Stochastic Interchange Scheduling in the Real-Time Electricity Market

0.18 MB·

by  Yuting Ji

Stochastic Interchange Scheduling in the Real-Time
Electricity Market
Yuting Ji, Tongxin Zheng, Senior Member, IEEE, and Lang Tong, Fellow, IEEE
Abstract—The problem of multi-area interchange scheduling The goal of this paper is to obtain the optimal interchange
in the presence of stochastic generation and load is considered. schedule in the presence of system and operation uncertainty.
A new interchange scheduling technique based on a two-stage
To this end, we propose a two-stage stochastic optimization
6 stochastic minimization of overall expected operating cost is
formulation aimed at minimizing the expected overall system
1 proposed. Because directly solving the stochastic optimization is
0 intractable, an equivalent problem that maximizes the expected cost.Theproposedoptimizationframeworktakesintoaccount
2 social welfare is formulated. The proposed technique leverages random fluctuations of load and renewable generations in the
theoperator’scapabilityofforecastinglocationalmarginalprices systems. Because directly solving the stochastic optimization
n
(LMPs) and obtains the optimal interchange schedule without
a is intractable, this paper presents an approach to transfer the
iterations among operators.
J stochastic optimization problem into an equivalent determin-
Index Terms—Inter-regional interchange scheduling, multi-
0 area economic dispatch, seams issue. istic problem that maximizes the expected economic surplus.
1 This transformation allows us to generalize the deterministic
I. INTRODUCTION TOsolutionbyintersectingexpecteddemandandsupplyfunc-
]
Y Since the restructuring of the electric power industry, in- tions, therefore avoiding costly iterative computationbetween
S dependent system operators (ISOs) and regional transmission operators.
. organizations(RTOs)havefacedtheseamsissuecharacterized
s A. Related Work
c by the inefficient transfer of power between neighboring
[ regions. Such inefficiency is caused by incompatible market There have been extensive studies on the seams issue. In

designs of independently controlled operating regions, incon- this paper, we do not consider inefficiencies arise from mar-
1
v sistencies of their scheduling protocols, and their different ket designs; we focus instead on optimizing the interchange
1 pricing models. The economic loss due to seams for the New schedule.We highlightbelow approachesmost relevantto the
9 York and New England customers is estimated at the level of technique developed here. For broadly related work, see [2],
1 $784 million annually [2]. [5]–[10] and references therein.
2
There has been recent effort in addressing the seams issue Mathematically, optimal interchange scheduling can be
0
. by optimizing interchange flows across different regions. In obtained from the multi-area Optimal Power Flow (OPF)
1
particular, a new interchange scheduling technique, referred problem,which is a decentralized optimization of power flow
0
6 to as Tie Optimization (TO), is proposed in [2] to minimize thatcanbesolvedusingvariousdecompositiontechniques[5].
1 the overall operating cost. The Federal Energy Regulatory A general approach is based on the principle of Lagrangian
: Commission (FERC) has recently approved the Coordinated Relaxation (LR) that decomposes the original problem into
v
i TransactionScheduling(CTS)thatallowsmarketparticipants’ smaller subproblems.Some of the earliest approachesinclude
X participation in TO. Implementations of various versions of the pioneer work of Kim and Baldick [11] and Conejo
r CTS are being carried out by several system operators in the and Aguado [12] that predate the broad deregulation of the
a
US [3] [4]. electricity market in the US. Multi-area OPF problems that
One of the main challenges in eliminating seams is the explicitly involve multiple ISOs have been widely studied
inherent delay between the interchange scheduling and the [13] [9]. In general, decentralized OPF based techniques
actual power delivery across regions. This is caused by the typically require iterations between ISOs where one control
lack of real-time information necessary for scheduling and center uses intermediate solutions from the other and solves
operation constraints. For example, the information used in its own dispatch problem. Although the convergence of such
CTS for interchangeschedulingis 75 minutes prior to the ac- techniquesisoftenguaranteedundertheDC-OPFformulation,
tualpowerdelivery.Withincreasingintegrationofrenewables, the numberof iterations can be large and the practical cost of
interchange scheduling needs to be cognizant of uncertainty communications and computations substantial. We note that
that arises between the time of interchange scheduling and the recent marginal decomposition technique [6] is shown to
that of power transfer. converge in a finite (but unknown) number of iterations.
The growth of renewable integration has brought new at-
ThisworkissupportedinpartbytheDoECERTSprogramandtheNationalScience tention to uncertainty in seams. Both stochastic optimization
FoundationunderGrantCNS-1135844.Partofthisworkappearedin[1].
