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        An Optimal Reactive Power Dispatch Model
           for Deregulated Electricity Markets
           I. El-Samahy, Student Member, IEEE, C. A. Cañizares, Fellow, IEEE, K. Bhattacharya, Senior
                                 Member, IEEE, and J. Pan, Senior Member, IEEE

                                                                             [3] further states that a reactive power provider should not be
   Abstract—This paper proposes a reactive power dispatch                    financially compensated when operating within a power factor
model that takes into account both the technical and economical              range of 0.95 lagging and 0.95 leading, but an Independent
aspects associated with reactive power provision in the context of           System Operator (ISO) may change this range at its discretion.
the new operating paradigms in deregulated electricity markets.                 Traditionally, reactive power dispatch has always been
The main objective of the proposed model is to minimize the total            viewed by researchers as an optimization problem wherein the
amount of dollars paid by the ISO to the generators for
                                                                             objective is to minimize real power transmission loss subject
providing the required reactive power support. The real power
generation is decoupled and assumed fixed during the reactive                to various system constraints such as nodal active and reactive
power dispatch procedures; however, due to the effect of reactive            power flow balance, bus voltage limits, and reactive power
power on real power, real power generation is allowed be re-                 limits [4]-[9]. Multi-objective optimization models have also
scheduled within given limits. The 32-bus CIGRE benchmark                    been proposed for the reactive power dispatch problem. In
system is used to illustrate the proposed reactive power dispatch            these models, reactive power is dispatched to achieve other
technique. The developed model is generic in nature and can be               objectives, in addition to the traditional loss minimization,
adopted for any electricity market structure.                                such as maximizing voltage stability margin [10], [11], or
                                                                             minimizing voltage and transformers taps deviations [12].
   Index Terms—Ancillary services, electricity markets,                         In the context of deregulated electricity markets, reactive
deregulation, reactive power dispatch, system operation.
                                                                             power dispatch corresponds to short-term allocation of
                                                                             reactive power required from suppliers based on current
                          I. INTRODUCTION                                    operating conditions [13]. The ISO is concerned with

O    PTIMAL dispatch of reactive power services has always
     been of great interest to researchers as well as system
     operators, especially after the restructuring of the power
                                                                             determining the optimal reactive power schedule for all
                                                                             providers based on a given objective that depends on system
                                                                             operating criteria. Different objective functions can be used by
industry. Reactive power is tightly related to bus voltages                  the ISO, besides the traditional transmission losses
throughout a power network, and hence reactive power                         minimization, such as minimization of reactive power cost
services have a significant effect on system security.                       [14]-[16], minimization of deviations from contracted
Insufficient reactive power supply can result in voltage                     transactions [17], minimization of the cost of adjusting
collapse, which has been one of the reasons for some recent                  reactive power control devices [18], or maximization of
major blackouts; for example, the US-Canada Power System                     system loadability to minimize the risk of voltage collapse
Outage Task Force states in its report that insufficient reactive            [19], [20].
power was an issue in the August 2003 blackout, and has                         Any of the aforementioned objectives can be adopted, but
recommended strengthening the reactive power and voltage                     since some of them are of a conflicting nature, the ISO needs
control practices in all North American Electric Reliability                 to choose a criterion that best suits a competitive market
Council (NERC) Regions [1].                                                  structure. In this paper, a cost-based reactive power dispatch
   In deregulated electricity markets, the Independent System                model is proposed that seeks to minimize the total cost of
Operator (ISO) is responsible for the provision of additional                reactive power dispatch from generators, considering the costs
services that are necessary to support the transmission of                   associated with the supply of this reactive power, including
electrical energy while maintaining secure and reliable                      opportunity costs resulting from active power re-scheduling
operation of the power system; these services are referred to
                                                                             when generator capability limits are reached, as well as the
as ancillary services. According to the Federal Energy
                                                                             cost associated with the use of balance services to supply the
Regulatory Commission (FERC) Order No.888, reactive
                                                                             required active power load levels. The proposed model
power supply and voltage control from generators is one of
six ancillary services that transmission providers must include              incorporates both the economic and technical aspects of
in an open access transmission tariff [2]. FERC Order 2003                   reactive power support. The economic aspect is considered by
                                                                             virtue of the objective function which minimizes the total cost
   This work was supported in part by ABB US Corporate Research, and         of reactive power purchase from service providers, while the
NSERC Canada.                                                                technical aspect is taken care of through the security
   I. El-Samahy, K. Bhattacharya, and C. A. Cañizares are with the           constraints such as the bus voltage limits, reactive power
Department of Electrical and Computer Engineering, University of Waterloo,
Waterloo, Ontario, Canada.                                                   capability limits, and transmission line power transfer limits.
Emails:     ieselsam@engmail.uwaterloo.ca;      kankar@ece.uwaterloo.ca;        The paper is organized as follows: In Section II, the
ccanizar@uwaterloo.ca.
                                                                                                                                 2

