{"id":1049,"date":"2020-09-07T14:02:05","date_gmt":"2020-09-07T13:02:05","guid":{"rendered":"http:\/\/pb.ee.pw.edu.pl\/pb\/iepe\/?post_type=chapter&p=1049"},"modified":"2020-09-10T16:03:07","modified_gmt":"2020-09-10T15:03:07","slug":"task-6","status":"publish","type":"chapter","link":"http:\/\/pb.ee.pw.edu.pl\/pb\/iepe\/chapter\/task-6\/","title":{"raw":"Task 6. Capability chart of the generation unit","rendered":"Task 6. Capability chart of the generation unit"},"content":{"raw":"

Aim of the task<\/h1>\r\nThe aim of the task is to learn about limitations related to admissible operating states of a generating source based on a synchronous generator. The area of admissible operating states will be determined by means capability chart.\r\n

Introduction<\/h1>\r\nReal and reactive power produced by the generator depends on [1]:\r\n
    \r\n \t
  1. the electromagnetic force (emf)\u00a0 \u00a0 which is proportional to the generator field current\u00a0 ,<\/li>\r\n \t
  2. voltage U<\/em> on the terminals of the step-up transformer,<\/li>\r\n \t
  3. power angle\u00a0 (where - the angle\u00a0 between the voltage U<\/em> and the generator emf ).<\/li>\r\n<\/ol>\r\n \r\n\r\nThe voltage of a generator operating in a power system cannot change much and must be held within typically 10% percent of the network rated voltage.\r\n\r\nLimits in the generator real and reactive power control are the result of the following constructional and operational constraints:\r\n
      \r\n \t
    1. Stator (armature) current I<\/em> must not cause overheating of the armature winding. Hence it must be smaller than a certain maximum value , i.e. .<\/li>\r\n \t
    2. Rotor (field) current \u00a0 must not cause overheating of the field winding. Hence it must be smaller than a certain maximum value , i.e.\u00a0 or\u00a0 .<\/li>\r\n \t
    3. The power angle must not be higher than a maximum value due to stable generator operation \u00a0.<\/li>\r\n \t
    4. The temperature in the end region of the stator magnetic circuit must not exceed a maximum value<\/li>\r\n \t
    5. The generator real power must be within the limits set by the turbine power, i.e. .<\/li>\r\n<\/ol>\r\n

      Capability chart<\/strong><\/h1>\r\nCapability chart of the synchronous generator can be determined analysing the following conditions:\r\n

      <\/h2>\r\n

      a) Condition (1) - stator (armature) current<\/h2>\r\nOn the P-Q plane, that condition (i) corresponds to circle with the radius \u00a0 and the centre at the origin. Assuming a given voltage U<\/em> and loading\u00a0 one gets\r\n\r\n\r\n\r\nwhich corresponds to a circle with a radius\u00a0 .\r\n\r\n \r\n

      b) Condition (2) - rotor (field) current<\/h2>\r\nIn the P-Q plane, it corresponds to a circle with with a radius \u00a0\u00a0 and a centre displaced from the origin along the reactive power axis Q<\/em> by the value .\r\n\r\n\r\n\r\n \r\n

      c) Condition (3) - the power angle<\/h2>\r\nOn the P-Q plane, condition (iii) concerning the maximum value of the power angle, corresponds to a straight line. The position and tangent of the line can be derived the following way:\r\n\r\n\r\n\r\nSubstituting \u00a0 leads to \u00a0\u00a0 where\u00a0\u00a0 \u00a0\u00a0 \u00a0 \u00a0 and\u00a0\u00a0\u00a0 \u00a0 \r\n\r\nAbove equation describes a straight line intersecting the reactive power axis at the angle \u00a0 and at the point: or\u00a0 .\r\n\r\n \r\n

      d) Condition (4) - the temperature<\/h2>\r\nThere is no simple mathematical formulation describing\u00a0 the constraint corresponding the end region heating limit. The relevant curve has to be determined experimentally by the manufacturer.\r\n

