Two-Stage Stochastic Integer Programs Z d min { c ⊺ z : A N z = b , z ∈ Z + } A 0 0 · · · 0 0 · · · 0 T W T 0 W · · · 0 A N := . . . . ... . . . . . . . . 0 0 · · · T W with N denoting the number of scenarios, d = m + Nn , c = ( c 0 , c 1 , . . . , c N ) ⊺ := ( h , π 1 q , . . . , π N q ) ⊺ b = ( a , ξ 1 , . . . , ξ N ) ⊺ .
We will ◮ extend the notion of S-vectors ◮ retain the pattern of the previous completion method ◮ work on pairs ( u , V u ) (defined below) instead of vectors. ◮ employ a generalization of the Gordan-Dickson Lemma, Maclagan’s Theorem (Proc. AMS, 2001), to ensure termination of the algorithm. ◮ see, that the block angular structure of the problem matrix induces a symmetry structure on the elements of the Graver set. ◮ see, that the Graver set vectors are formed by a comparably small number of building blocks. ◮ compute these building blocks without computing the Graver set. ◮ reconstruct an improving vector to a given non-optimal feasible solution, scenario by scenario, using building blocks only. ◮ find an optimal solution with comparably small effort, once the building blocks have been computed.
Lemma ( u , v 1 , . . . , v N ) ∈ ker( A N ) if and only if ( u , v 1 ) , . . . , ( u , v N ) ∈ ker( A 1 ) . Definition Let z = ( u , v 1 , . . . , v N ) ∈ ker( A N ) and call the vectors u , v 1 , . . . , v N the building blocks of z . Denote by G N the Graver test set associated with A N and collect into H N all those vectors arising as building blocks of some z ∈ G N . By H ∞ denote the set � ∞ N =1 H N .
Lemma ( u , v 1 , . . . , v N ) ∈ ker( A N ) if and only if ( u , v 1 ) , . . . , ( u , v N ) ∈ ker( A 1 ) . Definition Let z = ( u , v 1 , . . . , v N ) ∈ ker( A N ) and call the vectors u , v 1 , . . . , v N the building blocks of z . Denote by G N the Graver test set associated with A N and collect into H N all those vectors arising as building blocks of some z ∈ G N . By H ∞ denote the set � ∞ N =1 H N . The set H ∞ contains both m -dimensional vectors u associated with the first-stage and n -dimensional vectors v related to the second-stage in the stochastic program. For convenience, we will arrange the vectors in H ∞ into pairs ( u , V u ) . Definition For fixed u ∈ H ∞ , all those vectors v ∈ H ∞ are collected into V u for which ( u , v ) ∈ ker( A 1 ) .
Finiteness of H ∞ Definition We say that ( u ′ , V u ′ ) reduces ( u , V u ) , or ( u ′ , V u ′ ) ⊑ ( u , V u ) for short, if the following conditions are satisfied: ◮ u ′ ⊑ u , ◮ for every v ∈ V u there exists a v ′ ∈ V u ′ with v ′ ⊑ v , ◮ u ′ � = 0 or there exist vectors v ∈ V u and v ′ ∈ V u ′ with 0 � = v ′ ⊑ v .
Theorem (Maclagan 2001) Let I be an infinite collection of monomial ideals in a polynomial ring. Then there are two ideals I , J ∈ I with I ⊆ J . Definition We associate with ( u , V u ) , u � = 0 ,and with (0 , V 0 ) the monomial ideals I ( u , V u ) ∈ Q [ x 1 , . . . , x 2 m +2 n ] I (0 , V 0 ) ∈ Q [ x 1 , . . . , x 2 n ] and generated by all the monomials x ( u + , u − , v + , v − ) with v ∈ V u , and by all the monomials x ( v + , v − ) with v � = 0 and v ∈ V 0 , respectively. Lemma Let (( u 1 , V u 1 ) , ( u 2 , V u 2 ) , . . . ) be a sequence of pairs such that ( u i , V u i ) �⊑ ( u j , V u j ) whenever i < j . Then this sequence is finite. Theorem (Finiteness of H ∞ ) Given rational matrices A , T , and W of appropriate dimensions, and let H ∞ be defined as above. Then H ∞ is a finite set.
