Distributed Storage for Intermittent Energy Sources:
Control Design and Performance Limits
Yashodhan Kanoria
Stanford University
Joint work with:
Andrea Montanari, David Tse and Baosen Zhang
The challenge of renewables
• Intermittent
• Intrinsically distributed
How to manage?
The
paper
Strategy
Requirement
Spatial averaging
Transmission
Time averaging
Storage
Over-provisioning
Generation
Demand response
Signaling, incentives
The challenge of renewables
• Intermittent
• Intrinsically distributed
How to manage?
This
talk
Strategy
Requirement
Spatial averaging
Transmission
Time averaging
Storage
Over-provisioning
Generation
Demand response
Signaling, incentives
Questions?
• Optimal control of grid with transmission and storage?
• Infrastructure improvement:
– New transmission line? Upgrade in existing line?
– New storage facility?
– New generation?
Which will help more?
How to optimally allocate budget?
Our contribution
• Control design for discrete time model
• Analytical insights quantifying benefits of storage,
transmission and overprovisioning
Model
3
2
1
4
Model
3
2
1
4
= Electric load
Model
3
2
1
4
= Renewable sources
Model
𝑊ind3 (𝑡)
3
𝑊ind2 (𝑡)
2
1
4
𝑊ind1 (𝑡)
−
= 𝑊ind
𝑊ind4 (𝑡)
−
Model
𝑊ind3 (𝑡)
3
𝑊ind2 (𝑡)
2
1
4
𝑊ind1 (𝑡)
𝑊ind4 (𝑡)
= 𝑊ind
−
Model
𝐹ast 2 (𝑡)
𝑊ind3 (𝑡)
𝐹ast 2 (𝑡)
3
𝑊ind2 (𝑡)
2
𝐹ast1 (𝑡)
𝐹ast 2 (𝑡)
1
4
𝑊ind1 (𝑡)
= 𝐹ast
𝑊ind4 (𝑡)
= 𝑊ind
−
Model
= 𝑊ind
= 𝐹ast
𝐹ast 2 (𝑡)
𝑊ind3 (𝑡)
𝐹ast 2 (𝑡)
𝑊ind2 (𝑡)
𝐹23 (𝑡)
2
3
𝐹34 (𝑡)
𝐹ast1 (𝑡)
1
𝐹12 (𝑡)
𝐹ast 2 (𝑡)
𝐹24 (𝑡)
4
𝑊ind1 (𝑡)
𝑊ind4 (𝑡)
−
Model
= 𝑊ind
= 𝐹ast
𝐹ast 2 (𝑡)
𝑊ind3 (𝑡)
𝐹ast 2 (𝑡)
𝑊ind2 (𝑡)
2
𝐹ast1 (𝑡)
1
𝐹12 (𝑡)
1
2
𝐹23 (𝑡)
3
3
𝐹34 (𝑡)
2
𝐹ast 2 (𝑡)
𝐹24 (𝑡)
1
4
𝑊ind1 (𝑡)
𝑊ind4 (𝑡)
4
4
−
Model
= 𝑊ind
= 𝐹ast
𝐹ast 2 (𝑡)
𝑊ind3 (𝑡)
𝐹ast 2 (𝑡)
𝑊ind2 (𝑡)
𝑆ave3 (𝑡)
𝐹23 (𝑡)
𝐹34 (𝑡)
𝐹ast1 (𝑡)
𝐹12 (𝑡)
𝑆ave1 (𝑡)
𝑆ave2 (𝑡)
𝐹ast 2 (𝑡)
𝐹24 (𝑡)
𝑊ind1 (𝑡)
𝑆ave4 (𝑡)
𝑊ind4 (𝑡)
Energy pushed into
= 𝑆ave
−
Model
= 𝑊ind
= 𝐹ast
𝐹ast 2 (𝑡)
𝑊ind3 (𝑡)
𝐹ast 2 (𝑡)
𝑊ind2 (𝑡)
𝑆ave3 (𝑡)
𝐹23 (𝑡)
𝐹34 (𝑡)
𝐹ast1 (𝑡)
𝐹12 (𝑡)
𝑆ave1 (𝑡)
𝑆ave2 (𝑡)
𝐹ast 2 (𝑡)
𝐹24 (𝑡)
𝑊ind1 (𝑡)
𝑆ave4 (𝑡)
𝑊ind4 (𝑡)
Control inputs:
• 𝐹ast 𝑖 (𝑡)
• 𝑆ave𝑖 (𝑡)
Dependent variables:
• Flows 𝐹𝑖𝑗 (𝑡)
• Storage buffers 𝐵𝑖 (𝑡)
Constraints
• Storage capacity constraints
Hard constraint
• Transmission constraints
Soft constraint
Cost function
• 𝜀𝐹ast = Cost of fast generation
(free disposal of energy)
0
Fasti(t)
• 𝜀𝐹 = Cost of violating transmission constraints
-Ce
Ce
• Performance criterion: 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
Fe(t)
What’s the issue?
