‫‪M. A. El-Kady and M. S. Owayedh‬‬

                           ‫‪FRAMEWORK FOR OPTIMAL POWER FLOW‬‬

                                             ‫‪M. A. El-Kady * and M.S. Owayedh‬‬
                                                       ‫‪King Saud University‬‬
                                                     ‫‪Saudi Electricity Company‬‬
                                                       ‫‪Riyadh, Saudi Arabia‬‬

                                                                                      ‫ً‬     ‫ً‬      ‫ً‬
                ‫ﻳﻘﺪم هﺬا اﻟﺒﺤﺚ إﻃ ﺎرا ﻋﻤﻠﻴ ﺎ ﺟﺪﻳ ﺪا وﻃ ﺮق ﻣﻌﺎﻟﺠ ﺔ ﻧﻈﺮﻳ ﺔ وﺗﺤﻠﻴﻠﻴ ﺔ ﻟﻠﺘﻌﺎﻣ ﻞ ﻣ ﻊ اﻟﻤ ﺸﻜﻠﺔ اﻟﻤﻌﻘ ﺪة اﻟﺘ ﻲ‬
                ‫ﺗﺘﻄﻠ ﺐ اﻟﻤﻮازﻧ ﺔ ﺑ ﻴﻦ اﻻﻋﺘﺒ ﺎرات اﻻﻗﺘ ﺼﺎدﻳﺔ واﻷﻣﻨﻴ ﺔ ﻓ ﻲ ﺗ ﺸﻐﻴﻞ ﻧﻈ ﻢ اﻟﻄﺎﻗ ﺔ اﻟﻜﻬﺮﺑﺎﺋﻴ ﺔ. وﻟﻤﺠﺎﺑﻬ ﺔ ه ﺬا‬
                ‫اﻟﺘﺤﺪي ﺑﻄﺮﻳﻘﺔ ﻋﻤﻠﻴﺔ وﻓﻌﺎﻟﺔ ﺗﻘﺘﺮح هﺬﻩ اﻟﻮرﻗﺔ دﻣﺞ ﺗﻘﻴﻴﻢ اﻷﻣﻨﻴﺔ ﻟﻠﺸﺒﻜﺎت اﻟﻜﻬﺮﺑﺎﺋﻴﺔ ﻣ ﻊ ﻣﺘﻄﻠﺒ ﺎت اﻟﺘ ﺸﻐﻴﻞ‬
                ‫اﻻﻗﺘ ﺼﺎدي اﻷﻣﺜ ﻞ وذﻟ ﻚ ﺑﻮﺳ ﺎﻃﺔ اﺳ ﺘﺨﺪام اﻟﻄﺮﻳﻘ ﺔ اﻟﻤﻌﺮوﻓ ﺔ ﺑﺎﺳ ﻢ داﻟ ﺔ اﻟﻄﺎﻗ ﺔ اﻟﻌ ﺎﺑﺮة ﻣ ﻊ إدﺧ ﺎل ﺗﻜﻠﻔ ﺔ‬
                ‫اﻟﺘﺸﻐﻴﻞ اﻟﻜﻠﻴﺔ ﻟﻠﻨﻈﺎم آﺪاﻟﺔ رﺋﻴﺴﻴﺔ ﻳﺘﻢ ﺗﻘﻠﻴﻠﻬﺎ ﺑﻘﺪر اﻹﻣﻜﺎن ﻣﻊ اﻻﺣﺘﻔﺎظ ﺑﺎﻷﻣﻨﻴ ﺔ اﻟﺤﺮآﻴ ﺔ ﻟﻠ ﺸﺒﻜﺔ. وﻓ ﻲ ه ﺬا‬
                ‫اﻹﻃﺎر ﺗﻘﺪم اﻟﻮرﻗﺔ ﻣﺒﺪأﻳﻦ ﺟﺪﻳﺪﻳﻦ ﻟﻠﺘﻌﺎﻣﻞ ﻣﻊ اﻟﻤﺸﻜﻠﺔ وﻟﻠﺤﺼﻮل ﻋﻠ ﻰ وﺻ ﻒ أدق ﻷداء اﻟﻨﻈ ﺎم اﻟﻜﻬﺮﺑ ﺎﺋﻲ‬
                                                                                              ‫ﺗﺤﺖ ﻇﺮوف اﻟﺘﺸﻐﻴﻞ اﻟﻤﺨﺘﻠﻔﺔ.‬