Computation of H ∞ Idea: ◮ Retain the completion pattern of Graver set computation, but work with pairs ( u , V u ) instead. ◮ Define the two main ingredients, S-vectors and normalForm, that means the operations ⊕ and ⊖ , appropriately. ◮ Now, the objects f , g , and s all are pairs of the form ( u , V u ) . Algorithm (Extended normal form algorithm) Input: a pair s , a set G of pairs Output: a normal form of s with respect to G while there is some g ∈ G such that g ⊑ s do s := s ⊖ g return s
Algorithm (Compute H ∞ ) Input: a generating set F of ker( A 1 ) in ( u , V u ) -notation to be specified below Output: a set G which contains H ∞ G := F � C := { f ⊕ g } (forming S-vectors) f , g ∈ G while C � = ∅ do s := an element in C C := C \ { s } f := normalForm ( s , G ) if f � = (0 , { 0 } ) then C := C ∪ � { f ⊕ g } (adding S-vectors) g ∈ G ∪{ f } G := G ∪ { f } return G .
Choose as input the set of building blocks of all vectors in F ∪ { 0 } in ( u , V u ) -notation. Herein, F is a generating set for ker( A 1 ) over Z Z which contains a generating set for { (0 , v ) : Wv = 0 } ⊆ ker( A 1 ) consisting only of vectors with zero first-stage component. Definition (S-vectors, Reduction) Let ( u , V u ) ⊕ ( u ′ , V u ′ ) := ( u + u ′ , V u + V u ′ ) , where V u + V u ′ := { v + v ′ : v ∈ V u , v ′ ∈ V u ′ } . Moreover, let ( u , V u ) ⊖ ( u ′ , V u ′ ) := ( u − u ′ , { v − v ′ : v ∈ V u , v ′ ∈ V u ′ , v ′ ⊑ v } ) .
Feasibility at Building-Block Level Define the auxiliary cost function c ′ by � if z ( i ) 0 ≥ 0 ( c ′ ) ( i ) := 1 , for i = 1 , . . . , m + Nn . if z ( i ) − 1 < 0 1
Consider the two-stage program N min { 35 x 1 + 40 x 2 + 1 � 16 y ν 1 + 19 y ν 2 + 47 y ν 3 + 54 y ν 4 : N ν =1 x 1 + y ν 1 + y ν ξ ν ≥ 1 , 3 x 2 + y ν 2 + y ν ξ ν ≥ 2 , 4 2 y ν 1 + y ν ξ ν ≤ 3 , 2 y ν 1 + 2 y ν ξ ν ≤ 4 , 2 x 1 , x 2 , y ν 1 , y ν 2 , y ν 3 , y ν 4 ∈ Z Z + } R s is given by the scenarios ξ 1 , . . . , ξ N , all with equal Here, the random vector ξ ∈ I probability 1 / N . The realizations of ( ξ ν 1 , ξ ν 2 ) and ( ξ ν 3 , ξ ν 4 ) are given by uniform grids (of differing granularity) in the squares [300 , 500] × [300 , 500] and [0 , 2000] × [0 , 2000] , respectively. Timings are given in CPU seconds on a SUN Enterprise 450 , 300 MHz Ultra-SPARC. It took 3 . 3 seconds to compute H ∞ altogether consisting of 1438 building blocks arranged into 25 pairs ( u , V u ) . Aug ( H ∞ ) then gives the times needed to augment the solution x 1 = x 2 = y ν 1 = y ν 2 = 0 , y ν 3 = ξ ν 1 , and y ν 4 = ξ ν 2 , ν = 1 , . . . N to optimality. Example ( ξ 1 , ξ 2 ) -grid ( ξ 3 , ξ 4 ) -grid scen. var. Aug ( H ∞ ) CPLEX dualdec 1 5 × 5 3 × 3 225 902 1 . 52 0 . 63 > 1800 2 5 × 5 21 × 21 11025 44102 66 . 37 696 . 10 − 3 9 × 9 21 × 21 35721 142886 180 . 63 > 1 day − Although further exploration is necessary, the above table seems to indicate linear dependence of the computing time on the number N of scenarios, once H ∞ has been computed.
