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Optimum operation of a small autonomous system with unconventional energy sources
Electric Power Systems Research, 23 (1992) 93 102
93
Optimum operation of a small autonomous system with u n c o n v e n t i o n a l energy sources A. G. Bakirtzis and E. S. Gavanidou Aristotle University of Thessaloniki, Salonika (Greece)
(Received July 22, 1991)
Abstract This paper presents a method for the determination of the optimum operation of a small autonomous system with both conventional and unconventional energy sources and storage battery. The system generation consists of diesel generators, wind turbine generators and photovoltaic panels. This is the generation mix of the Greek island of Cythnos; it may be applied to other small islands. A stochastic dynamic programming based algorithm is employed to determine the optimal short-term generation scheduling and battery storage policy which minimize the fuel consumption for the next 24-hour period. An application of the method to the power system of Cythnos is also presented.
1. Introduction Utilization of unconventional energy sources for the electrical power supply to remote regions has proved to be a particularly promising application of wind and solar energy plants. Owing to the high production costs of electricity that result from a decentralized power supply (mostly by diesel generator units), large-scale utilization of unconventional energy sources is favoured in remote regions with appropriate weather conditions. This applies in particular to the Greek islands. A large unconventional penetration into a diesel power system faces technical and economic problems. The problem of the economic operation of a small autonomous system is addressed in refs. 1-3. In ref. 1 a method was presented for optimal generation scheduling in a small autonomous system with conventional (diesel) and unconventional energy sources and storage battery. The method relied on forecasts of the average hourly values of the load demand, the wind velocity and the global insolation for the next 24-hour period, and used a dynamic programming based algorithm to determine an optimal diesel unit commitment schedule and an optimal battery charge/discharge sequence so as to minimize the diesel fuel consumption the next day. The forecasts were assumed to be completely reliable. Further tests showed that, whereas the load demand prediction for the next 24 hours is 0378-7796/92/$5.00
satisfactory, the wind velocity and the global insolation can only be predicted within relatively large confidence intervals. Therefore the method in ref. 1 cannot be applied for the on-line management of a small autonomous system without further exploitation of the stochastic variation of the weather conditions. This paper addresses the problem of the optimum operation of a small autonomous system under stochastically varying weather conditions. The t-hour ahead forecasts are treated as random variables, not known at the beginning of the day when the system operation scheduling takes place. A stochastic dynamic programming based algorithm is used in order to determine the optimal diesel unit commitment and an adaptive battery storage policy that can follow the changing weather conditions. The developed algorithm gives a feedback control of the system, appropriate for on-line application.
2. System description Figure 1 shows a typical generation of a small autonomous system. The basic components of the system are the diesel plant and the unconventional sources, the wind park and the solar plant. A battery is also available for energy storage. A brief description of the basic parts of the system follows. © 1992 -- Elsevier Sequoia. All rights reserved
94 DIESEL
PLANT
This is the minimum diesel plant o u t p u t requirement. At hours of low load demand and high u n c o n v e n t i o n a l energy production, when the minimum diesel plant r e q u i r e m e n t is active, the excess u n c o n v e n t i o n a l energy production must be stored in the battery. If the b a t t e r y is fully c h a r g e d at that hour the excess unconventional energy is lost.
CD
WIND
PARK
(K)
'
2.2. The wind park The available wind park o u t p u t during hour t is a function of the wind velocity w(t) during that hour:
SOLAR PLANT
I
BATTERY
~
~fiw(t) = gw (w(t))
w
I Fig. 1. T h e g e n e r a t i o n system.
s y s t e m of a s m a l l a u t o n o m o u s
power
For maximum fuel saving, all the available wind park p o w e r should be utilized in order to satisfy the load demand and/or charge the battery. However, due to the minimum diesel plant o u t p u t requirement, some wind power may be lost, so that, in general, Pw(t) ~
2.1. The diesel plant The diesel plant consists of NI) units. The production cost of the ith diesel unit is described by a linear cost function: F; (PD;)
fNLCi + IC;PD,: = ~O
if if
P,)~ ~< PD; PD; = 0
~<
w h e r e NLCi is the unit no-load cost and ICi is the unit i n c r e m e n t a l cost. The diesel plant o u t p u t during h o u r t is NI)
P,(t) = ~ Prj;(t)
(1)
i=1
where
P_r)i ~ PDi(t) <~l~Di
(2a)
if unit i is committed, and
PDi(t) = 0
(2b)
if unit i is not committed. The p o w e r that the diesel plant must supply at a certain time equals the load demand at t h a t time minus the u n c o n v e n t i o n a l generation and the b a t t e r y discharge. The diesel plant controls the voltage level and the frequency of the system. Also, if a diesel unit operates at a small fraction of its nominal p o w e r o u t p u t its efficiency is considerably reduced. For these reasons, the diesel p l a n t o u t p u t must never fall below a certain value _PD: PD(t) ~ ~Pl)
(3)
(5)
2.3. The solar plant The available p h o t o v o l t a i c plant o u t p u t during hour t is a function of the global insolation P~ (t) = gs (s(t))
PI)i
(4)
(6)
In general, the usable solar plant o u t p u t is less t h a n t h a t available, due to the minimum diesel plant o u t p u t requirement: Ps (t) ~
(7)
The photovoltaic panels are connected t h r o u g h a DC/DC c o n v e r t e r to the b a t t e r y busbar. There is a maximum power t r a c k i n g control for the c o n v e r t e r so that maximum power can be drawn from the panels. The solar plant is connected to the grid t h r o u g h a DC/AC inverter and a step-up transformer.