Y.JiandL.TongarewiththeSchoolofElectricalandComputerEngineering,Cornell and robust optimization approaches have been considered
University,Ithaca,NY14853,USA(e-mail:[email protected];[email protected]). recently.In particular,Ahmadi-Khatir,Conejo, andCherkaoui
T. Zheng is with ISO New England Inc., Holyoke, MA 01040, USA (e-mail:
[email protected]). formulateatwo-stagestochasticmarketclearingmodelin[14]
for the multi-area energy and reserve dispatch problem. The
solution to the stochastic optimization is obtained based on
scenario enumerations, which requires a prohibitively high
Region 1 Region 2
computation effort. In [8], the day-ahead tie-flow scheduling
is considered in the unit commitment problem under wind
generation uncertainty. Specifically, a two-stage adaptive ro-
Figure 1: A 2-region system with an interface.
bust optimization problem is formulated with the goal of
minimizing the cost of the worst-case wind scenario, and
solvedbythecolumn-and-constraintgenerationalgorithm.The
present paper complements these existing results by focusing p1
Region 1
onthereal-timeinterchangeschedulinganddevelopatractable
stochastic optimization technique. q
A pragmatic approach to the seams problem, one that has
been adopted in practice and that can incorporate external
q
market participants, is the use of proxy buses representing p2
Region 2
the interface between neighboring regions. The technique
presented here falls into this category. Among existing prior
workistheworkbyChenet.al.[9]whereacoordinatedinter-
Figure 2: The single proxy bus representation.
changeschedulingschemeisproposedfortheco-optimization
of energy and ancillary services. The technique is based
on (augmented) LR involving iterations among neighboring net interchange1 q that minimizes the overall operating cost
controlcenters.TheworkclosesttooursaretheTOtechnique subject to generation and transmission constraints. Note that,
presented in [2] and the work of Ilic and Lang [10]. The exceptwhenthereisasingletielinethatconnectstworegions,
underlying principle of [2] and [10] is based on the eco- the proxyrepresentationis an approximation,and the optimal
nomics argument of supply and demand functions, which are interchangeschedulingbasedontheproxyrepresentationdoes
exchanged by the neighboring operators. For the (scalar) net not providethe optimal interchangeof the original system. In
interchange, such functions can be succinctly characterized, general, the optimal interchange via proxy representation is
and the exchange needs to be made only once; the need of strictly suboptimal when it is compared with multi-area OPF
iterations among control centers is eliminated. Our approach solutions.
is also based on the same economics argument with the
innovationon incorporatingsystem and operationuncertainty. B. Optimal Interchange Scheduling
Note that this type of approaches do not solve the multi-area
The interchange scheduling problem under the proxy bus
OPF problem except the special case when there is a single
model can be formulated as minimizing the generation costs
tie line connecting the two operating regions.
ofbothregionswithrespecttothepowerbalance,transmission
(internal and interface) and generator constraints. For sim-
II. DETERMINISTIC INTERCHANGESCHEDULING plicity, we make the following assumptions throughout the
paper: (i) the system is lossless, and (ii) the cost function
A. Proxy Bus Representation c (·),i∈{1,2},isquadraticintheformc (·)=g⊺H g +q g
i i i i i i i
where matrix H is positive definite. Under the single proxy
In practice, coordination between neighboring control re- i
bus system, the net interchange can be modeled explicitly as
gions and markets are typically through the use of proxy bus
an additional scalar variable in the optimization problem as
mechanism. As pointed out in [15], a proxy bus models the
follows:
locationatwhichmarginalchangesin generationareassumed
to occur in response to changes in inter-regionaltransactions.
(P ) min c (g )+c (g )
The proxy bus mechanism is utilized by all of the existing 1 1 1 2 2
q,g1,g2
LMP based markets for representing and valuing interchange subject to 1⊺(d −g )+q =0, (λ )
1 1 1
power [15]. 1⊺(d −g )−q =0, (λ )
2 2 2
Inthispaper,weconsiderapowersystemconsistingoftwo S1(d1−g1)+Sq1q ≤F1, (µ1)
independentlyoperated subsystems, as illustrated in Figure 1. S2(d2−g2)−Sq2q ≤F2, (µ2)
Each operator selects a proxy bus to represent the location q ≤Q, (µq)
of import or export in the neighboring region. Specifically, g1 ∈G1,
as shown in Figure 2, the operator from region 1 assumes a g2 ∈G2,
withdrawal q at proxy bus p and the operator from region 2
1 where
assumes an injection with the same quantity q at proxy bus
p .
2 1Thenetinterchange betweentwoneighboringregionsisthetotalamount
The interchangeschedulingis to determinethe value of the ofpowerflowingfromoneoperating regiontoanother.
c (·)real-time generation offer function for region i;
i π
d vector of forecasted load and renewable generation for
i π1(q)
region i;
q net interchange from region 1 to region 2, if q > 0;
π∗
from region 2 to region 1, otherwise; π2(q)
g vector of dispatches for region i;
i
F vector of transmission limits for region i;
i
q
Q interface limit; q∗ Q
TO
G generator constraints for region i;
i Figure 3: Illustration of TO.
S shift factor matrix of buses in region i to transmission
ij
lines in region j;
Note that function π (q) is ISO i’s incremental dispatch cost
S shift factor vector of buses in region i to the interface; i
qi at the interface at the net interchange q, which serves as a
λ shadow price for power balance constraint in region i;
i supplycurve for the exportingISO or a demandcurvefor the
µ shadow prices for transmission constraints in region i;
i importing ISO.
µ shadow price for the net interchange constraint.
q We use the graphical representation in [2] to illustrate the
The problem (P ) is a centralized formulation for deter-
1 basic principle of TO. As shown in Figure3, π (q) represents
i
mining the optimal interchangebetween region1 and 2. Such
the generation supply curve for region i, but π (q) is drawn
2
an optimization problem requiresa coordinatorwho have full
in a descending cost order. In this example, the direction
access to all related information of both regions which is
of interface flow2 is from region 1 to region 2; π (q) and
1
unsuitable in the present deregulated electricity markets.