II Unit Commitment – A Recurring Issue in Power Management
Step Back in Time for 101 years
Step Back in Time for 101 years 1915
Step Back in Time for 101 years 1915 Zschornewitz
Step Back in Time for 101 years 1915 Zschornewitz Biggest lignite-fired thermal power station of its time inaugurated.
Step Back in Time for 101 years 1915 Zschornewitz Biggest lignite-fired thermal power station of its time inaugurated.
Step Back in Time for 101 years 1915 Zschornewitz Biggest lignite-fired thermal power station of its time inaugurated. ◮ Build within one year (Groundbreaking March 24, 1915, First Turbine (16 MW) in Operation December 15, 1915, 1916: 8 × 16 MW installed)
Step Back in Time for 101 years 1915 Zschornewitz Biggest lignite-fired thermal power station of its time inaugurated. ◮ Build within one year (Groundbreaking March 24, 1915, First Turbine (16 MW) in Operation December 15, 1915, 1916: 8 × 16 MW installed)
VEAG Kraftw e rke u n d Ye rb u n d n etz Remarkable: In operation until 30.06.1992
VEAG Kraftw e rke u n d Ye rb u n d n etz Remarkable: In operation until 30.06.1992
Unit Commitment ◮ is the problem of determining switching and operational decisions, ◮ for a system of power producing units, over some time horizon, ◮ so that all relevant technological and economical conditions are met.
Unit Commitment ◮ is the problem of determining switching and operational decisions, ◮ for a system of power producing units, over some time horizon, ◮ so that all relevant technological and economical conditions are met. 1985 VEAG in (East Germany)
Unit Commitment ◮ is the problem of determining switching and operational decisions, ◮ for a system of power producing units, over some time horizon, ◮ so that all relevant technological and economical conditions are met. 2006 Virtual Power Plant 1985 VEAG in (East Germany)
Specification (Mixed-Integer Linear Program – When Deterministic) Unit Commitment for a hydro-thermal system (early VEAG + Vattenfall)
Specification (Mixed-Integer Linear Program – When Deterministic) Unit Commitment for a hydro-thermal system (early VEAG + Vattenfall) � � c ⊤ 1 ξ 1 + c ⊤ min 2 ξ 2 : A 1 ξ 1 + A 2 ξ 2 = b , ξ 1 ∈ X 1 , ξ 2 ∈ X 2 Variables: ◮ ξ 1 : start-up/shut-down for thermal units, ◮ ξ 2 : all remaining, i.e., power output, pumping/generating in pumped-storage (psp), water levels in psp, auxillary variables for modeling specific effects.
Specification (Mixed-Integer Linear Program – When Deterministic) Unit Commitment for a hydro-thermal system (early VEAG + Vattenfall) � � c ⊤ 1 ξ 1 + c ⊤ min 2 ξ 2 : A 1 ξ 1 + A 2 ξ 2 = b , ξ 1 ∈ X 1 , ξ 2 ∈ X 2 Variables: ◮ ξ 1 : start-up/shut-down for thermal units, ◮ ξ 2 : all remaining, i.e., power output, pumping/generating in pumped-storage (psp), water levels in psp, auxillary variables for modeling specific effects. Objective: ◮ affinely linear fuel costs for operation and piece-wise constant for switching of thermal units
Specification (Mixed-Integer Linear Program – When Deterministic) Unit Commitment for a hydro-thermal system (early VEAG + Vattenfall) � � c ⊤ 1 ξ 1 + c ⊤ min 2 ξ 2 : A 1 ξ 1 + A 2 ξ 2 = b , ξ 1 ∈ X 1 , ξ 2 ∈ X 2 Variables: ◮ ξ 1 : start-up/shut-down for thermal units, ◮ ξ 2 : all remaining, i.e., power output, pumping/generating in pumped-storage (psp), water levels in psp, auxillary variables for modeling specific effects. Objective: ◮ affinely linear fuel costs for operation and piece-wise constant for switching of thermal units Constraints: ◮ connecting units: load balances, reserve balances, ramping ◮ for individual units: output bounds, minimum up- and down-times, water management in psp,
Unit Commitment Under UNCERTAINTY in the 1970ies and 1980ies
Unit Commitment Under UNCERTAINTY in the 1970ies and 1980ies
Unit Commitment under Uncertainty Fourty Years Ago
Unit Commitment under Uncertainty Fourty Years Ago ◮ Before deregulation, power producers optimized costs by fuel cost minimization, with power demand as major source of uncertainty.