2.4. The storage battery The storage b a t t e r y is c o n n e c t e d to the solar plant t h r o u g h a DC/DC c o n v e r t e r and to the grid t h r o u g h the solar plant DC/AC inverter and transformer as s h o w n in Fig. 1. The b a t t e r y capacity is I?B and the depth of discharge is VB. Thus, the energy stored in the b a t t e r y at any time must satisfy the inequalities VB ~< VB(t) ~< VB
(8)
The b a t t e r y can be c h a r g e d either by the solar plant or by the grid. The b a t t e r y efficiency is ~h~ and is t a k e n into a c c o u n t during b a t t e r y charging
95
only. The energy balance in the battery during hour t is described by the following relations:
The output of the inverter is not allowed to exceed the step-up transformer rating, PST:
VB(t + 1) = VB(t) -- t/nPB(t)
(9a)
Pss(t)
(9b)
2.6. System operation At every hour of the system operation, the load demand must be supplied by the diesel plant, the unconventional sources and the battery:
Vs(t + 1) = Vs(t) - P s ( t ) --PB(t)
~/)Bc
if
PB(t) ~
if if
PB(t) < 0
PB(t) > 0
PB(t) < 0
(10a)
PB(t) > 0
(10b)
~
(14)
where the battery is charging when PB < 0 and discharges when PB > 0; PBc and/~Bo are the battery charge and discharge rates, respectively. The energy stored in the battery at the beginning of the day and the desired battery storage at the end of the day are also specified:
PL(t) = PD(t) A-Pw(t) + PsB(t)
VB(0) =VBo
(11)
VR(T) = VBT
(12)
The problem of the optimum operation of a small autonomous system can be stated as follows. Given the short-term forecasts (next 24 hours) and the forecast error standard deviations of (a) the hourly average load demand/~(t), GL(t), (b) the hourly average wind velocity &(t), Gw(t) and (c) the hourly average global insolation ~(t), as(t), determine (i) the hourly diesel unit commitment and dispatch and (ii) the hourly battery energy storage policy so as to minimize the total diesel cost over the next 24-hour period. The hourly wind park and solar plant output are determined by the hourly wind velocity and insolation levels so as to fully utilize the cost-free soft energy sources. In rare cases a reduction of the unconventional generation may be necessary in order to satisfy the minimum diesel plant output requirement. The above-mentioned problem is a dynamic optimization problem with objective the total diesel cost during the scheduling period:
2.5. The inverter The solar plant and the battery are connected to the grid with a DC/AC inverter and a step-up transformer as shown in Fig. 1. The inverter controls the flow of power, PsB, between the solar-plant-battery combination and the grid. The direction of the power flow PsB can be either from the inverter to the grid (Ps~ > 0) or vice versa (PsB < 0). Figure 1 shows the direction for positive PB and PsB- If the inverter efficiency is ~i and the step-up transformer is assumed 100% efficient, the following relations describe the power balance in the inverter: PsB(t) = ~i[Ps(t) + PB(t)]
if
PsB(t) > 0
(13a)
PsB(t) = 1 [Ps(t) + PB(t)] th
if
PsB(t) < 0
(13b)
(15)
where PsB(t) is the inverter output given in (13).
3. P r o b l e m s t a t e m e n t
The inverter efficiency is a function of the power delivered by the inverter, as shown in Fig. 2. subject to the system operating constraints (1)(15). The solution to the problem is discussed next.
q 1.0
~
q
0.9
4. S o l u t i o n o f the p r o b l e m
0.8
The solution to the optimum operation problem is outlined in Fig. 3. At the beginning of the day (twelve midnight), the scheduling of the next day's operation is performed (block 2). At this point it must be decided which diesel u.nits will be committed and what will be the battery usage during the next day so that the expected operation cost of the system is minimized. These
0.7
0
.
.
0.1
0.3
.
. 0.6
" P/PN
1.0
Fig. 2. I n v e r t e r efficiency as a function of the inverter power output.
96
.
;.°
HISTORYDATA~ 0_ °•"'°•..°
FORECASTS Determine
BLOCK 1
~
"
W(T), S(T), Z(T) Ow(t), Os(t), OL(T) for t=1,2,...T
"..
"•
.." •.
° °
"..
..." • ,~
•
"•'.
"j J . ;
".\ ..~
-°oo -
N
• . .....
OPERATION SCHEDULING Determine Optimal Unit Commitment Determine Optimal Battery Storage Policy in the form of a "Look up Table":
,
.
•
BLOCK 2
~
;2
time
VB*(T+I )=fB(VB(T) ,w(T),S(T),L(1) For T=O,I.... ~T-I
I
,:
20
2'4
(h)
F i g . 4. Load d e m a n d forecasts versus m e a s u r e m e n t s .
=.
.•••'.o°o°.°..
ON-LINE OPER.AII ~I M e a s u r e
VB(t) ,w(t) , s ( t ) , L ( t )
Read Optimal Unit Committment Read VB (t+]) from Look up Table Send Control Signals to: Diesel Plant Inverter
BLOCK 3
•
•••
.0.
•..'..
"..•...
.°
~¢~. E "'. >
• .
\
• ......
~.~/
'%
\,~"
/
~ ' ~
\
''''•'..'0.••...••,. °°• YES time
(h)
NO F i g . 5. W i n d v e l o c i t y forecasts v e r s u s m e a s u r e m e n t s . F i g . 3. F l o w c h a r t of the s o l u t i o n to t h e o p t i m u m o p e r a t i o n problem.
o_
decisions are made under the uncertainty associated with the forecasts of the wind velocity, the global insolation and the load demand (block 1). During the system on-line operation (block 3) the diesel unit commitment is kept fixed while the battery usage is adapted to the changing weather conditions and load demand as new measurements of these quantitites are available. A more detailed discussion of these three solution steps follows•
OE 8°=_ .E
o-
•
•
g
111
113
115 time
117
t/
(h)
F i g . 6. Global i n s o l a t i o n forecasts versus m e a s u r e m e n t s .
4.1. Forecasts
The forecasts of the next day's average hourly values of the wind velocity, the global insolation and the load demand are computed based on measurements of the average hourly values of the previous 30 days• B o x - J e n k i n s models [4] are used to compute the forecasts. These models give, in addition to the forecasts, the forecast error standard deviation which can be used to compute the confidence limits of the forecasts•
Sample results of the forecasts are s h o w n in Figs. 4, 5 and 6. In the Figures, the solid lines represent the forecasts, the dashed lines represent the measured values of the forecasted quantities and the dotted lines the 95% confidence limits of the forecasts (i.e. the region in which, with 95% probability, the measured values of the forecasted quantities will lie). The small confidence interval for the insolation in Fig. 6 was observed during summer months only.
