Distributionally Robust Approaches for Optimal Power Flow with Uncertain Reserves from Load Control Siqian Shen Industrial and Operations Engineering University of Michigan Joint work with Yiling Zhang (IOE) and Johanna Mathieu (EECS) June, 2015 1 / 28
Introduction Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? How to Solve CC-OPF? Notation Joint and Individual CC-OPF Models Solution Approaches Mixed-integer Linear programming (MILP) Approach (A1) Gaussian Approximation Approach (A2) Scenario Approximation Approach (A3) Distributionally Robust Optimization Approach (A4) Computational Results IEEE 9-Bus System IEEE 39-Bus System 2 / 28
Introduction Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? How to Solve CC-OPF? Notation Joint and Individual CC-OPF Models Solution Approaches Mixed-integer Linear programming (MILP) Approach (A1) Gaussian Approximation Approach (A2) Scenario Approximation Approach (A3) Distributionally Robust Optimization Approach (A4) Computational Results IEEE 9-Bus System IEEE 39-Bus System 3 / 28
Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? The Optimal Power Flow (OPF): minimize system-wide energy and reserve costs subject to the physical constraints of the system. More reserve needed: an increase in intermittent and uncertain power generation, i.e., wind and solar capacity Large amount of uncertainty in power systems motivates stochastic optimization approaches, i.e., CC-OPF. Past work: Focused on managing uncertainty stemming from renewable energy production and load consumption Our work: also the uncertain balancing reserves provided by load control 4 / 28
Introduction Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? How to Solve CC-OPF? Notation Joint and Individual CC-OPF Models Solution Approaches Mixed-integer Linear programming (MILP) Approach (A1) Gaussian Approximation Approach (A2) Scenario Approximation Approach (A3) Distributionally Robust Optimization Approach (A4) Computational Results IEEE 9-Bus System IEEE 39-Bus System 5 / 28
How to Solve CC-OPF? A robust reformulation of the scenario approach requires no knowledge of uncertain distributions but significant number of “uncertain scenarios” – data! Such data may be unavailable in practice. our goal: investigate the performance of a variety of methods to solve CC-OPF problems given limited information of uncertain distribution. 6 / 28
Introduction Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? How to Solve CC-OPF? Notation Joint and Individual CC-OPF Models Solution Approaches Mixed-integer Linear programming (MILP) Approach (A1) Gaussian Approximation Approach (A2) Scenario Approximation Approach (A3) Distributionally Robust Optimization Approach (A4) Computational Results IEEE 9-Bus System IEEE 39-Bus System 7 / 28
Notation Decision variables: energy production at generators P G generators’ up- and down-reserve capacities R G , R G loads’ up- and down-reserve capacities R L , R L “distribution vectors” d G , d G and d L , d L Other variables: actual generator reserves R G and load reserves R L real-time supply/demand mismatch P m Cost parameters: c = [ c 0 , c 1 , c 2 , c G , c G , c L , c L ] T Given data: loads forecast P f L and wind forecast P f W actual wind power � P W , actual load � P L P L , � actual minimum and maximum load [ � P L ] min/max generator production P G , P G 8 / 28
Introduction Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? How to Solve CC-OPF? Notation Joint and Individual CC-OPF Models Solution Approaches Mixed-integer Linear programming (MILP) Approach (A1) Gaussian Approximation Approach (A2) Scenario Approximation Approach (A3) Distributionally Robust Optimization Approach (A4) Computational Results IEEE 9-Bus System IEEE 39-Bus System 9 / 28
Joint and Individual CC-OPF Models [J-CC-OPF]: c T [ 1 , P G , P 2 min G , R G , R G , R L , R L ] (1) N W N L � � ( � P W,i − P f ( � P L,i − P f s.t. P m = W,i ) − L,i ) (2) i =1 i =1 N G N L � � d G,i + d L,i = 1 (3) i =1 i =1 N G N L � � d G,i + d L,i = 1 (4) i =1 i =1 R G = d G max {− P m , 0 } − d G max { P m , 0 } (5) R L = d L max { P m , 0 } − d L max {− P m , 0 } (6) � � Ax ≥ � � ≥ 1 − ǫ (7) P b x = [ P G , R G , R G , R L , R L , d G , d G , d L , d L ] ≥ 0 . (8) 10 / 28
Joint and Individual CC-OPF Models Constraints inside (7) Ax ≥ � � b = { P G ≤ P G + R G ≤ P G , P L + R L ≤ � P L ≤ � � P L , − R G ≤ R G ≤ R G , − R L ≤ R L ≤ R L , � � 0 − P line ≤ B flow ≤ P line } . (9) bus ˆ B − 1 P inj [I-CC-OPF]: min (1) s.t. (2)–(6), (8) � � A i x ≥ � � P b i ≥ 1 − ǫ i i = 1 , . . . , m. (10) 11 / 28
