) = USD 100. In our model we use cash flow discounting to calculate current value of future accounts receivable and accounts payable.

Model Inputs
Storage Contract
• Storage contract start date
• Storage contract end date
• Rented storage capacity
• Initial inventory
• Required inventory at contract end date
• Any additional cost as mentioned in the contract
• Delivery terms– Frequency of delivery
Injection Withdrawal Rate Limits
• Lower limit of inventory range
• Upper limit of inventory range
• Highest injection rate
• Highest withdrawal rate
Injection Withdrawal Cost
• Period start date
• Period end date
• Injection cost per unit
• Withdrawal cost per unit
Injection Withdrawal Loss
• Period start date
• Period end date
• Injection loss
• Withdrawal loss

Minimum Inventory Level Requirements
• Remaining Month
• Minimum required inventory level
Inventory Snapshot
• Observation/current date
• Inventory Quantity
• Publication/settlement schedule
Forward Prices
• Observation/current date (when model is run)
• location pricing group
• Remaining month
• Price
User-Specified Input
• Scenario Id
• Scenario Name
• Risk free interest rate

Model Outputs

• Decision Horizon – Daily for the leftover days in month (Current time)
• Injection quantity in that decision horizon
• Withdrawal quantity in that decision horizon
• Available Inventory at the end of decision horizon

• Decision Horizon – Remaining month (Current time)
• Injection quantity in that decision horizon
• Withdrawal quantity in that decision horizon
• Available Inventory at the end of decision horizon



Figure 1: Schematic Diagram of Optimization Model

Illustrative Example:

We have taken a fictitious example to show format of input and output data. This example is used to assess that whether model is working as per expectation or not, such as whether all constraints are being satisfied or not. Here is basic detail of the illustrative example. Detail input and output data is attached in below excel sheets.
Contract Start Date : June 1, 2015
Contract End Date : May 31, 2018
Storage Capacity : 100,000
Initial Inventory : 200
Required Inventory at Contract End : 200
Threshold Inventory to be Maintained : 200
Delivery Terms : Daily
Risk Free Interest Rate : 4 %


Graphical Representation of Input and Output data

Below graph plots forward prices. This is plot of input data. It can be observed that the prices are higher in winter and relatively lower in summer.


Figure 2: Graphical Representation of Forward Prices (Input Data)

Below graph plots variation in achievable injection and withdrawal rates as function of inventory. This is a plot of input data.


Figure 3: Graphical Representation of Inventory Ratchets (Input Data)

Below graph shows output injection and withdrawal schedule from the model. This is representation of output data.


Figure 4: Graphical Representation of Injection and Withdrawal Schedule (Output Data)

 

This graph below (figure 5) shows cash inflow and outflow because of injection and withdrawal and other expenses. This is representation of output data.

Figure 5: Graphical Representation of Cash Inflow and Outflow (Output Data)

Input vs Output Chart

 

The graph below shows input price and output injection and withdrawal schedule on same chart. It can be seen that the model recommends injection when prices are low and recommends withdrawal when prices are high.

Figure 6: Injection and Withdrawal Schedule vs Forward Prices (Input vs Output)

Scenario Comparison

           

We create new scenarios by modifying previous Illustrative example. Then we compare new scenario output with the base scenario output (We say Illustrative example as base scenario). Scenario comparison has been performed to assess that whether change in input data produce desirable/expected output or not. Scenario comparison can also be used as what-if analysis tool.  

Comparison by Forward Price

Below graph compares output injection and withdrawal schedule when there is change in forward price. On right side of the chart, we give price spike in the month of September, 2014. Because of price spike, model suggests withdrawal in September and then replenish the withdrawal inventory in adjacent future months.  

Figure 7: Scenario Comparison by Forward Prices (Input vs Output)

Comparison by Ratchets

Below graph compares output injection and withdrawal schedule when there is change in inventory ratchets. On right side of the chart, we have increased injection and withdrawal rate limits. We observe that model suggests higher injection and withdrawal in case of increased injection and withdrawal rate limits.