Unit Commitment under Uncertainty Fourty Years Ago ◮ Before deregulation, power producers optimized costs by fuel cost minimization, with power demand as major source of uncertainty. ◮ TV sets consumed more energy than today. Their operation had to be included when estimating power demand, at least during certain periods of the day.
Unit Commitment under Uncertainty Fourty Years Ago ◮ Before deregulation, power producers optimized costs by fuel cost minimization, with power demand as major source of uncertainty. ◮ TV sets consumed more energy than today. Their operation had to be included when estimating power demand, at least during certain periods of the day. ◮ In the 1970ies and 1980ies Heavyweight Boxing was a very popular spectator sport (Ali, Frazier, Foreman etc.), in West and East Germany.
Unit Commitment under Uncertainty Fourty Years Ago ◮ Before deregulation, power producers optimized costs by fuel cost minimization, with power demand as major source of uncertainty. ◮ TV sets consumed more energy than today. Their operation had to be included when estimating power demand, at least during certain periods of the day. ◮ In the 1970ies and 1980ies Heavyweight Boxing was a very popular spectator sport (Ali, Frazier, Foreman etc.), in West and East Germany. ◮ Time zone difference and duration of fight (knock-out: if at all and when) produced random variables that were hard to handle ...
Unit Commitment under Uncertainty Fourty Years Ago ◮ Before deregulation, power producers optimized costs by fuel cost minimization, with power demand as major source of uncertainty. ◮ TV sets consumed more energy than today. Their operation had to be included when estimating power demand, at least during certain periods of the day. ◮ In the 1970ies and 1980ies Heavyweight Boxing was a very popular spectator sport (Ali, Frazier, Foreman etc.), in West and East Germany. ◮ Time zone difference and duration of fight (knock-out: if at all and when) produced random variables that were hard to handle ... and (induced) water consumption was uncertain, too!
Unit Commitment under Uncertainty Fourty Years Ago ◮ Before deregulation, power producers optimized costs by fuel cost minimization, with power demand as major source of uncertainty. ◮ TV sets consumed more energy than today. Their operation had to be included when estimating power demand, at least during certain periods of the day. ◮ In the 1970ies and 1980ies Heavyweight Boxing was a very popular spectator sport (Ali, Frazier, Foreman etc.), in West and East Germany. ◮ Time zone difference and duration of fight (knock-out: if at all and when) produced random variables that were hard to handle ... and (induced) water consumption was uncertain, too!
Specification (continued) Unit Commitment with random load
Specification (continued) Unit Commitment with random load � � f ( ξ 1 , ω ) = [ c ⊤ c ⊤ 1 ξ 1 + min 2 ξ 2 : A 2 ξ 2 = b ( ω ) − A 1 ξ 1 , ω ∈ Ω ξ 2 ∈ X 2 � �� � � c ⊤ c ⊤ Q E ( ξ 1 ) := 1 ξ 1 + min 2 ξ 2 : A 2 ξ 2 = b ( ω ) − A 1 ξ 1 P ( d ω ) ξ 2 ∈ X 2 Ω
Specification (continued) Unit Commitment with random load � � f ( ξ 1 , ω ) = [ c ⊤ c ⊤ 1 ξ 1 + min 2 ξ 2 : A 2 ξ 2 = b ( ω ) − A 1 ξ 1 , ω ∈ Ω ξ 2 ∈ X 2 � �� � � c ⊤ c ⊤ Q E ( ξ 1 ) := 1 ξ 1 + min 2 ξ 2 : A 2 ξ 2 = b ( ω ) − A 1 ξ 1 P ( d ω ) ξ 2 ∈ X 2 Ω Variables: ◮ ξ 1 ∈ X 1 : start-up/shut-down for thermal units, ◮ ξ 2 ( ω ) : all remaining, i.e., power output, pumping/generating in pumped-storage (psp), water levels in psp, auxillary variables for modeling specific effects.