97
The forecasts of any of the forecasted quantities, for example the wind velocity, are given in the form {&(t), aw(t), t = 1, 2. . . . . T} where t~(t) is the forecast of the wind velocity w(t) that will be observed during the tth hour and aw(t) is the forecast error standard deviation. Assuming that the forecast error ew(t) is a normal random variable with zero mean and standard deviation aw(t), the wind velocity that will be observed during hour t,
w(t) = &(t) + ew(t) is a normal random variable with mean &(t) and standard deviation aw(t). That is, at the beginning of the day, w(t) is not known but its probability distribution is well defined. Since w(t) is a normal random variable, it will, with 95% probability, be in the interval &(t) _+2aw(t) which gives the 95% confidence limits of Fig. 5. Now, taking into account the fact that w(t) is positive, the 95% probability limits are modified to [w(t), ~(t)] where w(t) = max[0, &(t) - 2aw(t)]
(17a)
and &(t) = &(t) + 2aw(t)
(17b)
The probability density function of w(t) is shown in Fig. 7, where the amplitude of the impulse at zero equals the shaded area along the negative w(t) axis.
4.2. Operation scheduling The scheduling of the operation of the system is performed at the beginning of the day and consists of two steps.
Step 1. Determination of the optimal diesel unit commitment The diesel unit commitment is computed using a deterministic dynamic programming algorithm described in ref. 1 and is based on the forecasts &(t), ~(t) and £(t), t = 1 , . . . , T, available at the beginning of the day. The forecasts are used in f.(w)
-2Ow
0
W
Fig. 7. Wind velocity probability density function.
order to compute the projected load demand and unconventional generation and an optimal battery charge/discharge sequence, which give the projected diesel plant loading for every hour of the next day. The projected hourly diesel plant loading is then used to compute the optimal diesel unit commitment. Details of the method are given in ref. 1. In summary, the method developed in ref. 1 computes: (a) the optimal diesel unit commitment {UC*(t), t -- 1. . . . . T}
(18)
(b) the optimal sequence of battery storage levels {V~(t), t -- 1 , . . . , T}
(19)
which minimize the diesel operation cost of the next day under the assumption of zero forecast error (i.e. it is assumed that the values of the forecasted quantities that will be observed during the course of the day will be equal to their forecasts). The above sequence of decisions gives an 'open-loop' control of the system which is determined at the beginning of the day and cannot change during the course of the day, when new measurements are available, in order to follow the unpredictable changes of the weather conditions and the load demand. Since the unit commitment must be decided at the beginning of the day and there is limited flexibility in changing it during the course of the day, the open-loop control policy described above is used. Once the unit commitment schedule is determined, based on the available forecasts, it is kept fixed for the rest of the day. The battery usage can be further improved by replacing the open-loop optimal battery storage sequence (19) with a feedback battery storage policy which is sufficiently flexible to follow the changing weather conditions.
Step 2. Optimal battery storage policy In this step the optimal battery storage policy is computed, using a stochastic dynamic programming algorithm. Since there is a certain amount of uncertainty associated with the forecasts (w(t), s(t) and L(t) are now treated as random variables, not known at the beginning of the planning period), the objective in the determination of the battery storage policy is the minimization of the expected diesel cost:
98
w h e r e the e x p e c t a t i o n in (20) is t a k e n over the p r o b a b i l i t y distributions of all the forecasted quantities. A stochastic dynamic programming algorithm is used to minimize the expected operating cost (20) subject to the system operating c o n s t r a i n t s (1) - (15). (a) T h e d y n a m i c p r o g r a m m i n g state space. The energy stored in the b a t t e r y is discretized into Kj~ discrete energy storage levels b e t w e e n V~ and VBk=V~+(k--1)
AVB
2 = 1, 2 . . . . . A(t)
w h e r e w~(t) is the )~th wind velocity level during h o u r t and Aw is the wind velocity discretization step (e.g. Aw = 1 m/s). The p r o b a b i l i t y density function of w(t) is also discretized with the comp u t a t i o n of the probabilities Pr,~.(t) = Pr{w(t) = w~(t)}
~ = 1, 2 . . . . .
V ~ ( t ) = V.,~
and the average wind velocity, global insolation and load demand during hour t are w(t) = w~(t),
s(t) = s,,(t)
and L(t) = L,,(t)
k = 1 , 2 . . . . ,K~
w h e r e V~k is the k t h b a t t e r y storage level and A V~ is the b a t t e r y energy storage discretization step. As discussed in §4.1, the wind velocity w(t) is a random variable t a k i n g values in the interval [_w(t), w(t)] given in (17). The wind velocity w(t) is discretized into A(t) discrete wind levels: w j ( t ) = _w(t) + (~ - 1) Aw
of the scheduling period, given that the energy stored in the b a t t e r y at the beginning of hou{ t is
A(t)
The global insolation and the load d e m a n d as well as their p r o b a b i l i t y distributions are also discretized with steps As and AL, respectively: s,,(t) = s_(t) + (p - 1) As
IL = 1, 2 . . . . .
M(t)
Pr~(t) = Pr{s(t) = s,,(t)}
p = 1, 2 . . . . .
M(t)
L,,(t) = L(t) + (u - 1) AL
u = 1, 2 . . . . .
Y(t)
Pr)'(t) = Pr{L(t) = L,(t)}
u = 1, 2 . . . . .