      <\/h2>\r\n

      e) Condition (5) \u2013 turbine power<\/h2>\r\nFor steam turbines, the upper constraint \u00a0 is due to the maximum (rated) output of the turbine while the lower constraint \u00a0 is due to stable operation of burners at a low turbine output. On the P-Q plane the upper and lower limits correspond to straight vertical lines \u00a0 and .\r\n\r\n \r\n\r\nFig. 6.1 shows the synchronous generator capability chart determined for given voltage.\r\n\r\n \r\n\r\n\r\n

      Fig.6.1. Synchronous generator capability chart assuming a given voltage, where segment A-B\u00a0 corresponds to\u00a0\u00a0\u00a0\u00a0\u00a0 the power angle condition,\u00a0 segment B-C corresponds to the temperature condition, segments C-D and E-F corresponds to stator (armature) current condition, segment D-E corresponds to upper turbine power condition, segment G-A corresponds to lower turbine power condition [1]<\/p>\r\n \r\n\r\n \r\n

      Exercise<\/h1>\r\n \r\n\r\nContent and input data:<\/span>\r\n\r\nAnalysis refers to the synchronous generator described in Task 5.\r\n\r\n \r\n\r\nTo do:<\/span>\r\n\r\nCalculate and plot a capability chart of a synchronous\u00a0 generator.\r\n\r\nIn the chart, mark the point corresponding to load of the generator. Verify if the operation point of the generator (P<\/em>, Q<\/em>) is located in the admissible area.\r\n\r\nAdditional assumptions:\r\n
        \r\n \t
      • Calculation should be carried out in per units<\/strong> (base values correspond to rated values of the generator U<\/em>b<\/sub>= U<\/em>N<\/sub> =15,75 kV, S<\/em>b<\/sub>= S<\/em>N<\/sub> =250MVA)<\/li>\r\n \t
      • maximum value of the stator (armature) current is equal to current rated value I<\/em>MAX<\/sub>=I<\/em>n<\/sub><\/li>\r\n \t
      • turbine power limits: P<\/em>MAX<\/sub>=1,05 P<\/em>n<\/sub>, P<\/em>min<\/sub>=0,55 P<\/em>n<\/sub> or 0,45 P<\/em>n<\/sub> (where P<\/em>n<\/sub> - generator rated real power)<\/li>\r\n \t
      • generator-rated power<\/em> factor cosj<\/em>n = 0,85<\/li>\r\n \t
      • maximum value of excitation E<\/em>fMAX<\/sub> = E<\/em>qMAX<\/sub> = 3,5 or 3,6 or 3,7 or 3,8 or pu (for base values corresponding to rated values of the generator)<\/li>\r\n \t
      • power angle limit: = 120 or 110 or 100 deg<\/li>\r\n \t
      • Condition corresponding to temperature in the end region of the stator magnetic circuit is ignored in analyze.<\/li>\r\n<\/ul>\r\nHint:\r\n\r\nIn Task 5 reactances of synchronous generator were expressed in per unit but the base values were: U<\/em>b<\/sub>= 220 kV, S<\/em>b<\/sub>=100MVA (network base values). For capability chart calculation the base values related to rated values of generator U<\/em>n<\/sub> and S<\/em>n<\/sub> should be taken into account.\r\n\r\nRecalculation of the generator reactance:\r\n\r\nZ<\/em>b1<\/sub> \u2013 base impedance for U<\/em>b<\/sub> =220 kV and S<\/em>b<\/sub>=100 MVA\r\n\r\nX<\/em>d1<\/sub> \u2013 generator reactance X<\/em>d<\/sub> from task 5\r\n\r\nZ<\/em>b2<\/sub> \u2013 base impedance for U<\/em>b<\/sub> =15,750 kV and S<\/em>b<\/sub> =250 MVA\r\n\r\n\u00a0\u2013 transformer ratio ( 245\/15,75\u00a0 or = 250\/15,75 )\r\n\r\n \r\n

        References<\/h1>\r\n[1] Machowski J., Bialek J, Bumby J.,\u00a0 Power system dynamics: stability and control<\/em>. New York, USA: John Wiley & Sons; 2020.\r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n \r\n\r\n ","rendered":"