‫:‪* Address for correspondence‬‬
‫‪Professor M. A. El-Kady‬‬
‫‪Electrical Engineering Department‬‬
‫‪King Saud University‬‬
‫008 ‪P. O. Box‬‬
‫12411 ‪Riyadh‬‬
‫‪Saudi Arabia‬‬

                 ‫.5002 ‪Paper Received 14 September 2004; Revised 28 February 2005; Accepted 18 June‬‬

‫6002 ‪October‬‬                                                          ‫‪The Arabian Journal for Science and Engineering, Volume 31, Number 2B‬‬   ‫991‬
      M. A. El-Kady and M.S. Owayedh

                               This paper introduces a novel framework and methodologies which are
                          capable of tackling the complex issue of power system economy versus security
                          in a practical and effective manner. At heart of achieving such a challenging and
                          far-reaching objective is the incorporation of the Dynamic Security Assessment
                          (DSA) into production optimization techniques using the Transient Energy
                          Function (TEF) method. In addition, and in parallel with the already well-
                          established concept of system security, two new concepts pertaining to power
                          system performance will be introduced in this paper, namely the concept of
                          system dynamic susceptibility, which measures the level of system weakness to a
                          particular contingency and the concept of system consequent restorability, which
                          measures the extent of contingency severity in terms of the required subsequent
                          system restoration work should a particular contingency occur.
                          Key words: Economy, security, optimization, transient stability, dynamic security
                          assessment, transient energy margin, transient energy function method, optimal
                          power flow, power flow

200   The Arabian Journal for Science and Engineering, Volume 31, Number 2B                                   October 2006
                                                                                            M. A. El-Kady and M. S. Owayedh

                          SYSTEM SECURITY

      Due to the critical importance of electric energy and the rising cost of its production, power utilities around the
world are compelled to minimize production cost while, at the same time, operating within acceptable security limits
[1–3]. The complexity of the overall problem and the multi-disciplinary nature of the research involved have resulted in
division of the research into two main areas, namely power system security and operational economy. The objective in
the operational economy area is to determine the optimum schedule of utility generating units that minimizes the total
operation cost subject to equipment and system constraints [4,5]. On the other hand, the objective in the power security
domain is to ensure the system's ability to withstand some unforeseen, but probable, disturbances with the minimum
disruption of service or reduction of service quality [6,7]. Basically, power system security assessment can be broadly
divided into two sub-areas: Static Security Assessment (SSA) and Dynamic Security Assessment (DSA). The term
“static security” means that all limit violations reflect steady-state quantities such as steady state bus voltage violations
and steady-state transmission line over-loading. The analysis tools required are those related to steady-state analysis (i.e.
load flow and related sensitivity analysis methods). DSA, on the other hand, corresponds to the investigation of
disturbances, which may lead to transient instability (loss of synchronism among machines).
  Over the past two decades, the loading of transmission network and the amount of power transfer between
interconnected systems, in many power systems worldwide, has increased to the point where power system security
constraints start to influence the generation commitment and loading decisions. This has led researchers [2,8,9] to
develop techniques suitable for incorporating static security constraints into production optimization procedures known
as Optimal Power Flow (OPF). In real power systems, however, any re-distribution of generator power output to
minimize fuel cost (economic dispatch) would also influence the system dynamic behavior (stability) when a
contingency occurs (for example, a fault at a given bus of the network which is cleared by tripping a transmission line).
The proper inclusion of dynamic security constraints in the OPF formulation has so far been limited, mainly because of
the inherent problem complexity and the lack of appropriate methodologies This paper will introduce framework and
methodologies for incorporating dynamic security constraints in the production optimization techniques. In addition, and
in parallel with the already well-established concept of system security, two new and far-reaching concepts pertaining to
power system performance will be introduced, namely the concept of ‘system dynamic susceptibility’ and the concept of
‘system consequent restorability’.
     In general, and depending on the philosophy and mandate of the utility, there are basically two possible scenarios.
The first is to try to minimize the operating cost subject to minimum reliability requirements. The second is to maximize
reliability subject to a maximum cost (budget ceiling). Of course, the two solutions are theoretically and practically
     Therefore, in theory, there are two possible formulations involving the integration of both economy and security
functions in the power utility business. These alternative formulations are [1]:
         Alternative 1:
           Minimize: Production Cost
           Subject to: Minimum Security (Reliability) Requirements
         Alternative 2:
          Maximize: Security (Reliability)
          Subject to: Maximum Cost (Affordability Constraint).
     In practice, however, all power utilities are adopting the first formulation. That is, to focus primarily on minimizing
operating costs as long as the minimum acceptable security level is met. The above dilemma exists essentially because
the economy and security functions have totally different mandates. The economy group in the power utility is
concerned, in principle, with making money and generating profit by minimizing costs. It is their job mandate to
minimize the production cost to the extent possible. On the other hand, the group concerned with system security

