The first term is known as the instantaneous real/active power and the second term the instantaneous reactive power. The constants real/active power P and reactive powerQ characterise these sinusoidal terms:
P = VpIp
2 cos(θ−φ) =V Icos(θ−φ) (A.11)
Q= VpIp
2 sin(θ−φ) =V Isin(θ−φ) (A.12) Physically, real powerP represents the average power consumed/produced. It also represents the peak amplitude of the element of power that does work. Reactive powerQis the peak amplitude of the element of power that oscillates back and forth without doing any net work. It is the result of energy being periodically stored and released in reactive components like capacitors and inductors. The sign of reactive power modifies the phase of the element it represents by 180◦.
Together real and reactive power characterise instantaneous power, and they form a complex constant called the complex power S:
S=P +iQ (A.13)
The magnitude of the complex powerS =|S|is called the apparent power. The complex power consumed/produced by a two-terminal component can be calculated in phasor notation by multiplying the complex conjugate of the current by the voltage across the component:
S=IabVab (A.14)
Intuitively, one might expect this operation to produce a complex form of the instantaneous power, but this is not the case. The time dependence of the phasors cancel out when multiplied in this way, producing a complex constant. Note that if peak amplitudes for the current and voltage are used instead of RMS values, then a factor of 12 would need to be applied to the RHS of this equation.
A.1.1 Balanced 3-Phase
A balanced 3-phase system is one where the load and line impedances in each phase are identical. The voltages (currents) are the same in each phase, apart from being 120◦out of phase from each other. If the phases are labelled
a, b and c, and the neutral labelled n, the relations between the different line-to-neutral voltages (same for line currents) is given by:
Van=Vbn∠120◦ =Vcn∠240◦ (A.15)
The same relation holds for the line-to-line voltages:
Vab =Vbc∠120◦ =Vca∠240◦ (A.16)
which are related to the line-to-neutral voltages by:
Vab =
√
3Van∠30◦ (A.17)
The total complex power is the sum of power on each phase or just three times the power on one of those phases in a balanced system:
S= 3Sa= 3IaVan (A.18)
whereVan is the phaseato neutral voltage and Ia is the phaseacurrent.
The balanced 3-phase system can be treated as a single phase system by converting ∆ connected loads to their Y equivalent:
ZY = Z∆
3 (A.19)
There are a few options when it comes to dealing with the factor of three that appears in the total complex power equation. Either it can be used in all calculations, the power in just one phase can be used instead, or the factor of three can be absorbed by other terms. It does not really matter what approach is taken as long as it is applied consistently and the physical significance as it ties back to the 3-phase system is remembered. We adopt the latter approach, and have the voltage and current terms soak up the factor: Z=ZY (A.20) V= √ 3Van =Vab∠−30◦ (A.21) I=√3Ia (A.22)
After applying this transformation, Ohm’s law, Kirchhoff’s laws and com- plex power are calculated as for any single phase system, and the calculated powers are the total powers for the full 3-phase system.
Appendix B
Proofs
Lemma 1. Given the convex functions f : RM 7→ R and gj : RM 7→
R, if (∇f(x) − ∇f(y))T(x −y) = PNj=1(bj∇gj(y)− aj∇gj(x))T(x −y),
ajgj(x) = bjgj(y) = 0 and gj(x), gj(y) ≤0 for some x, y ∈ RM, aj, bj ≥ 0
and subgradients∇f(·)∈∂f(·),∇gj(·)∈∂gj(·), thenbjgj(x) =ajgj(y) = 0,
(∇f(x)− ∇f(y))T(x−y) = 0 and (a
j∇gj(x)−bj∇gj(y))T(x−y) = 0.