traditional reactive power dispatch model is discussed. The       economical and technical aspects associated with reactive
proposed dispatch model is then presented in Section III,         power service provision.
including general discussions regarding the overall dispatch
procedures. In Section IV, the proposed model is implemented           III. PROPOSED REACTIVE POWER DISPATCH MODEL
on the CIGRE 32-bus system and two cases are tested.              In practice, active power and reactive power have generally
Finally, in Section V, the main contributions of the proposed     been handled separately by most power system operators.
work are summarized and highlighted.                              Typically, active power dispatch is carried out based a Linear
                                                                  Programming (LP) model associated with an Economical
    II. TRADITIONAL REACTIVE POWER DISPATCH MODEL                 Load Dispatch (ELD) problem that maximizes social welfare,
The main objective of optimal reactive power dispatch has         while guaranteeing that system security constraints are met
always been to minimize the total transmission losses subject     [21]. Reactive power, on the other hand, is dispatched based
to system operational constraints. The following OPF model        on power flow studies and operational experience. However,
represents a traditional loss minimization reactive power         there are several complex issues involved in reactive power
dispatch model [5], [6]:                                          service provision in deregulated electricity market that call for
                                                                  further systematic procedures to arrive at better solutions.
                               ( (
Min. Loss = 0.5∑ Gi , j Vi 2 + V j2 − 2ViV j cos(δ j − δ i )))    Thus, reactive power should be dispatched from generators in
                                                                  an economical manner that minimizes the ISO’s payment
                        i, j
                                                                  burden, while considering system security constraints.
                                                           (1)       Figure 1 shows the proposed schematic procedure for short-
s.t. PGi − PDi = ∑ ViV jYij cos(θ ij + δ j − δ i )         (2)    term dispatch of reactive power services. In this model, the
                           j                                      ISO carries out the reactive power dispatch in a time frame of
     QGi − QDi = −∑ ViV j Yij sin (θ ij + δ j − δ i )      (3)
                                                                  one hour to half-hour ahead of real-time by solving an OPF
                                                                  that minimizes the cost of reactive power provision from
                               j
                                                                  generators, the cost of active power rescheduling, and the cost
     Vi min ≤ Vi ≤ Vi max                                  (4)    of real power balance, subject to power flow and security
                                                                  constraints. The payments to the service providers are
     QGg ≤ Q    max
                Gg                                         (5)
                                                                  calculated post real-time operation, based on the actual usage
     QGg ≥ QGg
            min
                                                           (6)    and dispatch requested by the ISO aggregated over a period of
                                                                  time.
         Pij ≤ Pijmax                                      (7)
                                                                  A. Real Power Market Clearing
where                                                             The flow chart in Fig. 1 illustrates how real and reactive
  PGi:    Active power generation at bus i in p.u.                power markets can be decoupled from each other, so that the
  QGi:    Reactive power generation at bus i in p.u.              ISO does not handle reactive power dispatch in the same time-
  PDi:    Active power demand at bus i in p.u.                    frame as that of real power market clearing. The decoupling of
  QDi:    Reactive power demand at bus i in p.u.                  real and reactive power [13], also suggested in [22]-[24],
  Vi:     Voltage magnitude in p.u. at bus i.                     implies that the OPF problem can be separated into two parts:
  δi:     Voltage angle in radians at bus i.                      An active power sub-problem that provides the real power
  Pij:    Power flow from bus i to bus j in p.u.                  dispatch and prices in real-time based on a cost minimization
  Yij:    Element ij of admittance matrix in p.u.;                (or social welfare maximization) model; and a Q- sub-
          Yij = Gij + j Bij = |Yij| ∠θij                          problem that provides reactive power dispatch levels required
In the above model, (2) and (3) represent the nodal active and    to achieve a certain objective as determined by the ISO,
reactive power flow equations, respectively. The voltage          which, in this paper, is the minimization of total payments,
limits are represented by (4); the generators’ reactive power     while taking into account system security constraints. It is
upper and lower limits are represented by (5) and (6),            important to note that the solution obtained from a coupled
respectively; and limits in the power transfers are represented   OPF, simultaneously dispatching real and reactive power, is
by (7).                                                           theoretically closer to the optimal; however, in addition to
   If the ISO seeks only to minimize losses to determine the      market power and price volatility issues that constrain the ISO
required reactive power support, it could end up with an          from handling them simultaneously, computational burden
expensive set of service providers, which is not preferred in a   also becomes an issue for practical sized power systems, since
market-based environment. Hence, a reactive power cost-           it requires solving a rather complex and large-scale non-linear
based dispatch model would be more suitable, from the             programming (NLP) model every few minutes (e.g. every 5
economical point of view, for a competitive market structure.     minutes in Ontario). Decoupling the OPF problem provides
The main objective of the ISO should always be to provide         the required flexibility for spot market applications, and
reactive power support at the least payment, while maintaining    alleviates the problems associated with model complexity,
system constraints that ensure a secure operation of the power    while retaining an acceptable level of accuracy.
system. In this context, a reactive power dispatch approach is       Most ISOs throughout the world use DC-based OPF auction
proposed in the next section that takes into account both the     models for active power market clearing and dispatch, with
                                                                  iterative mechanisms to guarantee system security [21]. In this
                                                                                                                                              3