October 2006                                                   The Arabian Journal for Science and Engineering, Volume 31, Number 2B   201
      M. A. El-Kady and M.S. Owayedh

      functions attempts, in most cases, to limit the freedom available to the economy group by raising concerns relating to
      system security issues. Indeed, the utility cannot have it both ways. If it opts for higher reliability, it must then pay more
      and, conversely, if it opts for less cost, then it would have to relax the reliability standards (i.e. take the risk) and lower
      the level of service quality offered to its customers. This, in essence, is the main reason for regulating the power utility
      business by local governments in some parts of the world, even though they may lose some advantages of the free
      market economy.
           Depending on the organizational structure of the power utility and the technology available, there are basically three
      possible scenarios for integrating economy and security functions these over listed follows:
      2.1. Loosely-Integrated Security and Economy Functions:
           In this case, it is assumed that both economy and security functions are totally separate in terms of individual
      decision making processes. Here, the economy group develops a production costing plan, say for the next 48 hours, and
      pass it on to the security group to check if it would violate any of the system security (static and/or dynamic) limits. If
      some security limits are violated, then the production plan is rejected and returned to the economy group for
      modifications. In technical terms, this would degrade the plan from optimal to sub-optimal status. The cycle is repeated
      until an acceptable plan is developed. We note here that the only signal that is given back from the security to the
      economy group is a ‘Yes / No’ signal and, therefore, this scenario reflects a measure of weak integration between the two
      functions. This scenario is depicted in Figure 1.

                                                                              Economy Function

                                              Yes / No                    Operating Parameter

                                                                               Security Function

                                           Figure 1. Loosely-integrated economy and security functions.

      2.2. Semi-Integrated Security and Economy Functions
            In this scenario, the utility tries to achieve more integration between the two functions, which, because of the
      technology and software limitations that may exist, can only be done partially. Instead of the ‘Yes / No’ signal, the
      security function would also give sensitivity information to the economy function to indicate how much the security
      margin is sensitive to changes in operating parameters. This information would tremendously decrease the number of
      iterations between the two groups as the economy group would, in this case, be able to estimate the impact of variations
      in system operating parameters (for example, a power plant output) on the security. This scenario is shown in Figure 2.

                                                                                   Economy Function

                                            Sensitivities of Dynamic
                                               Security Margin

                                                                                     Security Function

                                           Figure 2. Semi-integrated economy and security functions.

      C) Fully-integrated Security and Economy Functions:
           In this scenario, which is demonstrated in Figure 3, a full integration would be attempted between economy and
      security functions. Here, the problem is usually formed as a conventional economy function in which security
      constraint(s) are added to ensure that the resulting plan satisfies the security constraints as well. An example of this
      scenario would be the minimization of the total fuel cost subject to the system being stable.

202   The Arabian Journal for Science and Engineering, Volume 31, Number 2B                                              October 2006
                                                                                                M. A. El-Kady and M. S. Owayedh

                                        Economy-Security Function

                                        Minimize:        Production Cost
                                        w.r.t:           Operating parameters

                                       Subject to:       Power Flow Equations
                                                         Voltage and Flow Constraints
                                                         Dynamic Security Requirements

                                      Figure 3. Fully integrated economy and security functions.