π (q) serve asthe supplyanddemandcurverespectively.The
2
As in [7], the centralized problem (P1) can be written in a optimal schedule q∗ is set at the intersection of the two
hierarchical form of decentralized optimization as follows. TO
curves.Notethatifthisquantityexceedstheinterfacecapacity
(P ) min c (g∗(q))+c (g∗(q)) Q,thescheduleshouldbesetatthemaximumcapacityinstead.
2 1 1 2 2
q The interface transmission constraint, in this case, becomes
subject to q ≤Q, (µ )
q binding and price separation happens between markets. It
where g∗(q), i ∈ {1,2}, is the optimal dispatch for region i, should also be noted that import or export transactions are
i
given the interchange level q. settled at the real-time LMP which is calculated at the proxy
The regional dispatch problem for region 1 is specified as bus after the delivery.
According to [2], the interchange schedule of TO is the
(P ) min c (g )
21 g1∈G1 1 1 optimal solution to (P1) (as well as (P2)). This intuitive
subject to 1⊺(d −g )+q =0, (λ ) argument is a manifestation of a deeper connection between
1 1 1
S1(d1−g1)+Sq1q ≤F1, (µ1) social welfare optimization illustrated in Figure 3 and cost
minimizationdefinedby(P ).Inwhatfollows,wewillexploit
and for region 2 as 1
this connection in the presence of uncertainty.
(P ) min c (g )
22 2 2
g2∈G2 III. ASTOCHASTIC INTERCHANGESCHEDULING
subject to 1⊺(d −g )−q =0, (λ )
2 2 2 So far, we have described the interchange scheduling in a
S (d −g )−S q ≤F . (µ )
2 2 2 q2 2 2 deterministic system setting. We now focus on the incorpora-
Note that the optimization problem involves an outer prob- tion of randomload andgenerationin the schedulingscheme.
lem (P ) to optimize the interchange level q, and an inner
2 A. Stochastic Programming Formulation
problem that is naturally decomposedinto two regionalprob-
lems, both parameterized by q. In other words, the optimizer Stochastic optimization is the most common framework to
andtheassociatedLagrangianmultipliersfor(P )and(P ) model optimization problems involving uncertainty. Consider
21 22
are functions of q, i.e., g∗(q), λ∗(q), µ∗(q), i∈{1,2}. the case that load (or stochastic generation,treated as a nega-
i i i
tiveload)israndom.Theinter-regionalinterchangescheduling
C. Tie Optimization canbeformulatedasatwo-stagestochasticoptimizationprob-
ThekeyideaofTOistodeterminetheinterchangeschedule lem.The firststage involvesoptimizingthe netinterchangeto
byintersectingthedemandandsupplycurves.By interchange minimize the expected overall cost
demand/supply curve we mean the incremental cost of the (P ) min E [c (g∗(q,d ))+c (g∗(q,d ))]
regionaldispatchattheinterface,whichisessentiallytheLMP 4 q d1,d2 1 1 1 2 2 2
subject to q ≤Q, (µ )
at the proxy bus. Given the interchange level q, the LMP at q
the proxy bus for region i∈{1,2} is defined as and the second stage solves the regional optimal dispatch
problem given the interchange level q and the realization
π (q)=λ∗(q)+S⊺µ∗(q), (1)
i i qi i
where λ∗(q) and µ∗(q) are the Lagrangian multipliers as- 2Thedirectionofinterfaceflowcanbedeterminedbycomparingtheprices
i i atq=0:ifπ1(0)<π2(0),thepowerflowsfromregion1to2;otherwise,
sociated with the optimal solution of (P2i), for i ∈ {1,2}. thedirection ofinterface flowisopposite.
π interface constraint) is simply the intersection of the expected
Consumersurplus Ed1[π1(q,d1)] demand and supply curves. In general, the interchange that
maximizes the expected social welfare is given by
π∗
Producersurplus
(P ) max qE [π (x,d )]−E [π (x,d )]dx
Ed2[π2(q,d2)] 5 q 0 d2 2 2 d1 1 1
R
subject to q ≤Q. (µ )
q
q
q∗ Q
STO To solve (P ), each operator needs to compute, for each
5
Figure 4: Illustration of STO. interchangequantityq,theexpectedLMPatitsownproxybus.
Such a computation requires the conditional expectation of
of random load d and d , which are specified as (P ) futureLMPatthetimeofdelivery.Theconditionalexpectation
1 2 21
and (P ). Note that the optimal dispatch and the associated can be obtained through probabilistic LMP forecast using
22
Lagrangian multipliers of (P ) are parameterized by two models for load and generation. See, for example, [16]. It
2i
factors: the interchange level q and the load realization d . is also conceivable that the conditional expectations can also
i
So the LMP π (q,d ) at the proxy bus is a functionof both q be approximated via regression analysis. Once the expected
i i
and d . demandandsupplyfunctionsareobtained,solvingfortheop-
i
Directly solving this problem requires the distribution of timalinterchangequantitybecomesa one-dimensionalsearch.
the regional cost function c (q,d ) at each interchange level,
i i
C. Stochastic Tie Optimization
and a coordinator to determine the optimal schedule, neither
of which is achievable in the present deregulated electricity In this section, we establish formally the equivalence of
markets. In general, this two-stage stochastic optimization (P )and(P )wherethesolutionof(P )solvesthestochastic
4 5 5
problemisintractable,especiallywhentheloadandrenewable optimization problem (P ).