Y(t)
With the discretization of the b a t t e r y storage, the wind velocity, the global insolation and the load demand, the stochastic dynamic programming state space is formed. Each state in the state space is defined by five indices (t, k, .~, It, u) specifying time, b a t t e r y storage level, wind velocity level, global insolation level and load dem a n d level, respectively. The state probability Pr~(t) Pr~(t) Pr~.(t) is a s s o c i a t e d with each state of the state space (t, k, ),, p, u). This is the probability of o c c u r r e n c e of the specific levels of the f o r e c a s t e d quantities, assuming their statistical independence. (b) T h e r e c u r s i v e f o r m u l a . The stochastic dynamic p r o g r a m m i n g recursive formula is derived with the help of the following definitions: (1) RC(t, k, )., IL, u): minimum expected production cost from the beginning of h o u r t to the end
(2) PC(t, k, ~, g, u; t + 1, m): p r o d u c t i o n cost during h o u r t, given t h a t the b a t t e r y stored energy at the beginning of hour t is VB(t) = V~k
the average wind velocity, global insolation and load demand during hour t are w(t) = w~(t),
s(t) = s,,(t)
and L(t) = L,,(t)
and the b a t t e r y stored energy at the beginning of hour t + 1 is VB (t + 1) = V~,,
The o p t i m u m b a t t e r y storage policy is comp u t e d by applying the stochastic dynamic programming recursive formula RC(t, k, )~, p, u)
E
=rain
[PC(t,k,i,l~, u;t + l,m)
+RC(t + 1, m, r, z, v)]
(21)
with
E [PC(t, k, )., p, u; t + 1, m) + RC(t + 1, m, r, z, v)] 'tr, u. u{
=PC(t, k, 2, p, u; t + 1, m) +
~,
RC(t + 1, m, r, u, v)Pr[,~(t)Pr~(t)Pr~'(t)
{r, u, vl
where the first term is taken out of the expectation since it is independent of r, z and v. That is, the production cost during hour t does not depend on the wind velocity, the global insolation and the load level that will be observed during hour t ÷ 1 (Wr(t T 1), sz(t ÷ 1), and L~(t + 1)). The production cost PC(t, k, )., p, u; t + 1, m) is computed as follows. - The power delivered by the battery is computed using (9):
99
" 1 [VB(t) - VB(t + 1)]
PB (t) =
=l(k -m)AVB ~/B [VB(t) -- Vn(t + 1)1 =(k - m ) A V B
if m > k
ifm
- The unconventional generation is computed assuming w(t)= w~(t) and s(t)=s~(t). The load demand is L(t) = Lu(t). - The diesel plant output PD(t) is computed using (12) and if the minimum diesel plant output requirement is violated the unconventional generation is reduced. - If any of the constraints (10) and (14) is violated, the corresponding cost is set to infinity: PC(t, k, it, It, u; t + 1, m) = oo For the computation of PC(t, k, 2, ~, u; t + 1, m) the required diesel plant output PD(t) is allocated to the ND units of the diesel plant according to the unit commitment determined in step 1. (c) The dynamic programming results. The results of the dynamic p r o g r a m m i n g algorithm will be in the following form. For every state in the dynamic p r o g r a m m i n g state space defined by (t, k, 2, p, u) the optimum b a t t e r y discrete storage level m* for the next time interval t + 1 is computed and stored in the form of a 'look-up table' (m* is the value of m t h a t minimizes the expression in (21)). In other words, the optimal battery storage policy is computed in the form
V*m(t + 1) = fB(Vnk(t), w~.(t), s,(t), Lu(t), t) t=0 .... ,T-1
(22)
The optimal b a t t e r y storage policy (22), computed at the beginning of the day, is in the form of a 'look-up table' t h a t contains the optimum operating decision (V*m(t + 1)) for every h o u r and for all the possible operating states of the next day. The m e a n i n g of the b a t t e r y storage policy (22) can be better explained if we drop the discrete level indices (assuming very fine discretization) so t h a t (22) becomes
V*(t + 1) = fB(VB(t), w(t), s(t), L(t), t)
V*(t + 1) can only be d e t e r m i n e d during h o u r t w h e n the m e a s u r e m e n t s of w(t), s(t) and L(t) are available. This is a feedback control policy in which m e a s u r e m e n t s of the c u r r e n t operating state are used in order to t a k e the optimal decision. The decision is not t a k e n once at the beginning of the day, but decisions are t a k e n every h o u r during the system on-line operation, w h e n new m e a s u r e m e n t s are available. This is described next. 4.3. On-line operation The results of the system's operation scheduling, performed at the beginning of the day, are: (a) the optimal diesel unit commitment, derived as an optimal sequence of decisions (18) and (b) the optimal b a t t e r y storage policy, derived in terms of a 'look-up table' (22). During the tth h o u r of the system operation, when m e a s u r e m e n t s of VB(t), w(t), s(t) and L(t) are available, the optimum b a t t e r y storage level at the end of h o u r t, V*(t + 1), is read from the 'look-up table' formed during the operation scheduling phase. V*(t + 1), t o g e t h e r with the b a t t e r y storage level at the beginning of the hour, Vn(t), specify the desired b a t t e r y charging or discharging during h o u r t, PB(t). Measurements of the wind park and solar plant o u t p u t Pw(t) and Ps(t) are also available. Therefore, from (13) and (15), the desired inverter output PsB(t) and diesel plant output PD(t) are determined and appropriate reference signals are sent to the inverter and the diesel plant. The desired diesel plant o u t p u t PD(t) is allocated to the diesel units committed during h o u r t.
5.