• Centralized control problem
• 𝐹(𝑡) is linear function of controls
• Storage constraints cause thresholding
X
• Non-quadratic cost function
X
Idea: Use surrogate cost function
Var 𝐹ast + 𝜆 Var 𝐹 + 𝜆 Var 𝐵
(drop storage constraints for now)
The surrogate problem
•
•
•
•
•
•
‘State’ consisting of buffer sizes
‘Noise’ 𝑊ind 𝑡
State and noise fully observed
Linear state evolution
Quadratic cost
Assume 𝑊ind 𝑡 is Gaussian
We have an LQG problem!
• Can be solved numerically yielding control scheme
• Can be mapped back to original problem
The surrogate LQG problem
 Is the scheme any good?
 Any insights into roles of storage and transmission?
Specific networks
• Transmission network is almost a tree
• What happens on an infinite line?
• What about a two-dimensional grid?
1-D and 2-D grids
Assume:
 Storage 𝑆 at each node
 Capacity 𝐶 on each line
 𝑊ind𝑖 (𝑡) ~ N(0, 𝜎 2 ) iid
1-D and 2-D grids
𝑆 = Storage
𝐶 = Transmission
𝜎 2 = Var(𝑊ind)
We find, for original problem:
 Analytical expressions for cost of LQG-based schemes
 Fundamental limits
⇒ LQG is near optimal
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
Only 𝑺
Only 𝑪
𝜎2
𝐶
1-D grid
2-D grid
𝜎 2 = Var(𝑊ind)
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
Only 𝑺
Only 𝑪
𝜎2
𝐶
1-D grid
2-D grid
𝜎 2 = Var(𝑊ind)
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝐶 and 𝑆 together works best!
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝝈𝟐
𝑪
2-D grid
𝜎2
𝑆
𝑪
𝛔 𝐞𝐱𝐩 −
𝝈
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
Averaging over 2-D much more effective than 1-D
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝑪
𝛔 𝐞𝐱𝐩 −
𝝈
𝑪=𝑺
𝛔 𝐞𝐱𝐩 −
𝑪
𝝈
𝐶2
𝜎 exp − 2
𝜎
𝑆 seems to provide extra dimension for averaging!
Intuition: Time is like extra spatial dimension
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝜎 exp −
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
??
𝐶
𝜎
𝑆 = Storage
𝐶 = Transmission
Results 𝜺 = 𝜺𝑭𝐚𝐬𝐭 + 𝜺𝑭
𝜎 2 = Var(𝑊ind)
Only 𝑺
Only 𝑪
1-D grid
𝜎2
𝑆
𝜎2
𝐶
2-D grid
𝜎2
𝑆
𝐶
𝜎 exp −
𝜎
𝑪=𝑺
𝜎 exp −
𝐶
𝜎
𝐶2
𝜎 exp − 2
𝜎
Spatial averaging is much easier in 2-D
1-D
2-D
Segment of length 𝑙
Square of side 𝑙
Conclusions and future work
• Gaussian assumption optimistic
• But key insights should remain valid:
– 2-D averaging much better than 1-D
– S facilitates averaging over extra dimension
• In the paper: A little overprovisioning can go a long way
Thank you!