4
generation forecast follows a continuous distribution. The
Theorem 1. If the optimal dual solutions of (P ) and (P )
proposed scheduling technique, on the other hand, can solve 21 22
are unique for all q ≤Q, then (P ) and (P ) are equivalent
this problem without increasing the computation complexity 4 5
in the sense that they have the same optimizer.
of deterministic TO. Details are provided in the next two
subsections. Theorem 1 provides a new way, we call it Stochastic Tie
Optimization (STO), to solve the intractable problem (P ).
B. Social Welfare Optimization 4
This result is significant because the optimal interchange can
The main idea of solving (P4) is to exploit the connection be obtained from a deterministic optimization problem (P )
5
between cost minimization and social welfare optimization
which only requires the information of the expected supply
under uncertainty.With the randomnesspresent in the second
and demand curves. Since these price functions are non-
stageof(P4),itisnotobvioushowthetwo-stageoptimization confidential information, (P ) can be solved by one of the
5
problem can be transformed into a correspondingform of so-
operators if the other operator shares its price curve. In this
cialwelfareoptimization.Itturnsoutthattheoptimalsolution
way, operators do not need to iteratively update or exchange
can be obtained by solving a deterministic TO problem using
informationwithin the scheduling procedure.This property is
the expected demand and supply functions.
incontrasttomostdecompositionmethodswheresubproblems
We now present an optimization problem from the import-
are resolved and intermediate results are exchanged in each
exportperspective,buttakingintoaccountthatbothimportand
iteration.Because one-timeinformationexchangeis sufficient
exportregionsmustagreeontheforwardinterchangequantity
for the optimal schedule, operators do not need to repeatedly
in the presence of future demand and supply uncertainty.
solve regional OPF, which is computationally expensive for
Because the interchange quantity is fixed ahead of the actual
sufficiently large systems, within the scheduling procedure.
power delivery, each region may have to rely on its internal
Sucha propertysignificantlyreducesthecomputationcostsin
resources to compensate uncertainty in real time. To this
realtime,therebyprovidingthepotentialofhigherscheduling
end, it is reasonable for the export region to maximize its
frequency.
expected producer surplus and the importregion to maximize
Now we provide the proof of Theorem 1.
its expected consumer surplus.
Proof of Theorem 1: We first show the differentiability
Without loss of generality, let region 1 be the exporter. For
of the objective functions of (P ) and (P ). This follows
4 5
fixed interchange q, let π (q,d ) be the (random) LMP at the
i i immediately from the well known results in multiparametric
proxy bus. Then E [π (q,d )], as a function of interchange
d1 1 1 quadratic programming summarized in Lemma 1.
q, is the expected supply curve averaged over its internal
randomness.Similarly,Ed2[π2(q,d2)] is the expecteddemand Lemma 1 ([17]). If the dual problem of (P2i), i ∈ {1,2}, is
curveaveragedovertheinternalrandomnessinregion2atthe not degenerate for all q ≤Q, then
time of delivery. 1) the optimizer of (P ) and associated vector of La-
2i
As shown in Figure 4, the optimal interchange quantity grangianmultipliersarecontinuousandpiecewiseaffine
q∗ thatmaximizestheexpectedsocialwelfare(inabsenceof (affine in each critical region), and
STO
2) the optimalobjectiveof(P2i)is continuous,convexand π
piecewise quadratic (quadratic in each critical region). Ed1[π1(q,d1)]
J(Bq)y, iLsedmifmfeare1n,tiathbeleowbjiethctdiveerivfuatnicvteion of (P4), denoted by π∗ Ed2[π2(q,d2)]
∂ ∂
J′(q) = Ed1,d2(cid:20)∂qc1(g1∗(q,d1))+ ∂qc2(g2∗(q,d2))(cid:21) Ed2[π˜2(q,d2)]
= E [π (q,d )]−E [π (q,d )] qS∗CTS Q q
d1 1 1 d2 2 2
Figure 5: Illustration of SCTS.
where the second equality holds by the Envelope Theorem.
Lemma1also impliesthatπ (q,d )andπ (q,d ) arecontin-
1 1 2 2
uous functions, so the objective function of (P ), denoted by IV. STOCHASTICCTS
5
s(q), is differentiable with derivative
In this section, we incorporate external market participants
s′(q)=Ed1[π1(q♯,d1)]−Ed2[π2(q♯,d2)]. in STO, which generalizes the CTS proposal currently in
implementation.This generalization,we call it Stochastic Co-
Now we derive the connection between the optimal solu-
ordinatedTransactionScheduling(SCTS),is simplyreplacing
tions to (P ) and (P ) from the first order conditions. The
4 5
optimal solution q∗ to (P ) and the associated Lagrangian the supply and demand curvesused in CTS by their expected
4
multiplier µ∗ satisfy the first order condition for (P ): values.
q 4
As in CTS, market participants are allowed to submit re-
E [π (q∗,d )]−E [π (q∗,d )]+µ∗ =0. (2)
d1 1 1 d2 2 2 q queststobuyandsellpowersimultaneouslyoneachsideofthe
Similarly, the optimal solution q♯ to (P ) and the associated interface. Such request is called interface bid, which includes
5
Lagrangian multiplier µ♯ satisfy the first order condition for a price indicating the minimum expected price difference
q
(P ) between the two regions that the participant is willing to
5
accept, a transaction quantity and its direction.