The performance of the m e t h o d is demonstrated by a case study with the power system of T A B L E 1. Diesel u n i t d a t a i
_PDi (kW)
/~i (kW)
NLCi (S/h)
ICi (S/kWh)
1 2 3 4 5 6 7 8
50 50 30 30 30 30 30 30
530 530 200 200 100 100 100 100
3.57 3.57 3.57 3.57 1.07 1.07 1.07 1.07
0.0615 0.0615 0.0661 0.0661 0.0657 0.0657 0.0657 0.0657
(23)
T h a t is, the optimum b a t t e r y storage level at the beginning of the next h o u r is a function of the b a t t e r y storage level at the beginning of the c u r r e n t h o u r and the wind velocity, the global insolation and the load d e m a n d m e a s u r e d during the c u r r e n t hour. Since w(t), s(t) and L(t) are not k n o w n at the beginning of the scheduling period,
Results
Power demand (kW)
203.6 173.4 184.7 145.1 173.7 174.7 168.6 183.2 217.8 281.8 283.6 286.2 306.6 275.1 228.9 204.3 229.7 257.8 229.1 320.1 409.7 385.5 321.6 241.2
Hour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
95.8 73.4 84.7 45.1 73.7 74.7 93.3 70.3 99.9 274.2 199.1 176.4 262.7 182.7 96.7 95.8 194.9 194.4 152.7 274.7 338.6 327.6 266.3 175.4
Diesel plant output (kW)
100.0 100.0 100.0 100.0 100.0 100.0 65.8 88.2 59.2 9.2 6.6 42.1 5.3 28.9 36.8 48.7 32.9 52.6 76.3 75.0 71.1 57.9 55.3 65.8
Wind park output (kW)
0.0 0.0 0.0 0.0 0.0 0.1 11.8 27.7 44.9 59.0 69.3 74.0 75.2 69.7 59.8 46.1 29.5 13.1 0.1 0.0 0.0 0.0 0.0 0.0
Solar plant output (kW)
TABLE 2. O p t i m u m system o p e r a t i o n r e s u l t s
10.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 20.0 60.0 15.0 0.0 -33.3 0.0 45.0 20.0 26.7 0.0 0.0 -26.7 0.0 0.0 0.0 0.0
Battery output (kW)
7.8 0.0 0.0 0.0 0.0 0.0 9.6 24.8 58.7 1.6 78.0 67.7 38.6 63.4 95.4 59.8 1.9 10.7 0.1 -29.6 0.0 0.0 0.0 0.0
Inverter output (kW)
400.0 390.0 390.0 390.0 390.0 390.0 390.0 390.0 390.0 370.0 415.0 400.0 400.0 425.0 425.0 380.0 360.0 380.0 380.0 380.0 400.0 400.0 400.0 400.0
Battery stored energy (kWh) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 274.2 0.0 0.0 262.7 0.0 0.0 0.0 0.0 0.0 0.0 274.7 338.6 327.6 266.3 175.4
Unit 1 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 2 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 3 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 4 (kW) 95.8 73.4 84.7 45.1 73.7 74.7 93.3 70.3 99.9 0.0 99.1 76.4 0.0 82.7 96.7 95.8 94.9 94.4 52.7 0.0 0.0 0.0 0.0 0.0
Unit 5 (kW)
Diesel p l a n t u n i t c o m m i t m e n t and dispatch
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 100.0 0.0 100.0 0.0 0.0 i00.0 I00.0 100.0 0.0 0.0 0.0 0.0 0.0
Unit 6 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 7 (kW)
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 8 (kW)
($) (S/h) 7.4 5.9 6.6 4.0 5.9 6.0 7.2 5.7 7.6 20.4 15.2 13.7 19.7 14.2 7.4 7.4 15.0 14.9 12.2 20.5 24.4 23.7 19.9 14.4
304.6 294.1 286.9 277.1 271.5 264.0 258.9 250.9 246.5 242.7 223.9 208.2 198.2 180.4 168.3 161.2 158.1 145.3 126.7 114.8 92.8 67.2 40.4 14.4
Total cost cost
Fuel
101 TABLE 3. Summary of system operation results Per cent of total production Demand (kWh) Production (kWh) Diesel plant Wind park Solar plant Total Losses (kWh) Battery Inverter Total
5950.0
97.5
4140.0 1396.0 563.6 6099.6
67.9 22.9 9.2 100.0
63.3 86.3 149.6
1.0 1.5 2.5
Cost ($)
313.1
the Greek island of Cythnos. The system consists of a diesel plant, a wind park and a solar plant with storage battery. The diesel plant consists of eight units (2 × 630 kVA, 2 × 250 kVA, 4 × 125 kVA) with a total installed capacity of 2.26 MVA. Table 1 contains economic data for the diesel units. The wind park consists of five identical wind turbine generators (20kW each). The nominal park power output is reached at wind velocities of about 11 m/s. The solar plant with peak power of 100 kW feeds a battery unit (600 kWh). The battery is not allowed to discharge below 180kWh and its efficiency is 75%. The inverter efficiency as a function of the inverter output is given in Fig. 2. The solar plant transformer rating is 100 kW. The amount of energy stored in the battery at the end of the scheduling period must be equal to the initial battery energy (400 kWh). A minimum diesel plant output requirement of 30 kW is imposed. The optimal operation of the system for a typical summer day is derived. The forecasts and the measured values of the load demand, the wind velocity and the global insolation are shown in Figs. 4-6. The optimum operation of the island's energy resources is given in Table 2. Table 3 gives a summary of the contribution of the various energy resources used in order to cover the power demand and system losses. It is observed in Table 2 that the battery usage obeys the following two rules: (a) battery discharging takes place during hours when it helps to avoid the commitment of an additional diesel unit;
(b) battery charging takes place during hours when there is no need to commit an additional diesel unit for this purpose. The above two rules for the battery usage, first proposed in ref. 2 based on engineering judgement, are fully supported by the results of our method.
6. Conclusions A method for the determination of the optimum operation of an autonomous system has been presented. The method gives a feedback control of the system operation that can follow unpredictable changes in the weather conditions. The proposed method can be used for the optimization of the on-line operation of an autonomous system with conventional and unconventional energy sources.
Acknowledgements This research was supported financially by the Greek Ministry of Industry, Energy and Technology and the National Fellowship Foundation. The authors wish to thank the Public Power Corporation in Greece for the information given.
Nomenclature i L(t)
L(t) No
PB(t) PDi(t) PD(t) P_s(t) Ps(t) Pw(t) Pw(t)
s(t) ~(t) T t
V~(t) w(t) &(t)
diesel unit index, i = 1, 2. . . . , ND load demand during hour t load demand forecast for hour t number of diesel units = VB(t) -- VB(t + 1), battery discharging (PB > 0) or charging (PB < 0) during hour t output of ith diesel unit during hour t diesel plant output during hour t solar plant output during hour t available solar plant output during hour t wind park output during hour t available wind park output during hour t global insolation during hour t global insolation forecast for hour t scheduling horizon (T =24, for 24-hour scheduling) time index, t = 1, 2. . . . . T energy stored in battery at beginning of hour t wind velocity during hour t wind velocity forecast for hour t
102
~B
GL(t)
a~(t) am(t)
b a t t e r y efficiency i n v e r t e r efficiency s t a n d a r d d e v i a t i o n o f l o a d d e m a n d forec a s t e r r o r for h o u r t s t a n d a r d d e v i a t i o n of g l o b a l i n s o l a t i o n f o r e c a s t e r r o r for h o u r t s t a n d a r d d e v i a t i o n of w i n d v e l o c i t y forec a s t e r r o r for h o u r t
A bar above a variable denotes an upper bound while a bar below a variable denotes a lower bound.