Figure 8: Scenario Comparison by Inventory Ratchets (Input vs Output)

Comparison by Interest Rates

Below graph compares output injection and withdrawal schedule when there is change in interest rates.  We observe that higher interest rate encourages early withdrawal and discourages early injection. 

Figure 9: Scenario Comparison by Interest Rates (Input vs Output)

Comparison by Injection and Withdrawal Cost 

Below graph shows output injection and withdrawal schedule when there is change in per unit injection and withdrawal cost. Lower part of the graph shows injection and withdrawal schedule when injection and withdrawal costs are zero. We observe that zero injection and withdrawal cost encourages higher injection and withdrawal amount.

Figure 10: Scenario Comparison by Injection and Withdrawal Cost

Comparison by Injection and Withdrawal Loss 

Below graph shows output injection and withdrawal schedule when there is change in injection and withdrawal loss percentage. Lower part of the graph shows injection and withdrawal schedule when injection and withdrawal losses are zero. We observe that zero injection and withdrawal loss encourages higher injection and withdrawal amount.

Figure 11: Scenario Comparison by Injection and Withdrawal Loss

Assumptions
• Supply/demand of gas for injection/withdrawal is instantaneous
• Injection/withdrawal rate is independent of available inventory
• All costs and revenues are expressed in single currency
• Unit of measurements are consistent with each other
• Achievable injection and withdrawal rates are step function of available inventory
• Injection and withdrawal cost or injection and withdrawal schedule may vary across the months but will remain constant within a month.
Out of Scope
• Not concerned with buy/sell decisions. It is primarily (or only) concerned with inject/withdrawal decision.
• Derivative contracts or hedging are out of scope.
• Demand/supply constraint are out of scope.

Conclusion:

At high level, gas storage optimization appears as a problem of finding months pairs with maximum spread. In simplistic version of the model, gas should be injected at lowest price and withdrawn at highest price. But presence of complicating factors, such as variation in injection and withdrawal cost with time, variation in injection and withdrawal loss with time, variation in injection and withdrawal rate limits with inventory make it very difficult to solve gas storage optimization problem by simple analysis. We have formulated an optimization model which maximizes profit from the storage contract while also considering complicating factors outlined above. We have tested our model for different scenarios, and we find that the model gives best possible schedule for given input. We also validate our model by creating different scenarios, and comparing model output with expected output.

References:
• Alan Holland, Optimization of Injection/Withdrawal Schedules for Natural Gas Storage Facilities
• Nicola Secomandi, Optimal Commodity Trading with a Capacitated Storage Asset
• Patrick Henaff, Ismail Laachir, Francesco Russom, Gas storage valuation and hedging : A quantification of the model risk
• Hicham Zmarrou, Natural gas storage valuation reviewed
• Yun Li, Natural Gas Storage Valuation
• Christopher Athaide, A Primer on Natural Gas Storage Valuation
• Alexander Boogert, Cyriel de Jong, Gas Storage Valuation Using a Monte Carlo Method
• John Breslin, Les Clewlow, Tobias Albert, Calvin Kwok, Chris Strickland, Daniel Van Der Zee, Gas Storage : Rolling Intrinsic Evaluation

Mathematical model is provided at this link:

Mathematical%20model%20-%20GAS%20STORAGE%20OPTIMIZATION_July9_2014....

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Comment by Alex Callahan on March 1, 2017 at 7:30am

I agree with the previous poster in that this made a lot of sense for me (new to this process)...but like the previous poster, is there an excel model that pairs with this report?

Comment by Michael Conklin on July 8, 2016 at 5:53am

This was a great read.  Thank you! 

In the first paragraph under Illustrative Example, you mention "Detail input and output data is attached in below excel sheets," but I don't see the sheets anywhere.

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