E [π (q♯,d )]−E [π (q♯,d )]+µ♯ =0, (3)
d1 1 1 d2 2 2 q We use a similar graphical representation of STO to illus-
which is exactly the same as (2). trate the scheduling procedure of SCTS. As shown in Figure
Finally, we show q∗ = q♯. To prove this, we need the 5, Ed1[π1(q,d1)] is the expected supply curve of region 1,
monotonicityof price functionπi(q) (with fixed di as defined and Ed2[π˜2(q,d2)] is the adjusted curve of Ed2[π2(q,d2)]
in (1)) which is summarized in the following lemma whose by subtracting the aggregated interface bids πbid(q). The
proof is provided in the appendix. SCTS schedule is set at the intersection of Ed1[π1(q,d1)] and
E [π˜ (q,d )]. All the interface bids to the left of q∗ are
Lemma 2. If the dual problem of (P ),i ∈ {1,2}, has a d2 2 2 SCTS
2i accepted and settled at the real-time LMP difference.
unique optimal solution for all q ≤ Q, then π (q) is mono-
1 The scheduling and clearing procedure described above is
tonically increasing and π (q) is monotonically decreasing.
2 summarized as follows:
Below we show that in each of the following cases, either 1) share the expected LMP functions E [π (q,d )] and
the case itself is impossible or q∗ =q♯. E [π (q,d )]; d1 1 1
1) q∗ =q♯ =Q. The statement is trivially true. 2) dedt2erm2ineth2e directionof the interchangeflow bycom-
2) q∗ < Q and q♯ < Q. In this case, the interface paring E [π (0,d )] and E [π (0,d )];
d2 2 2 d1 1 1
constraint is not binding in either problem, so we have 3) construct the aggregated interface bid curve π (q)
bid
µ∗q = µ♯q = 0, which implies that Ed1[π1(q∗,d1)] = which is a stack of all interface bids with the direction
Ed2[π2(q∗,d2)] and Ed1[π1(q♯,d1)] = Ed2[π2(q♯,d2)]. determined in step 2) in an increasing order of the
By Lemma 2 and the preservation of monotonicity submitted price difference;
underexpectationoperation,thereisauniquesolutionto 4) calculatetheoptimalSCTSschedulefromthefollowing
Ed2[π2(q,d2)]−Ed1[π1(q,d1)]=0.Therefore,q∗ =q♯. optimization problem (P6).
3) q∗ < q♯ = Q. We construct a solution of (P ) using
q♯ andtheassociatedoptimalfunctionsdefined1in(P21) (P6) mqax 0qEd2[π2(x,d2)]−Ed1[π1(x,d1)]−πbid(x)dx
and (P22), i.e., q♯, µ♯q (which is zero), g1∗(q♯), g2∗(q♯), subject to qR≤Q.
λ∗(q♯), µ∗(q♯),λ∗(q♯),µ∗(q♯). Note that this solution
1 1 2 2
satisfies the first order conditionsfor (P1), so it is opti- Note that the only difference between STO and SCTS is
malto(P1).However,thiscontradictstheuniquenessof the inclusion of interface bids. All other components are
theoptimizerto(P1).Therefore,thiscaseisimpossible. identical. This implies that one-time information exchange is
4) q♯ < q∗ = Q. This case is also impossible. The proof sufficient; no iteration between operators is necessary during
follows the logic of that in case 3). the scheduling procedure when one operator submits its ex-
To sum up, (P ) and (P ) are equivalent in the sense that pected generationsupplycurve to the other who executesthis
4 5
they share the same optimal solution. scheduling and clearing procedure.
Table I: Comparison of TO and STO.
Region1 Region2
1 3 4 5 Scenario Method q∗ E[Cost(q∗)] E[∆π(q∗)]
TO 166.5 6794.2 −2.13
d5=250
STO 162.8 6790.8 0
0≤g1≤120 0≤g3≤200 0≤g4≤100 TO 147.5 4621.6 2.38
c1=0.01g12+10g1 2 c3=0.01g32+40g3c4=0.01g42+30g4 6 d5=250 d5=200 STO 151.4 4608.6 0
0≤g6≤200 55 55
d2=30 c6=0.01g62+45g6 π1TO π1TO
Figure 6V:.AE2-VrAegLioUnAT6-IbOuNs system. Price at the proxy bus34455050 ππ2T1SOTO Price at the proxy bus34455050 ππ2T1SOTO
Inthissection,wecomparetheperformanceoftheproposed 30 30
140 145 150 155 160 165 170 140 145 150 155 160 165 170
STO with that of TO on two systems: a 6-bus system and Interchange Interchange
the IEEE 118-bus system. In particular, we focus on the two (a)d5=250. (b)d5=200.
most common symptoms of seams: (i) the under-utilization Figure 7: Generation supply curves.