      The OPF problem was defined in the early sixties as an extension of the traditional economic dispatch to determine
the actual setting for control variables in a power system representing various constraints. Traditional scheduling fails to
take the precise operating conditions of the network into account. In the conventional OPF problem formulation [10], a
total cost function is minimized with respect to optimization variables and subject to system constraints representing
steady-state load-flow equations as well as upper and lower bounds on system variables. The optimization variables
include a set of control variables u that are adjusted to obtain the optimal operating point defined in terms of the state
variables x. The Transient Energy Function (TEF) method [11,12] has been used for many DSA studies as an alternative
to the conventional time-domain simulation. Recent developments in the TEF and the related sensitivity analysis have
made it a position candidate to meet the speed and dependability required for incorporation into production optimization
techniques. The avoidance of a lengthy step-by-step time domain solution and the provision of quantitative measure
(Energy Margin) are features that make the TEF method very attractive. In addition the TEF has also the flexibility to
obtain analytical sensitivity information on how the energy margin is affected by variations in system parameters and
     The proposed formulation will utilize the TEF method to produce an analytic measure (energy margin) of the
system transient stability. Using proper formulations and advanced sensitivity analyses, both economy and security
requirements will be integrated and included in one routine of optimization. Without loss of generality, the control
variables are assumed to contain voltage magnitude and active power generation at generator buses. In addition, upper
and lower bounds are imposed on the control variables. It should be noted that other control variables, including
transformer tap and phase-shifter settings, controlled VAR sources, and even demand powers (load shedding) could be
incorporated in the computational scheme.
    When the objective function represents the total fuel cost, the resulting Dynamically-Constrained Optimal Power
Flow (DCOPF) problem can then be stated as follows:
Minimize C (PGi ;i=1,2,..., NG)                                                                                                   (1)
with respect to |Vi|, PGi; i=1,2,..., NG
     Subject to:
Equality constraints (load flow equations)                                                                                        (2)
Upper & lower bounds on problem variables
      (static security constraints)                                                                                               (3)

EM ≥ EMM in (dynamic security constraint)                                                                                         (4)
   If the objective is to maximize dynamics security, then, the overall Cost Constraint Maximum Dynamic Security
(CCMDS) problem, which maximizes the transient energy margin, can then be stated as follows:
Minimize EM (|VGi|, PGi ; i=1,2,..., NG)                                                                                          (5)
with respect to |VGi|, PGi ; i =1,2,..., NG
Subject to:
Equality constraints (load flow equations)                                                                                        (6)

October 2006                                                       The Arabian Journal for Science and Engineering, Volume 31, Number 2B   203
      M. A. El-Kady and M.S. Owayedh

                Upper and lower bounds on problem variables
                (static security constraints)                                                                                (7)
          C ≤ CMAX (Production cost constraint)                                                                              (8)
           The production cost represents the sum of fuel costs associated with various dispatched generators. Mathematically,
      the objective function is as follows:
            C =   ∑C i (PGi )
                  i =1

      where Ci ( PGi ) = Ci 0 + Ci1 * PGi + Ci 2 * PGi is a cost function for generator.

      The energy margin is a non-linear function of the control variables of the form
                  EM = f ( PG,VG )
           The TEF can be formulated directly using the Center of Inertia (COI) frame of reference. Converting loads to
      constant shunt admittances and transforming rotor angles and speeds to the COI reference, the swing equation of the NG
      generators, which represent the system equilibrium condition, can be written in the following compact form [12,13]
                                    •                  M
                f (V ,θ , δ ) = M i ω i = Pm i − PG i − i PCOI = 0                                                       (11)

      where θi is the generators bus voltage angle, δi is the generator internal angles, ωi and ωi are the generator speed and its
      time-derivative, Mi is the moment of inertia, Pmi and PGi are the mechanical power input and generation power output
      respectively, ⏐Vi⏐ is the bus voltage magnitude and
                  MT = ∑ M i
                          i =1                                                                                               (12)
                  PCOI = ∑ ( Pmi − PGi )
                           i =1                                                                                              (13)
           The equilibrium points of the system are the points representing various solutions of the nonlinear swing equations
      (11). Among such equilibrium points, the Stable Equilibrium Point (SEP) and the controlling Unstable Equilibrium Point
      (UEP) are of interest for the purpose of the transient stability analysis. The only difference between the determination of
      the SEP and the UEP is the initial condition provided to the solution algorithm.
            For the SEP, the condition at fault clearing is used while, for the UEP, the ray point [12], which maximizes the
      position energy along the ray from post disturbance point to the controlling UEP, is normally used unless for stressed
      systems in which more robust techniques are needed to solve for the UEP [10]. Having solved for the SEP and the UEP,
      the transient energy function V (θ,ω) is expressed as