References 1 A. G. B a k i r t z i s a n d P. S. Dokopoulos. S h o r t t e r m g e n e r a t i o n s c h e d u l i n g in a s m a l l a u t o n o m o u s s y s t e m w i t h u n c o n v e n t i o n a l e n e r g y sources, IEEE Trans., P W R S - 3 (3) (1988) 1230 1236. 2 J. C h a d j i v a s i l i a d i s , G. H a c k e n b e r g , W. K t e i n k a u f a n d F. Raptis, P o w e r m a n a g e m e n t fbr t h e c o m p o u n d o p e r a t i o n of diesel g e n e r a t o r sets with wind e n e r g y a n d p h o t o v o l t a i c p l a n t s .
Europ. Wind Energy Conf. (EWEC), Rome, Italy, 1986. 3 (I. C. C o n t a x i s , J. K a b o u r i s a n d J. C h a d j i v a s i l i a d i s , O p t i m u m o p e r a t i o n of' a n a u t o n o m o u s e n e r g y system, Europ. Wind
Energy Conf. (EWEC), Rome, Italy, 1986. 4 G. P. Box a n d G. M. J e n k i n s , Time Series Analysis bbreeasting and Control, Holden-Day, S a n F r a n c i s c o , CA. 1970.
93
Optimum operation of a small autonomous system with u n c o n v e n t i o n a l energy sources A. G. Bakirtzis and E. S. Gavanidou Aristotle University of Thessaloniki, Salonika (Greece)
(Received July 22, 1991)
Abstract This paper presents a method for the determination of the optimum operation of a small autonomous system with both conventional and unconventional energy sources and storage battery. The system generation consists of diesel generators, wind turbine generators and photovoltaic panels. This is the generation mix of the Greek island of Cythnos; it may be applied to other small islands. A stochastic dynamic programming based algorithm is employed to determine the optimal short-term generation scheduling and battery storage policy which minimize the fuel consumption for the next 24-hour period. An application of the method to the power system of Cythnos is also presented.
1. Introduction Utilization of unconventional energy sources for the electrical power supply to remote regions has proved to be a particularly promising application of wind and solar energy plants. Owing to the high production costs of electricity that result from a decentralized power supply (mostly by diesel generator units), large-scale utilization of unconventional energy sources is favoured in remote regions with appropriate weather conditions. This applies in particular to the Greek islands. A large unconventional penetration into a diesel power system faces technical and economic problems. The problem of the economic operation of a small autonomous system is addressed in refs. 1-3. In ref. 1 a method was presented for optimal generation scheduling in a small autonomous system with conventional (diesel) and unconventional energy sources and storage battery. The method relied on forecasts of the average hourly values of the load demand, the wind velocity and the global insolation for the next 24-hour period, and used a dynamic programming based algorithm to determine an optimal diesel unit commitment schedule and an optimal battery charge/discharge sequence so as to minimize the diesel fuel consumption the next day. The forecasts were assumed to be completely reliable. Further tests showed that, whereas the load demand prediction for the next 24 hours is 0378-7796/92/$5.00
satisfactory, the wind velocity and the global insolation can only be predicted within relatively large confidence intervals. Therefore the method in ref. 1 cannot be applied for the on-line management of a small autonomous system without further exploitation of the stochastic variation of the weather conditions. This paper addresses the problem of the optimum operation of a small autonomous system under stochastically varying weather conditions. The t-hour ahead forecasts are treated as random variables, not known at the beginning of the day when the system operation scheduling takes place. A stochastic dynamic programming based algorithm is used in order to determine the optimal diesel unit commitment and an adaptive battery storage policy that can follow the changing weather conditions. The developed algorithm gives a feedback control of the system, appropriate for on-line application.
2. System description Figure 1 shows a typical generation of a small autonomous system. The basic components of the system are the diesel plant and the unconventional sources, the wind park and the solar plant. A battery is also available for energy storage. A brief description of the basic parts of the system follows. © 1992 -- Elsevier Sequoia. All rights reserved
94 DIESEL
PLANT
This is the minimum diesel plant o u t p u t requirement. At hours of low load demand and high u n c o n v e n t i o n a l energy production, when the minimum diesel plant r e q u i r e m e n t is active, the excess u n c o n v e n t i o n a l energy production must be stored in the battery. If the b a t t e r y is fully c h a r g e d at that hour the excess unconventional energy is lost.
CD
WIND
PARK
(K)
'
2.2. The wind park The available wind park o u t p u t during hour t is a function of the wind velocity w(t) during that hour:
SOLAR PLANT
I
BATTERY
~
~fiw(t) = gw (w(t))
w
I Fig. 1. T h e g e n e r a t i o n system.