Specification (continued) Unit Commitment with random load � � f ( ξ 1 , ω ) = [ c ⊤ c ⊤ 1 ξ 1 + min 2 ξ 2 : A 2 ξ 2 = b ( ω ) − A 1 ξ 1 , ω ∈ Ω ξ 2 ∈ X 2 � �� � � c ⊤ c ⊤ Q E ( ξ 1 ) := 1 ξ 1 + min 2 ξ 2 : A 2 ξ 2 = b ( ω ) − A 1 ξ 1 P ( d ω ) ξ 2 ∈ X 2 Ω Variables: ◮ ξ 1 ∈ X 1 : start-up/shut-down for thermal units, ◮ ξ 2 ( ω ) : all remaining, i.e., power output, pumping/generating in pumped-storage (psp), water levels in psp, auxillary variables for modeling specific effects. Objective: f ( ξ 1 , . ) random cost profile for operation and switching of thermal units inuced by start-up/shut-down scheme ξ 1 � Q E ( ξ 1 ) := f ( ξ 1 , ω ) P ( d ω ) − − − Expected Value – Risk Neutral Model Ω
Unit Commitment under Uncertainty over the Years
Unit Commitment under Uncertainty over the Years � � � � c ⊤ x + min q ⊤ y : Wy = h ( ω ) − Tx , y ∈ Y min : x ∈ X x y � �� � f ( x ,ω ) ◮ 1985: Load the only quantity with relevant uncertainty - Risk neutral models, only ! f ( x , z ( ω )) − total cost for up/down regime x under random load z ( ω )
Unit Commitment under Uncertainty over the Years � � � � c ⊤ x + min q ⊤ y : Wy = h ( ω ) − Tx , y ∈ Y min : x ∈ X x y � �� � f ( x ,ω ) ◮ 1985: Load the only quantity with relevant uncertainty - Risk neutral models, only ! f ( x , z ( ω )) − total cost for up/down regime x under random load z ( ω ) ◮ 2006: After deregulation omnipresent uncertainty at input (renewables) and output sides. - Risk aversion became more and more indispensable ! f ( x , z ( ω )) − total cost for aquisition x of a vpp under random power in- and outputs z
Unit Commitment under Uncertainty over the Years � � � � c ⊤ x + min q ⊤ y : Wy = h ( ω ) − Tx , y ∈ Y min : x ∈ X x y � �� � f ( x ,ω ) ◮ 1985: Load the only quantity with relevant uncertainty - Risk neutral models, only ! f ( x , z ( ω )) − total cost for up/down regime x under random load z ( ω ) ◮ 2006: After deregulation omnipresent uncertainty at input (renewables) and output sides. - Risk aversion became more and more indispensable ! f ( x , z ( ω )) − total cost for aquisition x of a vpp under random power in- and outputs z ◮ 2010: Congestion and capacity management under uncertain in- and outputs f ( x , z ( ω )) − x pre-commitment so that renewables’ inflow z compensated with minimal re-commitment/re-dispatch and without overloading grid components
III Some Thoughts on Suitable Mathematics
Viewpoints (I) Ill-posed optimization problem Destructive – remove stochasticity swiftly, min { f ( x , ω ) : x ∈ X } As long as ω is unknown, it makes no sense to address optimality. Remedy: Arrive at a deterministic problem by “removing ω in formal manner”. ◮ Replace ω by its expectation E [ ω ] and solve min { f ( x , E [ ω ]) : x ∈ X } ◮ Consider expected value E [ f ( x , ω )] and solve min { E [ f ( x , ω )] : x ∈ X } ◮ Apply a statistical parameter S and solve min {S [ f ( x , ω )] : x ∈ X }
(II) Optimizing or ranking in a family of random variables Constructive: Be happy about havng stochastic information on the uncertain problem ingredients. Make active use of it. { f ( x , . ) : Ω → R } x ∈ X Remedy: Arrive at a deterministic problem by implementing your attitude towards risk . ◮ Risk neutral: Apply expectation E to f ( x , ω ) and solve min { E [ f ( x , ω )] : x ∈ X } ◮ Risk averse by criterion: Apply some risk measure R and solve min {R [ f ( x , ω )] : x ∈ X } ◮ Risk averse by constraint: Rank according to some stochastic order. Introduce a benchmark random variable b ( ω ) leading to the constraint { x ∈ X : f ( x , ω ) � b ( ω ) }