of interface transmission, and (ii) the presence of counter-
intuitiveflowsfromthehighcostregiontothelowcostregion. 9. From Table I, the expected price difference at the level
In both examples, TO uses the certainty equivalent forecast of TO schedule in the first example is −2.13$/MWh, which
of the stochastic generation, i.e., the mean value, while STO means that the expected price of the importing region (region
uses the probabilistic forecast, i.e., the distribution. Various 2) is lower then that of the exporting region. This implies
scenarios are studied in these two examples. that the interchange is scheduled from a high cost region to
a low cost region, which is counter intuitive. On the other
A. Example 1: a 2-region 6-bus system
hand, in the second example, the expected price difference
Consider a 2-region 6-bus system as depicted in Figure 6. at the interchange level of TO schedule is 2.38$/MWh, i.e.,
Generatorincrementalcostfunctions,capacitylimits,andload the marginal price of the importing region is higher than that
levels(thedefaultvalues)arepresentedinthefigure.Alllines of the exporting region. With this price difference, increasing
are identical except for the maximum capacities: the tie lines the interchange level can further reduce the expected overall
(line 2-6 and line 3-4) and the internal transmission lines in cost, which implies the interchangecapacity is underutilized.
region 1 have the maximum capacities of 100 MW, and the Because the interchange level of STO schedule is optimal as
internallinesin region2havethe maximumcapacitiesof 200 itsdesign,anyschedulemorethanthisoptimallevelwillcause
MW. The system randomnesscomes from the wind generator the counterintuitiveflow,and anyschedule less than thatwill
atbus1inregion1.Theentirenetworkmodel(theshiftfactor lead to the interface under utilization.
matrix) is assumed to be known to both ISOs. By default, we 2) Impact of forecast uncertainty: The impact of the fore-
chosebus3astheproxybustorepresentthenetworkinregion cast uncertainty level was then investigated by varying the
1, and bus 6 to representthe network in region2. The impact standard deviation σ of the probabilistic wind production
of the location of proxy buses will be further investigated. forecast w ∼N(55,σ2). Loads were set at the default values
1) Abaseline: Wefirsttestedabaselinewiththeprobabilis- given in Figure 6. Results are presented in Figure 10.
tic wind forecast distribution N(55,102). Two levels of load TheinterchangelevelofTO scheduledoesnotchangewith
werechosentoillustratethetwosymptomsoftheinefficiency σ sinceitonlyusesthemeanvalue55ofthewindproduction
ofTOschedule:thefirstloadleveld5 =250isanexampleof forecast. STO, on the other hand, captures the uncertainty
thecounterintuitiveflowoccurrence,andthesecondloadlevel level of the probabilistic forecast and adjusts the interchange
d5 = 200 shows the case of the interface under utilization. scheduleaccordingly.Theexpectedoverallcostincreaseswith
Results are presented in Figure 7-9 and Table I.
Figure7showsthegenerationsupplycurvesofregion1and
6950 4800
region2underTOandSTOforthetwoexamples,respectively.
dfπthou1TercOetpciiorasonstxt,hydeainbsidtunrsciπbr6eu1SmTtuiOoseinnnistgNaltth(hce5eo5sef,tox1orp0efe2ccr)aet.segtSdeioidninncmce1reettahomneder5ene5ltiiasvolefncrotohtshreatenwupdsiooninwmdgenprterhotsoes- Expected overall cost666889050000 Expected overall cost444677505000
in region 2, the supply curves of region 2 for TO and STO 6750 4600
140 145 150 155 160 165 170 140 145 150 155 160 165 170
are the same in both examples. At the interchange level of Interchange Interchange
STO schedule, the expected overall system cost is minimized (a)d5=250. (b)d5=200.
in bothcasesas shownin Figure8,and theexpectedpricesat Figure 8: Expected overall cost: TO is marked by the blue square
thetwoproxybusesconvergeinbothcasesasshowninFigure and STO by the red circle.
15 10
Table II: Comparison of TO and STO.
Expected price difference1005 Expected price difference−−10055 pp==Sc((e00n..a15r,,i00o..95)) MSTTeTtOOhOod 222664q832∗...616 E111[C000449o777st559(108q...∗691)] E[∆--00π0..22(59q∗)]
−5140 145 150 155 160 165 170 −15140 145 150 155 160 165 170 STO 236 109797.1 0
Interchange Interchange TO 197 114806.8 0.28
p=(0.9,0.1)
(a)d5=250. (b)d5=200. STO 204 114805.8 0
Figure9:Expectedpricedifference:TOismarkedbythebluesquare
and STO by the red circle. 40 πTO 1.0488x 105
Interchange111116666602468 TSOTO Expected overall cost666666778888680246000000 TSOTO Price at the proxy bus3333356789 200πππ12T1S2SOTTOO 22In0terchang2e40 260 Expected overall cost1111111......000000.044444447778888468246 200 22In0terchang2e40 260
158 6740 (a)Generation supplycurves. (b)Expectedoverall cost.
0 5 10 15 20 0 5 10 15 20
Forecast uncertainty level (σ) Forecast uncertainty level (σ)
Figure 11: High wind scenario p=(0.1,0.9).
(a)Interchange. (b)Expected overallcost.