    P-dispatch                                          Predetermined
   instructions                                        price components


                          Q-capability                   Solve ORPD
   P-market                 curves                     model minimizing
   clearing                                            the ISO payments


                                                      Determine the required
                                                      reactive power support



                                                          Q-settlements
                                                          and payments


Fig. 1. Flow chart for the proposed reactive power dispatch model.

paper, for the purpose of numerical simulations, real power
market clearing and dispatch is obtained by means of an ac
                                                                                Fig. 2. Synchronous generator capability curve.
OPF model that minimizes the cost of energy production,
subject to system security constraints and some pre-dispatch                    area in Fig. 2 represents the “mandatory” base reactive power
information. The interested reader is referred to [25] for                      provision range set by the system operator. Any reactive
further discussions regarding the advantages and/or                             power provision required by the operator beyond this area is
disadvantages of these two auctions models, which are beyond                    eligible for payment, due to increased costs associated with
the scope of the current paper.                                                 losses in its windings. Such mandatory and ancillary
B. Reactive Power Capability Curves                                             classification of reactive power capability is in line with what
                                                                                most system operators have in place for reactive power
The real and reactive power output from a synchronous                           management (e.g. the IESO of Ontario requires a mandatory
generator is usually limited by the capability of its prime                     provision from all generators within the power factor range of
mover. When real power and terminal voltage are fixed, the                      0.9 lead and 0.95 lag).
armature and field winding heating limits determine the                            The generator’s reactive power capability curves impose
reactive power capability of a generator, as shown in Fig. 2                    the requirement that reactive power generated should be
[13]. The armature heating limit is a circle with a radius (VtIa)               within or on the limiting curve; hence, any reactive output
and centered at the origin, and expressed by the following                      increase requested by the ISO beyond QGA will require a
equation:                                                                       decrease in active power generation. Therefore, an
                                                                                opportunity cost payment is expected for the reactive power
      PG2 + QG ≤ (Vt I a ) 2
             2
                                                                          (8)   service in this case to account for the lost opportunity of the
                                                                                generator to sell its real power in the energy market and the
The field limit, on the other hand, is a circle with radius                     associated revenue loss. Thus, the following three regions for
(VtEaf/Xs) at (0, -Vt2/Xs), and expressed by the following                      reactive power generation can be identified in Fig. 2:
equation:                                                                            • Region I (QGmin ≤ QG = QG1 ≤ 0) refers to the under-
                                                                                          excitation region, in which the generator is required
                              2                   2                                       to absorb reactive power.
          ⎛     V 2 ⎞ ⎛V E                    ⎞                                      • Region II (0 ≤ QG = QG2 ≤ QGA) refers to the over-
      P + ⎜ QG + t ⎟ ≤ ⎜ t af
        2
          ⎜                                   ⎟                           (9)
                Xs ⎟ ⎜ Xs                     ⎟
       G
                                                                                          excitation region, in which the generator is required
          ⎝         ⎠ ⎝                       ⎠                                           to supply reactive power within its reactive power
                                                                                          capability limits.
where Vt is the voltage at the generator terminal bus, Ia is the
                                                                                     • Region III (QGA ≤ QG = QG3 ≤ QGB) refers to the loss
armature current, Eaf is the excitation voltage, and Xs is the
                                                                                          of opportunity region, in which the generator is asked
synchronous reactance, and PG and QG are the real and
                                                                                          to reduce its active power production in order to meet
reactive power outputs of the generator, respectively. The
                                                                                          the system reactive power requirements.
generator’s MVA rating is the point of intersection of the two
curves, and therefore its MW rating is given by PGR. At an
operating point A, with real power output PGA such that                         C. Reactive Power Dispatch Model
PGA<PGR, the limit on QG is imposed by the generator’s field                    It can be observed from Fig. 2, as well as from the previous
heating limit; whereas, when PGA>PGR, the limit on QG is                        discussions, that reactive power production capability of a
imposed by the generator’s armature heating limit. The shaded                   generator essentially depends on the current state of real
                                                                                                                                          4