Proof. We define Γ to be equal to the LHS of the first equation:
Γ := (∇f(x)− ∇f(y))T(x−y) = N X j=1 (bj∇gj(y)−aj∇gj(x))T(x−y) (B.1) Using the convexity off andgj we get the following bounds on Γ:
Γ≥(f(x)−f(y)) + (f(y)−f(x)) = 0 (B.2) Γ≤ N X j=1 bj(gj(x)−gj(y)) +aj(gj(y)−gj(x)) (B.3)
The conditionajgj(x) =bjgj(y) = 0 simplifies the second inequality to:
Γ≤
N
X
j=1
bjgj(x) +ajgj(y) (B.4)
The conditions gj(x), gj(y) ≤0 and aj, bj ≥ 0 imply that bjgj(x) ≤ 0 and
ajgj(y) ≤0, but (B.2) and (B.4) requires that they sum to a non-negative
value. This is only possible if the first result holds:
bjgj(x) =ajgj(y) = 0 (B.5)
This means that Γ≤0, which combined with (B.2) implies Γ = 0, which is the second result. Using the convexity ofgj and applying the first result we
get:
(bj∇gj(y)−aj∇gj(x))T(x−y)
≤bj(gj(x)−gj(y)) +aj(gj(y)−gj(x)) = 0 (B.6)
The definition of Γ and the result Γ = 0 requires that these non-positive terms sum to zero, which is only possible if the final result holds:
(bj∇gj(y)−aj∇gj(x))T(x−y) = 0 (B.7)
Lemma 2 (No Violation). If all agents satisfy the RPAR, then for each agent i: ∆λT∆Pi = 0.
Proof. The power conservation rule requires:
A X i=1 Pi = 0 A X i=1 Pi∗= 0 (B.8)
Taking the difference and multiplying through by ∆λ:
A
X
i=1
∆λT∆Pi = 0 (B.9)
This is a sum of non-positive values because all agents satisfy the RPAR: ∆λT∆Pi ≤0. Therefore for each agenti:
∆λT∆Pi = 0 (B.10)
Lemma 3(Feasible Powers Adversary). Assume a convex agentiwith fixed power Pˆi ∈ RT that has non-zero power requirements in the first time step of the horizon Pˆi,1 6= 0. For any untruthful pure strategy that changes the
feasible set of powers for the agent at a future time step t∈ {2, . . . , T} (i.e. from Pi,t ∈ {Pˆi,t} to Pi,t ∈[
¯
Pi,P¯i] for some
¯
Pi ≤P¯i), there exists a convex
adversary which will increase the costs for the agent relative to their truthful strategy.
Proof. We build an environment where there is only one other agentj, the adversary, with powerPj ∈RT and convex cost function:
127 for some constants a and b. Because there are only two agents, the power conservation constraint requiresPj =−Pi. We set the power bounds on the
adversary so that they are looser than those for the fixed power agent (i.e. they are never active). The KKT conditions for the adversary at the two time steps of interest 1 andtare (the rest are trivially satisfied):
∇1fj(Pj) +λ1= 0 (B.12)
∇tfj(Pj) +λt= 0 (B.13)
Evaluating and substituting−Pi forPj:
2(−Pˆi,1−aPi,t+b) +λ1 = 0 (B.14)
2a(−Pˆi,1−aPi,t+b) +λt= 0 (B.15)
The value of Pi,t can take on one of three values. First assume λt = 0
(bounds from fixed power agent not active). This gives the valuePi,t0 :
2a(−Pˆi,1−aPi,t0 +b) = 0 (B.16)
=⇒ Pi,t0 = (−Pˆi,1+b)/a (B.17)
Note that for any ˆPi,1 anda6= 0 we can choose absuch thatPi,t0 ∈R. Since
fj is convex, the value of Pi,t is:
Pi,t = min(max(Pi,t0 ,P¯i),P¯i) (B.18)
The cost for the agent in the first time step is:
ci,1=λ1Pˆ1 (B.19)
Eliminatingλ1 and Pi,t:
ci,1 = 2 ˆPi,1( ˆPi,1+amin(max(Pi,t0 ,¯Pi),P¯i)−b) (B.20)
For the case that the agent tells the truth ¯Pi =
¯
Pi = ˆPi,t:
ˆ
ci,1 = 2 ˆPi,1( ˆPi,1+aPˆi,t−b) (B.21)
The cost change relative to this truthful case is:
ci,1−cˆi,1 = 2aPˆi,1(min(max(Pi,t0 ,P¯i),P¯i)−Pˆi,t) (B.22)
For given ¯
Pi ≤ P¯i and ˆPi,1 we can always select an a and Pi,t0 where this
change is greater than zero, unless ˆPi,1 = 0 (which the lemma prohibits) or
¯
Nomenclature
Acronyms
AC Alternating current
AC Exact AC (line model)
ACT Australian Capital Territory
ADMM Alternating direction method of multipliers
AEMO Australian energy market operator
AM Additive model