Figure 10: Impact of the forecast uncertainty (σ).
circlesinFigure11-13,andthepricesconvergeattheschedule
ofSTO,asshowninTableII.AsforTOschedules,thecounter
the forecast uncertainty, which is observed in both TO and
intuitiveflowsareobservedinthehighwindandmediumwind
STO. When there is no uncertainty (σ =0), the schedules of
scenario,andtheinterfaceunderutilizationhappensinthelow
TO and STO are the same and so do their costs.
wind scenario.
B. Example 2: a 2-region 118-bus system 2) Impactofinterfacecongestion: Toinvestigatetheimpact
of the interface congestion,we tested all three wind scenarios
We divided the standard IEEE 118 bus system3 into two
again under the same setting except the interface capacity
regions: region 1 includes bus 1-12 and region 2 bus 13-118.
which was set as 250 MW in this case.
Generatorincrementalcostfunctions,capacitylimits,andload
From the results shown in Table III, the presence of the
levels are the default values given in MATPOWER [18]. We
interface constraint only influences the performances in the
imposed the maximum capacity of 50 MW on line 4, 6, 58
high wind scenario. The price separation happens in both TO
and60.Theinterfacetransmissionwas notlimitedby default,
andSTO,becausethebindinginterfaceconstraintpreventsthe
but the impact of the interface constraint will be studied. Bus
economic interface flow.
6 and 42 were selected as the proxy buses to represent the
adjacent region’s network.
To introduce randomness in the system, we assumed that 40 x 105
three wind generators, located at bus 6, 42, and 60, produce πTO
1 1.0984
cwtphooienwnsdewi:srtiwasncdo=cofpr(ard1oi0ndp,gur1oct0btoi,ao1abn0idl)iibtsyaycnrmdewtaew,sd′sai=snftudr(ni1bctu0thit0eoi,on2np0.pr0oS,bap2nae0bdc0iilfi)tiws.ctaWoilcleyle,fcvodoereenlnscsoaiodtsef-t Price at the proxy bus333789 πππ2T1S2SOTTOO Expected overall cost111...000999888123
1.098
ered three scenarios: a high wind scenario p = (0.1,0.9), a
36
200 220 240 260 200 220 240 260
mediumwindscenariop=(0.5,0.5),andalowwindscenario Interchange Interchange
p = (0.9,0.1). TO uses the mean value (91,181,181), (a)Generation supplycurves. (b)Expectedoverall cost.
(55,105,105) and (19,29,29) for each respective scenario. Figure 12: Medium wind scenario p=(0.5,0.5).
1) A baseline: In this case, we verified the optimality of
STO schedule with the presence of discrete randomness. All 40.5 πTO x 105
ft1hu1rnS-ec1ietn3iwcoaeninnsaddlalTsrgcaeebennlceaeorrnIiaItot.iisnouwnoeucrosesttaenfsdutendpc.tiieRocneesswuailstrseeaqarufefiasndheroa.wtiTcnh,eitnhepFeiprgrfuiocrree- Price at the proxy bus3389334..89055 πππ12T1S2SOTTOO Expected overall cost1111....1111444488882468
mances of TO and STO schedule are similar to that in the
37.5 1.148
6-bussystemexample.Theexpectedoverallcostisminimized 200 220 240 260 200 220 240 260
Interchange Interchange
at the STO schedulein all threecases, as indicated by the red (a)Generation supplycurves. (b)Expectedoverall cost.
3Allbusandbranchindices arereferredto[18]. Figure 13: Low wind scenario p=(0.9,0.1).
Table III: Impact of interface congestion.
By Lemma 1, π (q) is affine in each critical region, so the
i
Scenario Method q∗ E[Cost(q∗)] E[∆π(q∗)] derivative of πi(q) exists. In addition, ci(gi∗(q)) is quadratic,
p=(0.1,0.9) TO 250 104754.8 0.6 which implies that the second derivative of ci(gi∗(q)) (the
STO 250 104754.8 0.6 derivative of π (q)) with respect to q is positive. Therefore,
i
TO 242.6 109798.1 -0.29 π (q) is monotonicallyincreasing and π (q) is monotonically
p=(0.5,0.5) 1 2
STO 236 109797.1 0 decreasingwithineachcriticalregion.Lemma1indicatesthat
p=(0.9,0.1) TO 197 114806.8 0.28 πi(q) is continuous for all q ≤ Q, so the monotonicity of
STO 204 114805.8 0
π (q) is preserved for all q ≤Q.
i
REFERENCES
3) Impact of proxy bus location: We finally tested the
[1] Y. Ji and L. Tong, “Stochastic coordinated transaction scheduling,” in
impact of proxy bus location in the medium wind scenario, Proc.ofIEEEPESGeneralMeeting, 2015,pp.1–5.
i.e. p = (0.5,0.5). Tie line bus and internal bus (wind bus) [2] ISO New England and New York ISO. Inter-regional interchange
scheduling (IRIS) analysis and options. [Online]. Available: http://
were selected as the proxy buses. From the results presented
www.iso-ne.com/pubs/whtpprs/iris white paper.pdf
[3] FERC Approves Coordinated Transaction Scheduling Between New
Table IV: Impact of the proxy bus location. York ISO and ISO New England, http://www.iso-ne.com/nwsiss/pr/
2012/final iso ne nyiso cts.pdf.