Solution by Scenario Decomposition � �� � � c ⊤ c ⊤ Q E ( ξ 1 ) := 1 ξ 1 + min 2 ξ 2 : A 2 ξ 2 ( ω ) = b ( ω ) − A 1 ξ 1 P ( d ω ) ξ 2 ∈ X 2 Ω Assume the rhs b ( ω ) is the only random ingredient, and let it follow a finite discrete probability distribution with scenarios b 1 , . . . , b ω , . . . , b S and probabilities π 1 , . . . , π ω , . . . , π S Then min { Q E ( ξ 1 ) : ξ 1 ∈ X 1 } is equivalent to the following large-scale block angular mixed-integer linear program S � � c ⊤ π ω c ⊤ min 1 ξ 1 + 2 ξ 2 ω : A 1 ξ 1 + A 2 ξ 21 = b 1 ω =1 . . ... . . . . A 1 ξ 1 + A 2 ξ 2 ω = b ω . . ... . . . . A 1 ξ 1 + A S ξ 2 S = b S � ξ 1 ∈ X 1 , ξ 2 ω ∈ X 2 , ω = 1 , . . . , S
Scenario Decomposition Basic Idea: Lagrangean Relaxation of Nonanticipativity[ Carøe/Sch.1999 ]:
Scenario Decomposition Basic Idea: Lagrangean Relaxation of Nonanticipativity[ Carøe/Sch.1999 ]: Introduce copies ξ 11 , . . . , ξ 1 ω , . . . , ξ 1 S of ξ 1 and add ξ 11 = . . . = ξ 1 ω = . . . = ξ 1 S . Then apply Lagrangean Relaxation on the chain of identites.
Scenario Decomposition Basic Idea: Lagrangean Relaxation of Nonanticipativity[ Carøe/Sch.1999 ]: Introduce copies ξ 11 , . . . , ξ 1 ω , . . . , ξ 1 S of ξ 1 and add ξ 11 = . . . = ξ 1 ω = . . . = ξ 1 S . Then apply Lagrangean Relaxation on the chain of identites. NA This includes solving the Lagrangean Dual which is a non-differentiable convex optimization problem.
Procedure:
Procedure: ◮ Objective function of the Lagrangean dual involves a minimization which is separable with respect to the scenarios.
Procedure: ◮ Objective function of the Lagrangean dual involves a minimization which is separable with respect to the scenarios. ◮ Regaining primal feasibility after maximization in the Lagrangean dual benefits from simplicity of relaxed constraints (chain of identities).
Procedure: ◮ Objective function of the Lagrangean dual involves a minimization which is separable with respect to the scenarios. ◮ Regaining primal feasibility after maximization in the Lagrangean dual benefits from simplicity of relaxed constraints (chain of identities). ◮ Duality gap inevitable, unless problem “is not really integer”. If gap inacceptable then imbedding into branch-and-bound on ξ 1 ∈ X 1 .
Procedure: ◮ Objective function of the Lagrangean dual involves a minimization which is separable with respect to the scenarios. ◮ Regaining primal feasibility after maximization in the Lagrangean dual benefits from simplicity of relaxed constraints (chain of identities). ◮ Duality gap inevitable, unless problem “is not really integer”. If gap inacceptable then imbedding into branch-and-bound on ξ 1 ∈ X 1 . This works nicely as long as mixed-integer linear programming formulation has the block structure
III Congestion Management in Power Nets
Load Flow Models – AC, DC, Ohmic Losses Graph G = ( V , E ) (undirected) with nodes v ∈ V = { 1 , . . . , n } , edges e ∈ E ⊆ V × V .
