Page 320 - Handbook of Electrical Engineering

306 HANDBOOK OF ELECTRICAL ENGINEERING

unstable operation will exist (i.e. averaged over several cycles of the alternating current delivered from
the generators). The elements that connect all the generators in the equations are the electrical power
terms P elec (P mech will change due to the governor action sensing the change of shaft speed). The
P elec terms are connected and balanced through algebraic equations that represent the power balance
and exchange that occurs in the static electrical interconnecting network, e.g. cables, overhead lines,
transformers, loads.
Hence the simultaneous solution of the generator prime-mover equations also requires the
simultaneous solution of the algebraic power transfer equations of the electrical network. Digital
computers must be used for the accurate solution of these complex equations. Manual solution is
almost impossible, even for relatively simple situations. An excellent treatment of these complex
equations for multi-machine systems is given in Reference 13, which lends itself to being reasonably
easy to program in a digital computer. The reference also compares the benefits and disadvantages
obtained when the mathematical modelling of the generators becomes very detailed.

11.11.2.3 Limit of transient stability

In the same way that steady state stability was assessed by concentrating on the variations of the
rotor angle δ g , so also is the limit of transient stability assessed. However, the situation is not so
exact. The transient variation of δ g for any one machine can exceed 90 degrees, and even reach 120
degrees, before unstable operation occurs. The limit of transient stability can therefore exceed 90
degrees and is influenced by several factors:

• The inertia constant (H) of the machines.
• Effectiveness of the electromagnetic rotor damping.
• The pre-disturbance operating conditions and how close they are to the rated conditions.
• The amplitude of the disturbance.
• The time function of the disturbance, e.g. step function such as a fault, slowly changing function
such as a motor start.
• The ‘tightness’ or ‘looseness’ of the interconnections in the system (see sub-section 11.11.1.2).
• The time constants and gains of the control systems used in the automatic voltage regulators,
governors and prime-movers.
• The non-linear limits imposed on the control systems, e.g. constraints on excitation current, valve
limits on fuel valves.
• The dynamic characteristics of motor loads.
• The mixture ratio of dynamic to static loads.
• Operating power factors before the disturbance is applied.

11.11.2.4 Applications
In the oil, gas and petro-chemical industries, the need for stability studies is primarily due to the fact
that most plants have their own power generation facilities which are occasionally interconnected
between themselves or with a large public utility. In either case, the stable performance of the system
is of great importance, otherwise unwarranted shutdowns can occur with a resulting loss of production.