Proxy q∗ E[Cost(q∗)] [4] FERCApprovesCoordinatedTransactionSchedulingforPJMandNY-
Bus TO STO TO STO ISO,http://www.pjm.com/∼/media/about-pjm/newsroom/2014-releases/
20140313-coordinated-transaction-scheduling-PJM-NYISO.ashx.
Internal (1,118) 244.1 238.1 109780.5 109779.7
[5] A. J. Conejo, E. Castillo, R. Minguez, and R. Garcia-Bertrand, De-
Bus (6,42) 242.6 236 109798.1 109797.1 compositiontechniquesinmathematicalprogramming:engineeringand
(8,30) 253 271 109569.8 109565.3 science applications. SpringerScience &BusinessMedia, 2006.
Tie (11,13) 78.4 126.5 110595.1 110540.4 [6] F. Zhao, E. Litvinov, and T. Zheng, “A marginal equivalent decom-
Line (12,14) 28.8 -15.4 110865.5 110828.6 position method and its application to multi-area optimal power flow
problems,” IEEE Transactions on Power Systems, vol. 1, no. 29, pp.
Bus (12,16) 34 4.7 110847.2 110833.4
53–61,2014.
(12,117) 16.8 -23.3 110890.5 110785.7 [7] R.Baldick and D.Chatterjee, “Coordinated dispatch ofregional trans-
missionorganizations: Theoryandexample,” Computers &Operations
Research,vol.41,pp.319–332, 2014.
in Table IV, we observethatthe interchangescheduleand the
[8] Z. Li, W. Wu, M. Shahidehpour, and B. Zhang, “Adaptive robust tie-
associated expected cost are very sensitive to the location of line scheduling considering wind power uncertainty for interconnected
the proxy bus in both TO and STO. With different selections powersystems,”IEEETransactions onPowerSystems,2015.
[9] J. Chen, J. S. Thorp, and T. D. Mount, “Coordinated interchange
of the proxy bus, the direction of the interchange schedule
scheduling and opportunity cost payment: a market proposal to seams
canbedifferent,forexample,(1,118)and(12,117).Although issues,” in Proc. of the 37th Annual Hawaii International Conference
there are several considerations that can guide the choice of onSystemSciences, 2004.
[10] M.IlicandJ.Lang.Methods ofSelecting theDesiredNetInterchange
proxybuslocation[15],notheoreticalresultsshowauniversal
(DNI) Across Multi-Control Areas: Demonstration of Seams Solution
selection rule. for Large-Scale NPCC . [Online]. Available: http://www.ferc.gov/
CalendarFiles/20120627090023-Wednesday SessionA Ilic.pdf
VI. CONCLUSION [11] B. H. Kim and R. Baldick, “Coarse-grained distributed optimal power
flow,”IEEETransactionsonPowerSystems,vol.12,no.2,pp.932–939,
This paper presents a stochastic interchange scheduling 1997.
technique that incorporates load and renewable generation [12] A. J. Conejo and J. A. Aguado, “Multi-area coordinated decentralized
dcoptimalpowerflow,”IEEETransactions onPowerSystems,vol.13,
uncertainties. Using the forecast of the expected LMP at the
no.4,pp.1272–1278,1998.
proxy bus, the proposed approach obtains the optimal inter- [13] M.D.Cadwalader,S.M.Harvey,S.L.Pope,andW.W.Hogan,“Market
change schedule from a deterministic optimization problem coordination of transmission loading relief across multiple regions,”
Cambridge, MA: Center for Business and Government, Harvard Uni-
that maximizes the expected economic surplus. The essence
versity, 1998.
of this technique is providing a way to reduce a two-stage [14] A.Ahmadi-Khatir, A.J.Conejo,andR.Cherkaoui,“Multi-area energy
stochastic optimizationproblemintoa deterministicoptimiza- and reserve dispatch under wind uncertainty and equipment failures,”
IEEE Transactions on Power Systems, vol. 28, no. 4, pp. 4373–4383,
tion problem with an one-dimensional decision. In addition,
2013.
theproposedtechniquedoesnotrequireanyiterationbetween [15] S.Harvey,“Proxybuses,seamsandmarkets[draft],”2003,http://www.
operators during the scheduling procedure. A one-time infor- hks.harvard.edu/hepg/Papers/Harvey Proxy.Buses.Seams.Markets 5-
23-03.pdf.
mation exchange is sufficient for the optimal scheduling.
[16] Y. Ji, L. Tong, and R. J. Thomas, “Probabilistic forecast of real-time
lmpandnetworkcongestion,” arXivpreprintarXiv:1503.06171, 2015.
APPENDIX [17] F. Borrelli, A. Bemporad, and M. Morari. (2014) Predictive control
for linear and hybrid systems. [Online]. Available: http://www.mpc.
Proof of Lemma 2: Denote the Lagrangian function for berkeley.edu/mpc-course-material
(P ) by L ,i ∈ {1,2}. By Lemma 1, c (g∗(q)) is convex [18] R.D.Zimmerman.MATPOWER:AMATLABpowersystemsimulation
2i 2i i i package. [Online]. Available: http://www.pserc.cornell.edu/matpower/
and quadratic in each critical region, so the derivative exists.
By the Envelope Theorem,
∂c (g∗(q)) ∂L ∂c (g∗(q)) ∂L
1 1 = 21 =π (q), 2 2 = 22 =−π (q).
∂q ∂q 1 ∂q ∂q 2