                                         NG                  NG
                  V ( θ , ω ) = 0.5     ∑M
                                         i =1
                                                     iωi -
                                                             i =1
                                                                    mi   ( θ i −θ s )

                                    N G θi
                                +   ∑∫P
                                    i =1 θ s
                                                Gi   dθ i                                                                    (14)

      in which the three RHS terms represent the kinetic energy, position energy, and magnetic and dissipation energy of the
      system, respectively. The stability assessment is done by comparing two values of the transient energy V. These are the
      values of V computed at fault clearing Vcl and the critical value Vcr which is the position energy at the controlling UEP,
      for the particular disturbance under investigation. Substituting for Vcr and Vcl in (14) and using the concept of kinetic
      energy correction [12], the energy margin can be obtained as

204   The Arabian Journal for Science and Engineering, Volume 31, Number 2B                                           October 2006
                                                                                                                                                                                                                                  M. A. El-Kady and M. S. Owayedh

                                                     NG                                                     NG
              EM = 0.5 ∑ M eq (ω cl ) 2 - ∑ Pmi ( θ iu - θ icl )
                                                     i=1                                                    i=1
                                          NG              θ iu                                                                                                                                                                                                                                                         (15)
                               +∑                     ∫θ    cl
                                                                 PG i dθ i ,
                                          i =1             i

in which ωcl and θcl are calculated using either the step-by-step method or directly assuming constant acceleration. The
dissipation energy term can be evaluated only if the system trajectory is known. The fairly accurate approximation,
assuming linear angle path as suggested by Athay [14], is used in the present analysis. The security constrained optimal
power flow problem has been solved by different techniques in the literature [15–17]. In the present work, the above
optimization problem is solved using standard non-linear programming techniques [18]. In this paper the Gradient
Projection method of Rosen is adopted for the solution of the optimization problem.
     In this section, practical applications of DCOPF and CCMDS are presented. The power system used in the
applications is the interconnected Saudi Electricity Company (SEC) power grid. This power system consists of two main
regions, namely the SEC-C (Central Region) and SEC-E (Eastern Region). The two SEC systems are interconnected
through two 380 kV and one 230 kV double-circuit lines. In the original (unreduced) load-flow system model, the
interconnected SEC bulk electricity system comprises 150 generator buses, 637 load buses, a total of 1168 transmission
lines, and transformers. In order to prepare a number of meaningful system models, which are suitable for the present
stability studies, the original base-case underwent a series of carefully performed static network reductions. The first
reduced network model comprises 119 buses (19 generators, 100 loads), 334 lines, and 122 transformers. This system
model will be referred to as the 19-Generator model. The nineteen generators are distributed as 11 in the SEC-C area, 8
in the SEC-E area, as shown in Figure 4. The system under investigation is the reduced 3-Generator system model
created from the 19-Generators SEC model. The contingency considered is a 3-phase fault at bus #30 on the SEC-E side
of the 380 kV tie-line between SEC-E and SEC-C, cleared after 0.08 seconds by tripping the double-circuit 380 kV line
(between bus #1 and bus # 30).
     Table 1 summarizes the results of five different operating objectives, namely: (1) Original base-case; (2) Cost
minimization without dynamic security constraints (conventional OPF); (3) Cost minimization with dynamic security
constraints (DCOPF); (4) CCMDS (maximize un-normalized EM with an upper bound on production cost of 370
kSR/hr; (5) DCOPF with the EM fixed at the initial base case value; and (6) CCMDS with total production cost fixed at
the initial base case value.
                                           6                                                                      SEC CENTRAL                                                                                           83                                        81        SEC EAST
                                                                                                                                                                                                   112                             110
                               41                                                                                                                                                                   47                                                                                 82               44
                                                                                            2                      4                                               1
                                                                                                                                                                                                   87                                   85     84                            48
                                                                                                                                                                                                              GAZLAN                                                                         45     46
                                                                 8                          7                           15                  14                5        23                    BERI             51                                      103
                                                                                                                                                                                                                                        NORTH AREA                            49
                                                                     PP8B                                                                                                                                                                                    61                 62
                               75                                                     74                                                                               63                                                                                              89               92        68          67
                                                                          PP8A                                                                                                                                                                 119
         78                         39          37                                                                                                                                                                                                           90                    91
         80              78
                                                                                                                   RIYADH AREA                                                                                               50                                                                              93
              79                                                                         114
                                                                                                                                                                                                          86                                 107             106            105             104                   94
                                                                                                                               113               19                          73
                                                                                                                                     116                                                                 56
                                                                                                                                                                                                                                                    108                 DAMAMAM AREA
                                                                                                                              88                                                             26 30
                                                                                     98         97          96     95                                                                                                                                 99
                          35                                                                                                                                                                                                                         109
                                                                                                                                                                                        21                                                                                                         71
                                                                                                                                      100                                                29    28
                                      43                                                                    PP4
                                   QPP3                                                                                                                                                                                                                                     65               59
                                                42                         76                          64
                                                                                                                                                                  20                                                                                         57                   60
                                                                     PP7B                       PP7A                                                                                                               54
                                                                                                                                                                  10                                                          55                      70                                     69
                   QASSIM AREA                                                                                          117          102         101
                                                                                                                                                                       27                     18                                                                                                   58
                                                                                                                                                          LAYLA                                                53
                                                                                                             33                                                   32        31                       24
                                                                                                                       118                                                                    25                        11         22
                                                                                                                                                                                                                                                     72                                SOUTH AREA
                                                                                     3                             AL-KAHARJ AREA