s y s t e m of a s m a l l a u t o n o m o u s
power
For maximum fuel saving, all the available wind park p o w e r should be utilized in order to satisfy the load demand and/or charge the battery. However, due to the minimum diesel plant o u t p u t requirement, some wind power may be lost, so that, in general, Pw(t) ~
2.1. The diesel plant The diesel plant consists of NI) units. The production cost of the ith diesel unit is described by a linear cost function: F; (PD;)
fNLCi + IC;PD,: = ~O
if if
P,)~ ~< PD; PD; = 0
~<
w h e r e NLCi is the unit no-load cost and ICi is the unit i n c r e m e n t a l cost. The diesel plant o u t p u t during h o u r t is NI)
P,(t) = ~ Prj;(t)
(1)
i=1
where
P_r)i ~ PDi(t) <~l~Di
(2a)
if unit i is committed, and
PDi(t) = 0
(2b)
if unit i is not committed. The p o w e r that the diesel plant must supply at a certain time equals the load demand at t h a t time minus the u n c o n v e n t i o n a l generation and the b a t t e r y discharge. The diesel plant controls the voltage level and the frequency of the system. Also, if a diesel unit operates at a small fraction of its nominal p o w e r o u t p u t its efficiency is considerably reduced. For these reasons, the diesel p l a n t o u t p u t must never fall below a certain value _PD: PD(t) ~ ~Pl)
(3)
(5)
2.3. The solar plant The available p h o t o v o l t a i c plant o u t p u t during hour t is a function of the global insolation P~ (t) = gs (s(t))
PI)i
(4)
(6)
In general, the usable solar plant o u t p u t is less t h a n t h a t available, due to the minimum diesel plant o u t p u t requirement: Ps (t) ~
(7)
The photovoltaic panels are connected t h r o u g h a DC/DC c o n v e r t e r to the b a t t e r y busbar. There is a maximum power t r a c k i n g control for the c o n v e r t e r so that maximum power can be drawn from the panels. The solar plant is connected to the grid t h r o u g h a DC/AC inverter and a step-up transformer.
2.4. The storage battery The storage b a t t e r y is c o n n e c t e d to the solar plant t h r o u g h a DC/DC c o n v e r t e r and to the grid t h r o u g h the solar plant DC/AC inverter and transformer as s h o w n in Fig. 1. The b a t t e r y capacity is I?B and the depth of discharge is VB. Thus, the energy stored in the b a t t e r y at any time must satisfy the inequalities VB ~< VB(t) ~< VB
(8)
The b a t t e r y can be c h a r g e d either by the solar plant or by the grid. The b a t t e r y efficiency is ~h~ and is t a k e n into a c c o u n t during b a t t e r y charging
95
only. The energy balance in the battery during hour t is described by the following relations:
The output of the inverter is not allowed to exceed the step-up transformer rating, PST:
VB(t + 1) = VB(t) -- t/nPB(t)
(9a)
Pss(t)
(9b)
2.6. System operation At every hour of the system operation, the load demand must be supplied by the diesel plant, the unconventional sources and the battery:
Vs(t + 1) = Vs(t) - P s ( t ) --PB(t)
~/)Bc
if
PB(t) ~
if if
PB(t) < 0
PB(t) > 0
PB(t) < 0
(10a)
PB(t) > 0
(10b)
~
(14)
where the battery is charging when PB < 0 and discharges when PB > 0; PBc and/~Bo are the battery charge and discharge rates, respectively. The energy stored in the battery at the beginning of the day and the desired battery storage at the end of the day are also specified:
PL(t) = PD(t) A-Pw(t) + PsB(t)
VB(0) =VBo
(11)
VR(T) = VBT
(12)
The problem of the optimum operation of a small autonomous system can be stated as follows. Given the short-term forecasts (next 24 hours) and the forecast error standard deviations of (a) the hourly average load demand/~(t), GL(t), (b) the hourly average wind velocity &(t), Gw(t) and (c) the hourly average global insolation ~(t), as(t), determine (i) the hourly diesel unit commitment and dispatch and (ii) the hourly battery energy storage policy so as to minimize the total diesel cost over the next 24-hour period. The hourly wind park and solar plant output are determined by the hourly wind velocity and insolation levels so as to fully utilize the cost-free soft energy sources. In rare cases a reduction of the unconventional generation may be necessary in order to satisfy the minimum diesel plant output requirement. The above-mentioned problem is a dynamic optimization problem with objective the total diesel cost during the scheduling period:
2.5. The inverter The solar plant and the battery are connected to the grid with a DC/AC inverter and a step-up transformer as shown in Fig. 1. The inverter controls the flow of power, PsB, between the solar-plant-battery combination and the grid. The direction of the power flow PsB can be either from the inverter to the grid (Ps~ > 0) or vice versa (PsB < 0). Figure 1 shows the direction for positive PB and PsB- If the inverter efficiency is ~i and the step-up transformer is assumed 100% efficient, the following relations describe the power balance in the inverter: PsB(t) = ~i[Ps(t) + PB(t)]
if
PsB(t) > 0
(13a)
PsB(t) = 1 [Ps(t) + PB(t)] th
if
PsB(t) < 0
(13b)
(15)
where PsB(t) is the inverter output given in (13).
3. P r o b l e m s t a t e m e n t
The inverter efficiency is a function of the power delivered by the inverter, as shown in Fig. 2. subject to the system operating constraints (1)(15). The solution to the problem is discussed next.
q 1.0
~
q
0.9
4. S o l u t i o n o f the p r o b l e m
0.8
The solution to the optimum operation problem is outlined in Fig. 3. At the beginning of the day (twelve midnight), the scheduling of the next day's operation is performed (block 2). At this point it must be decided which diesel u.nits will be committed and what will be the battery usage during the next day so that the expected operation cost of the system is minimized. These
0.7
0
.
.
0.1
0.3
.
. 0.6
" P/PN
1.0
Fig. 2. I n v e r t e r efficiency as a function of the inverter power output.
96
.
;.°
HISTORYDATA~ 0_ °•"'°•..°
FORECASTS Determine
BLOCK 1
~
"
W(T), S(T), Z(T) Ow(t), Os(t), OL(T) for t=1,2,...T
"..
"•
.." •.
° °
"..
..." • ,~
•
"•'.
"j J . ;
".\ ..~
-°oo -
N
• . .....
OPERATION SCHEDULING Determine Optimal Unit Commitment Determine Optimal Battery Storage Policy in the form of a "Look up Table":
,
.
•
BLOCK 2
~
;2
time
VB*(T+I )=fB(VB(T) ,w(T),S(T),L(1) For T=O,I.... ~T-I
I
,:
20
2'4
(h)
F i g . 4. Load d e m a n d forecasts versus m e a s u r e m e n t s .
=.
.•••'.o°o°.°..
ON-LINE OPER.AII ~I M e a s u r e
VB(t) ,w(t) , s ( t ) , L ( t )
Read Optimal Unit Committment Read VB (t+]) from Look up Table Send Control Signals to: Diesel Plant Inverter
BLOCK 3
•
•••
.0.