power generation (PG). Hence, prior knowledge of PG is                 where
essential in order to calculate reactive capability limits. The           ρB1:   Price of the upward balance services PB in $/MW.
values of PG for the generators are obtained from real power              ρB2:   Price of the downward balance services PB in $/MW.
market clearing information, as shown in Fig. 1 and discussed             ρ0g:   Availability price for generator g in $.
in Section III.A.                                                         ρ1g:   Price of Losses in the under-excitation region for
   A cost-based Q-dispatch model is proposed here, which                         generator g in $/MVar.
takes into account both economic and technical issues                      ρ2g: Price of losses in the over-excitation region for
associated with reactive power service provisions in a                           generator g in $/MVar.
competitive electricity market. The model is formulated as                 ρ3g: Loss of opportunity price for generator g in
follows:                                                                         $/MVAr/MVar.
            ⎛ ρ 0 g + ρ 2 g ⋅ QG 2 g − ρ1g ⋅ QG1g           ⎞              ΔPGi Reduction in active power at bus i due to increase in
Min   ∑⎜ + ρ
       ⎜               ⋅ QG3 g − 0.5ρ3 g ⋅ (QG3 g − QGA )
                                                        2
                                                            ⎟
                                                            ⎟   (10)
                                                                                 reactive power beyond heating limits.
       g
            ⎝     2g                                        ⎠              PGoi: Pre-determined active power dispatch at bus i.
       + ∑ ρ B1 ⋅ PB1g + ∑ ρ B 2 ⋅ PB 2 g                                  PBi1: Upward balance service at bus i.
            g                     g                                        PBi2: Downward balance service at bus i.
                                                                           ci: Maximum allowed level of active power reduction at
s.t. PGio − ΔPGi + PBi − PDi                                                     bus i.
                                                                           PGxg: New active power dispatch for generator g.
      = ∑ ViV jYij cos(θ ij + δ j − δ i )                       (11)
            j                                                          In the above model, the objective function (10) comprises two
      QGi − QDi = −∑ ViV jYij sin (θ ij + δ j − δ i )           (12)   major terms: The first represents the payments associated with
                              j
                                                                       reactive power provided from generators, wherein QG1, QG2
                                                                       and QG3 denote the reactive power generation if the generator
      Vi min ≤ Vi ≤ Vi max                                      (13)   is operating in Region I, II, and III respectively, in addition to
                                                                       an availability payment to account for the capital cost of
       Pij ≤ P      max
                   ij                                           (14)
                                                                       reactive power production. The predetermined reactive power
       ΔPGi ≤ ci PGio                                           (15)   price components associated with each region of operation
                                                                       (ρ0g, ρ1g, ρ2g , and ρ3g) are input to the dispatch model as per Fig.
       PBmini ≤ PB1, 2i ≤ PBmaxi
         1, 2               1, 2                                (16)   1. A detailed discussion on these price components and how
                                                                       they can be calculated can be found in [25], where the authors