                                                                                                       Figure 4. SEC 19-Generators System Model.

October 2006                                                                                                                                                  The Arabian Journal for Science and Engineering, Volume 31, Number 2B                                                                                           205
      M. A. El-Kady and M.S. Owayedh

                                                   Table 1. Security/Economy Applications

                           #                 Objective                    P. Cost (kSR/hr)   Energy Margin
                           1    Initial Base Case                             362.78             3.203
                           2    OPF                                           346.89             <0.00
                           3    DCOPF                                         356.58             0.000
                           4    CCMDS                                         370.00             29.84
                           5    DCOPF (Initial EM)                            358.16             3.203
                           6    CCMDS (initial P. Cost)                       362.78             10.42

           The optimization was performed by adjusting six operating parameters representing the real power and voltage at
      three major power plants (Qurraiah, PP8X, and Qassim). The results of Table 1 show that, for the operating scenarios
      considered, a mere cost minimization would lead to an unstable system if the anticipated contingency occurred. It is also
      shown from Table 1 that, while the CCMDS solution produces a very stable system with EM value close to 30, the
      resulting production cost is about 3.7 percent higher than the DCOPF solution. The Study Scenarios #5 and #6 represent
      interesting scenarios, which underline the importance of using advanced optimization methodologies in the modern
      system operation environment. The Study Scenario #6, for example, demonstrates that it is possible to increase dynamic
      system security without necessarily incurring any additional operating costs. In this case, the energy margin was
      improved from 3.2033 to 10.424 per-unit while maintaining the total production cost at 362.78 kSR/hr. On the other
      hand, the Study Scenario #5 demonstrates that it is possible to reduce the system operating cost without necessarily
      reducing the level of the dynamic system security. In this case, the total production cost was reduced from 362.78 kSR/hr
      to only 358.16 kSR/hr while maintaining the energy margin at its base-case value of 3.2033. Table 2 shows the values of
      some key control variables for both the conventional OPF (without security constraint) and the cost minimization with
      dynamic security constraints (DCOPF) solutions.
            System performance assessment is not, and should not be, confined to only the transient stability analysis and the
      dynamic security assessment, which has so far been the case. In this respect, it is well known in the literature that a
      traditional time-domain transient stability run would provide a basic stable/unstable answer. A dynamic security
      assessment, on the other hand, would extend the information provided by also giving the associated degree (or level) of
      system stability (or instability). This is done, in essence, by computing the value energy margin corresponding to the
      operating scenario under consideration. An advancement to the state-of-the-art is made by extending further the above
      two concepts to serve two very important objectives in the practical power system operation, namely:
      (i) To determine the rate of deterioration in the dynamic security with respect to certain contingencies, and
      (ii) To determine the true severity of a particular contingency in terms of its consequent impact on system restoration.
         In the present work, the first objective will be termed dynamic susceptability, while the second objective will be
      termed as consequent restorability. These two new far-reaching concepts, together with the more classical term of
      “probability of contingency” should complete, to a large extent, the general domain of power system dynamic
      performance assessment. Because of the nature of such an advancement, as conceptual rather than methodological, a
      comprehensive definition and an associated practical example would be sufficient to explain each concept, as in the
      following two subsections. An application is then presented to demonstrate the overall assessment of system dynamic
      performance on a particular operating scenario in the SEC-C power system.