        Aim of the task<\/h1>\n

        The aim of the task is to learn about limitations related to admissible operating states of a generating source based on a synchronous generator. The area of admissible operating states will be determined by means capability chart.<\/p>\n

        Introduction<\/h1>\n

        Real and reactive power produced by the generator depends on [1]:<\/p>\n

          \n
        1. the electromagnetic force (emf)\u00a0 \"image\"\u00a0 which is proportional to the generator field current\u00a0 \"image\" ,<\/li>\n
        2. voltage U<\/em> on the terminals of the step-up transformer,<\/li>\n
        3. power angle\u00a0 \"image\" (where \"image\"– the angle\u00a0 between the voltage U<\/em> and the generator emf \"image\" ).<\/li>\n<\/ol>\n

           <\/p>\n

          The voltage of a generator operating in a power system cannot change much and must be held within typically 10% percent of the network rated voltage.<\/p>\n

          Limits in the generator real and reactive power control are the result of the following constructional and operational constraints:<\/p>\n

            \n
          1. Stator (armature) current I<\/em> must not cause overheating of the armature winding. Hence it must be smaller than a certain maximum value \"image\" , i.e. \"image\" .<\/li>\n
          2. Rotor (field) current \"image\"\u00a0 must not cause overheating of the field winding. Hence it must be smaller than a certain maximum value \"image\" , i.e.\u00a0 \"image\" or\u00a0 \"image\" .<\/li>\n
          3. The power angle must not be higher than a maximum value due to stable generator operation \"image\"\u00a0.<\/li>\n
          4. The temperature in the end region of the stator magnetic circuit must not exceed a maximum value<\/li>\n
          5. The generator real power must be within the limits set by the turbine power, i.e. \"image\" .<\/li>\n<\/ol>\n

            Capability chart<\/strong><\/h1>\n

            Capability chart of the synchronous generator can be determined analysing the following conditions:<\/p>\n

            <\/h2>\n

            a) Condition (1) – stator (armature) current<\/h2>\n

            On the P-Q plane, that condition (i) corresponds to circle with the radius \"image\"\u00a0 and the centre at the origin. Assuming a given voltage U<\/em> and loading\u00a0 \"image\" one gets<\/p>\n

            \"image\"<\/p>\n

            which corresponds to a circle with a radius\u00a0 \"image\".<\/p>\n

             <\/p>\n

            b) Condition (2) – rotor (field) current<\/h2>\n

            In the P-Q plane, it corresponds to a circle with with a radius \"image\"\u00a0\u00a0 and a centre displaced from the origin along the reactive power axis Q<\/em> by the value \"image\" .<\/p>\n

            \"image\"<\/p>\n

             <\/p>\n

            c) Condition (3) – the power angle<\/h2>\n

            On the P-Q plane, condition (iii) concerning the maximum value of the power angle, corresponds to a straight line. The position and tangent of the line can be derived the following way:<\/p>\n

            \"image\"<\/p>\n

            Substituting \"image\"\u00a0 leads to \"image\" \u00a0\u00a0 where\u00a0\u00a0 \u00a0\u00a0 \"image\"\u00a0 \u00a0 and\u00a0\u00a0\u00a0 \u00a0 \"image\"<\/p>\n

            Above equation describes a straight line intersecting the reactive power axis at the angle \"image\"\u00a0 and at the point: \"image\" or\u00a0 \"image\".<\/p>\n

             <\/p>\n

            d) Condition (4) – the temperature<\/h2>\n

            There is no simple mathematical formulation describing\u00a0 the constraint corresponding the end region heating limit. The relevant curve has to be determined experimentally by the manufacturer.<\/p>\n

            <\/h2>\n

            e) Condition (5) \u2013 turbine power<\/h2>\n

            For steam turbines, the upper constraint \"image\" \u00a0 is due to the maximum (rated) output of the turbine while the lower constraint \"image\"\u00a0 is due to stable operation of burners at a low turbine output. On the P-Q plane the upper and lower limits correspond to straight vertical lines \"image\"\u00a0 and \"image\" .<\/p>\n