BOM Bureau of meteorology
DC Direct current
DC Linear DC (line model)
DET Distributed energy technology
DF Dist-flow (line model)
DLC Direct load control
DLMP Dynamic locational marginal prices
DR Demand response
DSM Demand-side management
EV Electric vehicle
EMS Energy management system
FIT Feed-in tariff
GAM Generalised additive model
GLM Generalised linear model 129
HVAC Heating, ventilation and air conditioning
K Quadratic approximation (line model)
KCL Kirchhoff’s current law
KKT Karush-Kuhn-Tucker
LDC Linear DC (line model)
MDP Markov decision process
MILP Mixed-integer linear programming
MIP Mixed-integer programming
MIQP Mixed-integer quadratic programming
MPC Model predictive control
NEM National energy market
NSW New South Wales
OPF Optimal power flow
PV Photovoltaics
QA Quadratic approximation (line model)
QC Quadratic constraint (line model)
QP Quadratic programming
SH Single horizon (problem)
RD Relax and decide (method)
RH Receding horizon
RP Relax and price (method)
RTP Real-time pricing
TOU Time-of-use (pricing)
TRH Terminating receding horizon (problem)
UR Unrelaxed (method)
131
Symbols
A Number of agents
A Area
A Connection constraint matrix
A Anticipated cost error set
α Anticipated cost error
α Relax and price penalty parameter
B Susceptance c Cost d Device d Component D Device set D Component set E Energy E Connection set η Efficiency f Cost function g Constraint function G Conductance γ KKT multiplier h Power function
h Connection constraint function
h Horizon
I Global irradiance
I Current
k Iteration
K Known random parameter index set
κ Specific heat capacity
L Number of parameters
L Lagrangian
λ KKT multiplier
λ Dynamic price
Λ Relax and price penalty matrix
m Number of sampled scenarios
m Mass
M Number of variables
µ KKT multiplier
n Bus/node
n Number of inter-phase terminal constraints
N Number of constraints
ν Big-M style values
ω Angular frequency
p Probability
P Real power
φ Current phase angle
ψ Price/cost coefficient
Ψ Price/cost coefficient
q Stochastic programming subproblem function
Q Reactive power
r Device parameters
r Residual
R Thermal resistance
133 ρ Step/penalty parameter s Scenario s Scale parameter S Scenario set S Complex power t Time step
T Number of time steps
T Horizon length
T Temperature
T Terminal set
τ Time
θ Voltage phase angle
V Voltage
w Random parameter value
W Random parameter
X Reactance
List of Publications
Publications
Paul Scott and Sylvie Thi´ebaux. Distributed Multi-Period optimal power flow for demand response in microgrids. In ACM e-Energy, Bangalore, India, jul 2015. “Best Paper Award”.
Paul Scott, Sylvie Thi´ebaux, Menkes van den Briel, and Pascal Van Hen- tenryck. Residential demand response under uncertainty. InInternational Conference on Principles and Practice of Constraint Programming (CP), pages 645–660, Uppsala Sweden, sep. 2013.
Menkes van den Briel, Paul Scott, and Sylvie Thiebaux. Randomized load control: A simple distributed approach for scheduling smart appliances. In
International Joint Conference on Artificial Intelligence (IJCAI), Beijing, China, August 2013.
Other Papers
Carleton Coffrin, Dan Gordon, and Paul Scott. NESTA, the NICTA energy system test case archive. In arXiv, CoRR, USA, nov 2014.
Paul Scott and Sylvie Thi´ebaux. Distributed Multi-Period optimal power flow for demand response in microgrids. In OptMAS, Istanbul, Turkey, may 2015.
Paul Scott, Sylvie Thi´ebaux, Menkes van den Briel, and Pascal Van Hen- tenryck. Residential demand response under uncertainty. In Green CO- PLAS, Beijing, China, aug 2013.
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