206   The Arabian Journal for Science and Engineering, Volume 31, Number 2B                                           October 2006
                                                                                           M. A. El-Kady and M. S. Owayedh

                                       Table 2: Application of OPF and DCOPF
                         VARIABLE       POWER                    BASE            OPTIMIZATION
                                         PLANT                   CASE            OPF     DCOPF
                                          PP8X                   1.035          1.050     1.050
                                          QUR                    1.020          1.030     1.041
                                          QPP3                   1.020          1.050     1.050
                                           GAZ                   1.015          1.050     1.050
                                           PP5                   1.000          1.004     1.009
                                          PP7A                   1.000          1.034     1.011
                          VG (PU)         SHED                   1.015          1.050     1.044
                                         FARAS                   1.000          1.028     1.022
                                           PP9                   1.020          1.050     1.033
                                          PP8A                   1.020          1.050     1.031
                                          QPP2                   1.045          1.050     1.050
                                          PP7B                   1.000          1.050     1.011
                                          PP8B                   1.020          1.050     1.032
                                          PP8X                   795.4          498.2     685.0
                                          QUR                    1792.4         2400.0   2143.0
                                          QPP3                   313.5           54.0      63.9
                                           GAZ                   3903.5         469.1     4139.1
                                           PP5                   406.6           60.0     270.3
                                          PP7A                   840.0          644.1     752.1
                          PG (MW)         SHED                   874.4          985.0     985.0
                                         FARAS                   549.6          725.0     725.0
                                           PP9                   398.4          110.7     285.1
                                          PP8A                   302.4          289.4     273.3
                                          QPP2                    72.0           90.0      90.0
                                          PP7B                   846.0          329.1     311.9
                                          PP8B                   302.4          115.3     221.3
                            ENERGY MARGIN                       19.0775          < 0.0    3.000
                            TOTAL COST (kSR)                    316.724         291.81   303.328
5.1. Dynamic Susceptibility
     Dynamic susceptibility of a given power system with respect to a particular contingency is defined as the combined
effects of: (1) how far the system operating point is from the insecurity boundary; and (2) how fast the deterioration in
dynamic security would be as one of the operating parameters changes. The key difference between dynamic security
and dynamic susceptibility concepts is that the former deals with incidental value of system security level (for example, a
value of the energy margin describing the degree of system stability, or instability), while the latter deals with the trend
(sensitivity) of the system security to varying parameters. For example, a higher gradient of energy margin with respect
to a particular operating parameter (e.g., power output from a particular generator) would indicate that the system is
susceptible to that parameter.
      A demonstration of this concept is given in Figure 5, which depicts the EM as a function of the East-Center tie-line
flow for four contingency scenarios representing faults at buses number 1, 24, 43, and 51, respectively, in the 19-
Generator system model. Based on the above definition, it is clear that the system is more dynamically susceptible to the
first and second contingency for increasing levels of the tie-line flow, and to the third contingency for decreasing levels
of the tie-line flow.

October 2006                                                  The Arabian Journal for Science and Engineering, Volume 31, Number 2B   207
      M. A. El-Kady and M.S. Owayedh

                                                                                       SCENARIO I                      SCENARIO II
                                                                                       SCENARIO III                    SCENARIO IV

                                 ENERGY MARGIN (UN-NORMALIZED)






                                                                    800.0      900.0   1000.0 1100.0 1200.0 1300.0 1400.0        1500.0   1600.0
                                                                                            EAST-TO-CENTER FLOW (MW)

                                                                            Figure 5. Demonstration of Dynamic Susceptibility.