Load Flow Models – AC, DC, Ohmic Losses Graph G = ( V , E ) (undirected) with nodes v ∈ V = { 1 , . . . , n } , edges e ∈ E ⊆ V × V . AC Model with Voltage in Polar Coordinates
Load Flow Models – AC, DC, Ohmic Losses Graph G = ( V , E ) (undirected) with nodes v ∈ V = { 1 , . . . , n } , edges e ∈ E ⊆ V × V . AC Model with Voltage in Polar Coordinates For all nodes v , voltage as complex number U v e j θ v with modulus U v and voltage angle θ v , for slack node U 1 = 1 , θ 1 = 0 .
Load Flow Models – AC, DC, Ohmic Losses Graph G = ( V , E ) (undirected) with nodes v ∈ V = { 1 , . . . , n } , edges e ∈ E ⊆ V × V . AC Model with Voltage in Polar Coordinates For all nodes v , voltage as complex number U v e j θ v with modulus U v and voltage angle θ v , for slack node U 1 = 1 , θ 1 = 0 . Moreover, for all e = vl ∈ E . ◮ Difference of Voltage Angles θ vl = θ v − θ l ,
Load Flow Models – AC, DC, Ohmic Losses Graph G = ( V , E ) (undirected) with nodes v ∈ V = { 1 , . . . , n } , edges e ∈ E ⊆ V × V . AC Model with Voltage in Polar Coordinates For all nodes v , voltage as complex number U v e j θ v with modulus U v and voltage angle θ v , for slack node U 1 = 1 , θ 1 = 0 . Moreover, for all e = vl ∈ E . ◮ Difference of Voltage Angles θ vl = θ v − θ l , ◮ Active Load Flow p vl , ◮ Reactive Load Flow q vl .
Load Flow Models – AC, DC, Ohmic Losses Graph G = ( V , E ) (undirected) with nodes v ∈ V = { 1 , . . . , n } , edges e ∈ E ⊆ V × V . AC Model with Voltage in Polar Coordinates For all nodes v , voltage as complex number U v e j θ v with modulus U v and voltage angle θ v , for slack node U 1 = 1 , θ 1 = 0 . Moreover, for all e = vl ∈ E . ◮ Difference of Voltage Angles θ vl = θ v − θ l , ◮ Active Load Flow p vl , ◮ Reactive Load Flow q vl . For all edges in E (AC) Load Flow Equations U 2 p vl = v g vl − U v U l g vl cos θ vl − U v U l b vl sin θ vl ∀ vl ∈ E U v U l b vl cos θ vl − U v U l g vl sin θ vl − U 2 v ( b vl + b 0 q vl = vl ) ∀ vl ∈ E
Load Flow Models DC
Load Flow Models DC Simplification
Load Flow Models DC Simplification ◮ θ vl ≈ 0 ∀ vl ∈ E hence sin θ vl ≈ θ vl und cos θ vl ≈ 1
Load Flow Models DC Simplification ◮ θ vl ≈ 0 ∀ vl ∈ E hence sin θ vl ≈ θ vl und cos θ vl ≈ 1 ◮ U v = 1 ∀ v ∈ V
Load Flow Models DC Simplification ◮ θ vl ≈ 0 ∀ vl ∈ E hence sin θ vl ≈ θ vl und cos θ vl ≈ 1 ◮ U v = 1 ∀ v ∈ V ◮ No reactive power components.
Load Flow Models DC Simplification ◮ θ vl ≈ 0 ∀ vl ∈ E hence sin θ vl ≈ θ vl und cos θ vl ≈ 1 ◮ U v = 1 ∀ v ∈ V ◮ No reactive power components. From AC equations, only the first remains and becomes: DC Load Flow Equation p vl = b vl ( θ l − θ v ) for all vl ∈ E
Load Flow Models DC Load Flow with Ohmic Losses
Load Flow Models DC Load Flow with Ohmic Losses 1/g kl 2 M kl 0 2 2 kl Loss on vl ∈ E ν vl = g vl ( U 2 v + U 2 l ) − 2 g vl U v U l cos ( θ v − θ l ) ∀ vl ∈ E
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