                                                                       proposed a reactive power market model that addresses
                  ⎛ (V I ) 2 − Q 2                                     several issues associated with reactive power management and
                  ⎜     t a      Gg
                  ⎜                                                    pricing. Determining reactive power prices in a different
                            if QGg ≥ QGAg & PGgo > PRg                 timeframe than that of active power minimizes price volatility
                  ⎜
                  ⎜            2
                                                                       risks, and thus help reduce market inefficiencies.
                                          2 2
       PGxg       ⎜ ⎛ Vt Eaf ⎞ − ⎛ Q + Vt ⎞
                 = ⎜          ⎟ ⎜          ⎟           (17)
                                                                          Under certain loading conditions, some generators may be
                  ⎜ ⎜ Xs ⎟ ⎜ G Xs ⎟                                    asked to supply reactive power in Region III, in which case
                  ⎜  ⎝        ⎠ ⎝          ⎠                           the generators will be required to reduce their real power
                  ⎜         if QGg ≥ QGAg & PGgo < PRg                 generation in order to meet the system reactive power
                  ⎜                                                    requirement. Consequently, a re-schedule of their real power
                  ⎜ PGog otherwise                                     dispatch (ΔPG) will be called for, and a balance service (PB)
                  ⎝                                                    might be required to compensate for this real power deviation
                                                                       from the already dispatched values (PGo). Thus, the second
                                                                       term in (10) seeks to minimize the payments associated with
       PGog − PGg = ΔPGg                                        (18)
                                                                       these energy balance services, from available providers, which
       PGoi − ΔPGi + PBi ≤ P           max
                                      Gi                        (19)   might be an upward or downward balance services within the
                                                                       limits defined in (16). The prices for these balance services
      Q    min
           Gg    ≤ QG1g ≤ QGblead g                                    (ρB1 and ρB2) are pre-determined from the energy balance
                                                                       market. Accordingly, the nodal active power flow equation
      QGblag g ≤ QG 2 g ≤ QGAg                                  (20)   (11) is modified to include both ΔPG and PB1,2. To minimize
      QGAg ≤ QG 3 g ≤ QGBg                                             the effect on real power dispatch (assumed known from a
                                                                       separate auction), the reduction in real power will only be
       QG1g ⋅ QG 2 g = 0                                               allowed up to a certain level (e.g. 5-10%) as per constraint
                                                                       (15); in principle, this value could be set to zero, which reverts
       QG 2 g ⋅ QG 3 g = 0                                      (21)   the problem back to the classical reactive power dispatch.
       QG1g ⋅ QG 3 g = 0                                                  The required reduction in the real power a certain
                                                                       generators (ΔPG) is determined by (17) and (18), where PGx is
       QGg = QG1g + QG 2 g + QG 3 g                             (22)   the new value of the real power after re-scheduling. Observe
                                                                                                                                      5