      5.2. Consequent Restorability
            Consequent restorability of a given power system with respect to a particular contingency is defined as the degree of
      difficulty which is encountered in attempting to restore the system, or its supplied load, to its initial state should the
      contingency occur. A demonstration of this concept is best provided through a practical operating case scenario in the
      SEC power system. This case scenario involves a fault close to bus 78 in the Qassim area as shown in Figure 4. The fault
      is cleared by, in one contingency case by disconnecting the double circuit line between buses 78 and 79 while, in the
      second contingency case, the fault is cleared by disconnecting the double circuit line between buses 78 and 80. The load
      lost in the two contingency cases is practically the same, that is 17 MW and 20 MW, respectively. The ease of load
      restorability is, however, very different in the two cases. It is possible to restore the lost load in the first contingency case
      within hours, through the 33 kV network, using the normally opened circuits.
          For the second contingency case, on the other hand, the load restoration might take up to several days, depending on
      the extent of the circuit damage. Note, in this case, that there are no low voltage network links to other generation
      sources due to the remoteness of the substation involved.
      5.3. Example of Dynamic Performance Assessment
          An example is presented here, which demonstrates a comprehensive assessment of the power system dynamic
      performance. The purpose of this example is to explain how different, and often contradicting, criteria could be analyzed
      to assess the overall dynamic performance of the system more comprehensively. In this case the following contingency
      and performance criteria are considered:
      (i) Severity of contingency in terms of amount of load lost,
      (ii) Probability of contingency,
      (iii) Difficulty of consequent load restoration.
            The case study under consideration pertains to the Al-Kharj area of the SEC-C power gird. The load for the
      Al-Kharj area is about 119 MW and only one power plant (Layla) is located in the area according to the 19-Generator
      system model of Figure 4. As the production cost of the Layla power plant is the most expensive in the network, it is
      economical to generate the minimum possible power from this plant. Now, consider the following two fault scenarios:

208   The Arabian Journal for Science and Engineering, Volume 31, Number 2B                                                                        October 2006
                                                                                             M. A. El-Kady and M. S. Owayedh

(i) Fault on circuit 13-33 cleared by disconnecting that circuit,
(ii) Fault on transformer 101-102 cleared by disconnecting that transformer.
    The analysis of the system for the above two fault scenarios reveals the following interesting facts concerning the
dynamic system performance:
1.   The severity of second fault scenario is less than the severity of the first fault scenario, according to the severity
     criterion described before (the load lost is 13 MW for the second fault scenario as compared to 119 MW for the first
     fault scenario).
2.   The consequent restoration of the second fault scenario is less difficult than the first fault scenario, as it is always
     possible to restore the lost load for the second scenario case within half an hour (the time required to synchronize
     and load the gas turbine generating unit). On the other hand, it would not be possible to restore the full lost load for
     the first fault scenario since the amount of the load lost is more than the maximum capacity of the power plant (90
3.   In addition, it is well known that the probability of the first fault scenario, which represents a common faults on an
     overhead transmission line, is much higher than the probability of the second fault scenario, which represents a rare
     internal failure in a transformer.
     The above assessment is summarized in Table 3. On the basis of this assessment, it can safely be concluded that the
dynamic performance of the system is more degraded in the first fault than in the second in regard to the criteria of
contingency severity, consequent restorability, and probability of contingency.
                                 Table 3. Assessment of Dynamic System Performance
                 FAULT                          FAULT SCENARIO 1                    FAULT SCENARIO 2
                 SEVERITY                                HIGH                                LOW
                 RESTORABILITY                        DIFFICULT                              EASY
                 PROBABILITY                             HIGH                                LOW


  This paper introduces a framework and methodologies, which are capable of tackling the complex issue of economy
versus security in a practical and effective manner. In this regard two formulation were presented, namely the DCOPF
and its dual formulation, the CCMDS. This frame work can be further developed in integration of dynamic security
constraints into other cost-based planning and operation algorithms and computer programs such as unit commitment,
seasonal maintenance schedules and fuel inventory.
     In addition, and in parallel with the already well-established concept of system security, two new and far-reaching
concepts pertaining to power system performance were introduced, namely the concept of system dynamic susceptibility,
and the concept of system consequent restorability. The concepts of dynamic susceptibility and consequent restorability
could be extended further to form quantitative measures which can easily be implemented in operation and planning
studies, provided that the required data is available.

October 2006                                                    The Arabian Journal for Science and Engineering, Volume 31, Number 2B   209
      M. A. El-Kady and M.S. Owayedh

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210   The Arabian Journal for Science and Engineering, Volume 31, Number 2B                                                 October 2006

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