             <\/p>\n

            Fig. 6.1 shows the synchronous generator capability chart determined for given voltage.<\/p>\n

             <\/p>\n

            \"image\"<\/p>\n

            Fig.6.1. Synchronous generator capability chart assuming a given voltage, where segment A-B\u00a0 corresponds to\u00a0\u00a0\u00a0\u00a0\u00a0 the power angle condition,\u00a0 segment B-C corresponds to the temperature condition, segments C-D and E-F corresponds to stator (armature) current condition, segment D-E corresponds to upper turbine power condition, segment G-A corresponds to lower turbine power condition [1]<\/p>\n

             <\/p>\n

             <\/p>\n

            Exercise<\/h1>\n

             <\/p>\n

            Content and input data:<\/span><\/p>\n

            Analysis refers to the synchronous generator described in Task 5.<\/p>\n

             <\/p>\n

            To do:<\/span><\/p>\n

            Calculate and plot a capability chart of a synchronous\u00a0 generator.<\/p>\n

            In the chart, mark the point corresponding to load of the generator. Verify if the operation point of the generator (P<\/em>, Q<\/em>) is located in the admissible area.<\/p>\n

            Additional assumptions:<\/p>\n

              \n
            • Calculation should be carried out in per units<\/strong> (base values correspond to rated values of the generator U<\/em>b<\/sub>= U<\/em>N<\/sub> =15,75 kV, S<\/em>b<\/sub>= S<\/em>N<\/sub> =250MVA)<\/li>\n
            • maximum value of the stator (armature) current is equal to current rated value I<\/em>MAX<\/sub>=I<\/em>n<\/sub><\/li>\n
            • turbine power limits: P<\/em>MAX<\/sub>=1,05 P<\/em>n<\/sub>, P<\/em>min<\/sub>=0,55 P<\/em>n<\/sub> or 0,45 P<\/em>n<\/sub> (where P<\/em>n<\/sub> – generator rated real power)<\/li>\n
            • generator-rated power<\/em> factor cosj<\/em>n = 0,85<\/li>\n
            • maximum value of excitation E<\/em>fMAX<\/sub> = E<\/em>qMAX<\/sub> = 3,5 or 3,6 or 3,7 or 3,8 or pu (for base values corresponding to rated values of the generator)<\/li>\n
            • power angle limit: \"image\"= 120 or 110 or 100 deg<\/li>\n
            • Condition corresponding to temperature in the end region of the stator magnetic circuit is ignored in analyze.<\/li>\n<\/ul>\n

              Hint:<\/p>\n

              In Task 5 reactances of synchronous generator were expressed in per unit but the base values were: U<\/em>b<\/sub>= 220 kV, S<\/em>b<\/sub>=100MVA (network base values). For capability chart calculation the base values related to rated values of generator U<\/em>n<\/sub> and S<\/em>n<\/sub> should be taken into account.<\/p>\n

              Recalculation of the generator reactance:<\/p>\n

              Z<\/em>b1<\/sub> \u2013 base impedance for U<\/em>b<\/sub> =220 kV and S<\/em>b<\/sub>=100 MVA<\/p>\n

              X<\/em>d1<\/sub> \u2013 generator reactance X<\/em>d<\/sub> from task 5<\/p>\n

              Z<\/em>b2<\/sub> \u2013 base impedance for U<\/em>b<\/sub> =15,750 kV and S<\/em>b<\/sub> =250 MVA<\/p>\n

              \"image\"\u00a0\u2013 transformer ratio ( 245\/15,75\u00a0 or = 250\/15,75 )<\/p>\n

               <\/p>\n

              References<\/h1>\n

              [1] Machowski J., Bialek J, Bumby J.,\u00a0 Power system dynamics: stability and control<\/em>. New York, USA: John Wiley & Sons; 2020.<\/p>\n

               <\/p>\n

               <\/p>\n

               <\/p>\n

               <\/p>\n

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