that ΔPG will only have a non-zero value if the generator is
operating in Region III, i.e. if the generator hits its field limit
(PGA<PGR) or armature limit (PGA>PGR); otherwise PGx in (17)
will be equal to PGo and hence, according to (18), ΔPG will be
zero. Constraint (19) ensures that the total real power for each
generator, including the rescheduling and the balance service
if applicable, does not exceed its maximum value.
   The three regions of reactive power production identified
from the generator’s capability characteristic (see Fig. 2) are
introduced through (20)-(22). It is to be noted that the two
constraints (21) and (22) guarantee that only one of the three
regions (out of QG1, QG2 and QG3) will be selected at a time,
for each generator. This is a non-convex NLP problem with
complementarity constraints, which requires special solvers
and/or solution techniques. In [25], the authors proposed an
iterative approach that can be used for solving this
optimization problem without the need for binary variables to
select one region out of the three operating regions.
   The proposed reactive power dispatch model is supposed to
run in a 30 min to 1 h window, and the solution yields the
required reactive power support that minimizes the payment
by the ISO, while considering system security constraints             Fig. 3. CIGRE 32-bus system.
represented through voltage limits (13) and transmission line         for these generators is zero. The prices of the energy balance
power flow limits (14).                                               services ρB1 and ρB2 are predetermined from the energy
                                                                      balance auction, and are assumed here to be equal to 10 and
          IV. IMPLEMENTATION AND TEST RESULTS                         15 $/MW, respectively, for both cases. The four reactive
The results of applying the proposed reactive power dispatch          power price components defined earlier in Section III.C are
model (10)-(22) to the CIGRE 32-bus test system (Fig. 3 [25])         also given in Table I, where they are fixed for both of the two
are presented and discussed in this section. Without loss of          case studies. All power values in all tables are given in p.u. for
generality, power flow limits are assumed for all transmission        a 100 MVA base.
elements simply based on their voltage ratings; thus, 2000               The solution of the model yields the required reactive
MW for 400 KV lines, 350 MW for 220 KV, and 250 MW for                power support from each generator; the amount of real power
130 KV. All generators are assumed to be eligible for                 to be re-scheduled in order to meet the system reactive power
financial compensation in all of the three regions of                 requirements; the amount of energy balance services needed
operations, i.e. QGblead and QGblag in Fig. 2 are assumed to be       to compensate for the losses as well as the change in real
equal to zero for all generators without any loss of generality.      power resulting from this reactive power dispatch; and the
The optimization models, which are essentially NLP                    total payment of the ISO to the service providers.
problems, are modeled in GAMS [27] and solved using the
                                                                      A. Case I
MINOS solver.
   Two cases are considered here to examine the proposed              Table II shows the solution for this low-loading level case,
reactive power dispatch model:                                        indicating first the required reactive power generation for each
• Case I: “Low” (80%) loading condition.                              generator. Observe that each generator is operating in only
                                                                      one of the three regions discussed earlier in Section III.B. It
• Case II: “High” (110%) loading condition.
                                                                      can be seen from these results that there are no generators
   In both cases, the input to the dispatch model is the list of
                                                                      required to operate in Region III; hence, no real power
predetermined price components for each generator [25]; real
                                                                      rescheduling is required and the value of ΔPG is zero for all
power obtained from the energy market clearing process that
                                                                      the generators. This is expected as the system is lightly
took place prior to the reactive power dispatch procedures;
                                                                      loaded, and thus reactive power requirements are not as high.
reactive power capability curves for each of the available
                                                                      A small amount of balance service, from generator 4072, is
generators, which are functions of real power as per (9) and
                                                                      required to compensate for the system losses. The value of the
(9); and the available energy balance services for each
                                                                      objective function in this case is $2793, denoting the total
generator. Table I includes these input parameters for each of
                                                                      amount to be paid by the ISO for the generators.
the 20 generators for the two case studies. The available real
power for energy balance is zero for a generator if its real          B. Case II
power generation is at its maximum limit, indicating that this        Table III shows the solution for this high-loading level case.
generator cannot provide any additional real power. For the           Observe that now three generators (1012, 1022, and 1043) are
high loading conditions (Case II), more generators are                required to operate in Region III, since the total system
working at their maximum real power capacity and hence they           demand has increased; thus, these generators need to reduce
cannot participate in the energy balance service auction; PBmax       their real power generation in order to meet the reactive power
                                                                                                                                                       6