Introduction Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? How to Solve CC-OPF? Notation Joint and Individual CC-OPF Models Solution Approaches Mixed-integer Linear programming (MILP) Approach (A1) Gaussian Approximation Approach (A2) Scenario Approximation Approach (A3) Distributionally Robust Optimization Approach (A4) Computational Results IEEE 9-Bus System IEEE 39-Bus System 12 / 28
Solution Approaches: Mixed-Integer Linear programming (MILP) Approach (A1) Known as Sample Average Approximation (SAA) approach Reformulate individual chance constraints (10) � � A i x ≥ � � b i ≥ 1 − ǫ i i = 1 , . . . , m as P A s i x ≥ b s i − My i s ∀ s ∈ Ω , i = 1 , . . . , m (11) � s ∈ Ω p s y i s ≤ ǫ i , ∀ i, and y i s ∈ { 0 , 1 } ∀ s, i, (12) where M is a large scalar coefficient. Associate each s ∈ Ω with a binary logic variable y i s such that y i s = 0 indicates that A s i x ≥ b s i . y i s = 1 indicates that A s i x < b s i . 13 / 28
Solution Approaches: Gaussian Approximation Approach (A2) Consider an equivalent of individual chance constraints (10) � � A i x ≥ � � P b i ≥ 1 − ǫ i i = 1 , . . . , m � � � A ′ x ≤ b ′ i ¯ ≥ 1 − ǫ i i = 1 , . . . , m, (13) P i Assume the uncertainty is Gaussian distributed: � A ′ i ∼ N ( µ i , Σ i ) . Then, � i ∼ N ( µ T x T Σ i ¯ A ′ x − b ′ x − b ′ , ¯ i ¯ i ¯ x ) . We rewrite (13) as � i − µ T x ≥ Φ − 1 (1 − ǫ i ) b ′ x T Σ i ¯ i ¯ ¯ x i = 1 , . . . , m. (14) The above are second-order cone constraints if Φ − 1 (1 − ǫ i ) ≥ 0, i.e., 1 − ǫ i ≥ 0 . 5. 14 / 28
Solution Approaches: Scenario Approximation Approach (A3) Replace each chance constraint in (10) � � A i x ≥ � � b i ≥ 1 − ǫ i i = 1 , . . . , m with P A s i x ≥ b s i ∀ s ∈ Ω ap . (15) Both A1 and A2 require full distributional knowledge, while A3 requires large sample sizes and significant computation. 15 / 28
Solution Approaches: Distributionally Robust Optimization Approach (A4) The DR variant of (10): f ( ξ ) ∈D P ξ ( � A ξ i x ≥ � b ξ inf i ) ≥ 1 − ǫ i ∀ i = 1 , . . . , m. (16) The confidence set (description in a general way) Given samples { ξ i } N i =1 of ξ , we first calculate the empirical mean and � N � N i =1 ξ i and Σ 0 = 1 1 i =1 ( ξ − µ i 0 )( ξ − µ i 0 ) T , and covariance matrix as µ 0 = N N then build a confidence set � ξ ∈S f ( ξ ) dξ = 1 ( E [ ξ ] − µ 0 ) T (Σ 0 ) − 1 ( E [ ξ ] − µ 0 ) ≤ γ 1 D = f ( ξ ) : . E [( ξ − µ 0 )( ξ − µ 0 ) T ] � γ 2 Σ 0 16 / 28
Solution Approaches: Distributionally Robust Optimization Approach (A4) � H i � p i (Duality theory) Let r i , , and G i be the dual variables associated p T q i i with the three constraints in the above confidence set D , respectively. The individual chance constraints (16) are equivalent to γ 2 Σ 0 · G i + 1 − r i + Σ 0 · H i + γ 1 q i ≤ ǫ i y i (17) � G i � � � 2 ¯ 1 A x − p i 0 i � (18) 2 ( ¯ 1 y i + ( ¯ i ) T µ 0 − ¯ − p T A x i ) T A x b x 1 − r i i i � G i � � H i � − p i p i � 0 , � 0 , y i ≥ 0 , i = 1 , . . . , m, (19) − p T p T 1 − r i q i i i where operator “ · ” in constraint (17) represents Frobenius inner product of two matrices (i.e., A · B = tr( A T B )). This is a semi-definite program and can be solved by commercial solvers. Importantly, note that the above approaches for bounding the unknown f ( ξ ) are general and allow the uncertainty ξ to be time-varying, correlated, and endogenous . 17 / 28
Introduction Why Chance-Constrained Optimal Power Flow (CC-OPF) Problem? How to Solve CC-OPF? Notation Joint and Individual CC-OPF Models Solution Approaches Mixed-integer Linear programming (MILP) Approach (A1) Gaussian Approximation Approach (A2) Scenario Approximation Approach (A3) Distributionally Robust Optimization Approach (A4) Computational Results IEEE 9-Bus System IEEE 39-Bus System 18 / 28
Computational Results: IEEE 9-Bus System �� �� �� �� �� �� �� �� �� �� �� �� Figure: IEEE 9-bus system, with added wind generation. 19 / 28
Computational Results: IEEE 9-Bus System Table: Results to IEEE 9-Bus system with 1 − ǫ i = 95% Obj. Rel(%) CPU avg min max avg min max avg min max A1 J-CC-OPF 1349 1328 1363 77 8 95 2 1 4 I-CC-OPF 1346 1336 1357 72 46 90 5876 131 32817 A2 I-CC-OPF 1349 1340 1358 82 65 94 1 1 1 A3 I-CC-OPF 1408 1371 1525 100 99 100 55 54 57 A4 I-CC-OPF 1393 1365 1458 100 98 100 5 4 6 20 / 28
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