                              TABLE I                                                                     TABLE III
               INPUT PARAMETERS FOR THE DISPATCH MODEL                                                SOLUTION FOR CASE II

               Case I               Case II        VAR Price Components          Bus       QG1        QG2       QG3        QG        ΔPG       PB1
  Bus                                                 for both Cases             4072       0        2.653       0       2.653        0         0
             PGo    PBmax     PGo       PBmax      ρ0    ρ1     ρ2    ρ3         4071       0          0         0         0          0         0
  4072      11.75   4.11      18.9      6.63      .78   .59    .74   .35         4011       0          0         0         0          0       0.251
  4071       4.7     0        4.7         0       .78   .59    .74   .35         4012       0          0         0         0          0         0
  4011      3.97    1.39      6.47      2.27      .78   .59    .74   .35         4021      -0.3        0         0        -0.3        0         0
  4012      5.42    1.90      7.52        0       .78   .59    .74   .35         4031       0          0         0         0          0         0
  4021      2.82     0        2.82        0       .92   .91    .91   .36         4042       0          0         0         0          0         0
  4031      3.29     0        3.29        0       .92   .91    .91   .36         4041       0          0         0         0          0         0
  4042      2.50    0.87      6.58        0       .92   .91    .91   .36         4062       0          0         0         0          0         0
  4041      2.82     0        2.82        0       .92   .91    .91   .36         4063       0        2.447       0       2.447        0         0
  4062      2.99    1.05      5.64        0       .85   .53    .81   .26         4051       0        1.549       0       1.549        0         0
  4063      9.68    1.60      11.3        0       .85   .53    .81   .26         4047       0          0         0         0          0         0
  4051      5.83    0.75      6.58        0       .85   .53    .81   .26         2032       0        0.175       0       0.175        0         0
  4047      1.20    0.42      11.3        0       .85   .53    .81   .26         1013       0          0         0         0          0         0
  2032      7.99     0        7.99        0       .92   .91    .91   .36         1012       0          0       2.729     2.729        0         0
  1013      4.74    0.90      4.75      0.89      .78   .59    .74   .35         1014       0          0         0         0          0         0
  1012      7.52     0        7.52        0       .78   .59    .74   .35         1022       0          0       0.853     0.853        0         0
  1014      3.72    1.30      4.45      1.56      .78   .59    .74   .35         1021       0          1         0         1          0         0
  1022      2.35     0        2.35        0       .92   .91    .91   .36         1043       0          0       0.959     0.959      0.125       0
  1021      4.71    0.93      4.80      0.84      .92   .91    .91    36         1042       0          0         0         0          0         0
  1043      0.58    0.20      1.88        0       .85   .53    .81   .26
  1042      3.76     0        3.76        0       .85   .53    .81   .26
                                                                                                       V. CONCLUSIONS
                                TABLE II                                      A cost-based reactive power dispatch model for competitive
                            SOLUTION FOR CASE I                            electricity markets is proposed in this paper. The model seeks
                                                                           to minimize the total payments of the ISO, and hence the
     Bus         QG1         QG2       QG3        QG     ΔPG     PB1       objective function from minimization comprises three terms:
     4072          0        0.299       0       0.299     0     0.064
                                                                           the cost associated with reactive power service provision; the
     4071          0          0         0          0      0       0
     4011          0        5.095       0       5.095     0       0        cost of real power rescheduling resulting from reactive power
     4012       -0.524        0         0       -0.524    0       0        dispatch; and the cost of providing additional energy balance
     4021        -0.3         0         0        -0.3     0       0        services needed to compensate for the system losses and the
     4031        -0.4         0         0        -0.4     0       0        change in real power generation.
     4042          0          0         0          0      0       0           The results show that as the system demand increases, there
     4041         -2          0         0         -2      0       0
     4062          0          0         0          0      0       0
                                                                           is more need for reactive power to maintain the system
     4063          0        0.543       0       0.543     0       0        voltage as well as power transfer limits within the assigned
     4051          0        0.236       0       0.236     0       0        limits, thus ensuring secure power system operation, as
     4047          0          0         0          0      0       0        expected. The region of reactive power operation is
     2032          0        1.685       0       1.685     0       0        determined by the generator capability curves, which also
     1013          0          0         0          0      0       0
                                                                           relate reactive power to real power; as the system reactive
     1012          0          0         0          0      0       0
     1014          0          0         0          0      0       0        power requirements increase, some generators will reach their
     1022          0          0         0          0      0       0        capability limits (field or armature thermal limits) and thus
     1021          0        1.156       0       1.156     0       0        reduce their real power output. These generators are eligible
     1043        -0.2         0         0        -0.2     0       0        for additional opportunity-loss payment for not being able to
     1042        -0.4         0         0        -0.4     0       0        sell their real power to the market.
requirements. For generators 1012 and 1022, no re-scheduling                                           VI. REFERENCES
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                                                                                                                                                               7

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