•..'..
"..•...
.°
~¢~. E "'. >
• .
\
• ......
~.~/
'%
\,~"
/
~ ' ~
\
''''•'..'0.••...••,. °°• YES time
(h)
NO F i g . 5. W i n d v e l o c i t y forecasts v e r s u s m e a s u r e m e n t s . F i g . 3. F l o w c h a r t of the s o l u t i o n to t h e o p t i m u m o p e r a t i o n problem.
o_
decisions are made under the uncertainty associated with the forecasts of the wind velocity, the global insolation and the load demand (block 1). During the system on-line operation (block 3) the diesel unit commitment is kept fixed while the battery usage is adapted to the changing weather conditions and load demand as new measurements of these quantitites are available. A more detailed discussion of these three solution steps follows•
OE 8°=_ .E
o-
•
•
g
111
113
115 time
117
t/
(h)
F i g . 6. Global i n s o l a t i o n forecasts versus m e a s u r e m e n t s .
4.1. Forecasts
The forecasts of the next day's average hourly values of the wind velocity, the global insolation and the load demand are computed based on measurements of the average hourly values of the previous 30 days• B o x - J e n k i n s models [4] are used to compute the forecasts. These models give, in addition to the forecasts, the forecast error standard deviation which can be used to compute the confidence limits of the forecasts•
Sample results of the forecasts are s h o w n in Figs. 4, 5 and 6. In the Figures, the solid lines represent the forecasts, the dashed lines represent the measured values of the forecasted quantities and the dotted lines the 95% confidence limits of the forecasts (i.e. the region in which, with 95% probability, the measured values of the forecasted quantities will lie). The small confidence interval for the insolation in Fig. 6 was observed during summer months only.
97
The forecasts of any of the forecasted quantities, for example the wind velocity, are given in the form {&(t), aw(t), t = 1, 2. . . . . T} where t~(t) is the forecast of the wind velocity w(t) that will be observed during the tth hour and aw(t) is the forecast error standard deviation. Assuming that the forecast error ew(t) is a normal random variable with zero mean and standard deviation aw(t), the wind velocity that will be observed during hour t,
w(t) = &(t) + ew(t) is a normal random variable with mean &(t) and standard deviation aw(t). That is, at the beginning of the day, w(t) is not known but its probability distribution is well defined. Since w(t) is a normal random variable, it will, with 95% probability, be in the interval &(t) _+2aw(t) which gives the 95% confidence limits of Fig. 5. Now, taking into account the fact that w(t) is positive, the 95% probability limits are modified to [w(t), ~(t)] where w(t) = max[0, &(t) - 2aw(t)]
(17a)
and &(t) = &(t) + 2aw(t)
(17b)
The probability density function of w(t) is shown in Fig. 7, where the amplitude of the impulse at zero equals the shaded area along the negative w(t) axis.
4.2. Operation scheduling The scheduling of the operation of the system is performed at the beginning of the day and consists of two steps.
Step 1. Determination of the optimal diesel unit commitment The diesel unit commitment is computed using a deterministic dynamic programming algorithm described in ref. 1 and is based on the forecasts &(t), ~(t) and £(t), t = 1 , . . . , T, available at the beginning of the day. The forecasts are used in f.(w)
-2Ow
0
W
Fig. 7. Wind velocity probability density function.
order to compute the projected load demand and unconventional generation and an optimal battery charge/discharge sequence, which give the projected diesel plant loading for every hour of the next day. The projected hourly diesel plant loading is then used to compute the optimal diesel unit commitment. Details of the method are given in ref. 1. In summary, the method developed in ref. 1 computes: (a) the optimal diesel unit commitment {UC*(t), t -- 1. . . . . T}
(18)
(b) the optimal sequence of battery storage levels {V~(t), t -- 1 , . . . , T}
(19)
which minimize the diesel operation cost of the next day under the assumption of zero forecast error (i.e. it is assumed that the values of the forecasted quantities that will be observed during the course of the day will be equal to their forecasts). The above sequence of decisions gives an 'open-loop' control of the system which is determined at the beginning of the day and cannot change during the course of the day, when new measurements are available, in order to follow the unpredictable changes of the weather conditions and the load demand. Since the unit commitment must be decided at the beginning of the day and there is limited flexibility in changing it during the course of the day, the open-loop control policy described above is used. Once the unit commitment schedule is determined, based on the available forecasts, it is kept fixed for the rest of the day. The battery usage can be further improved by replacing the open-loop optimal battery storage sequence (19) with a feedback battery storage policy which is sufficiently flexible to follow the changing weather conditions.
Step 2. Optimal battery storage policy In this step the optimal battery storage policy is computed, using a stochastic dynamic programming algorithm. Since there is a certain amount of uncertainty associated with the forecasts (w(t), s(t) and L(t) are now treated as random variables, not known at the beginning of the planning period), the objective in the determination of the battery storage policy is the minimization of the expected diesel cost:
98
w h e r e the e x p e c t a t i o n in (20) is t a k e n over the p r o b a b i l i t y distributions of all the forecasted quantities. A stochastic dynamic programming algorithm is used to minimize the expected operating cost (20) subject to the system operating c o n s t r a i n t s (1) - (15). (a) T h e d y n a m i c p r o g r a m m i n g state space. The energy stored in the b a t t e r y is discretized into Kj~ discrete energy storage levels b e t w e e n V~ and VBk=V~+(k--1)
AVB
2 = 1, 2 . . . . . A(t)
w h e r e w~(t) is the )~th wind velocity level during h o u r t and Aw is the wind velocity discretization step (e.g. Aw = 1 m/s). The p r o b a b i l i t y density function of w(t) is also discretized with the comp u t a t i o n of the probabilities Pr,~.(t) = Pr{w(t) = w~(t)}
~ = 1, 2 . . . . .
V ~ ( t ) = V.,~
and the average wind velocity, global insolation and load demand during hour t are w(t) = w~(t),
s(t) = s,,(t)
and L(t) = L,,(t)
k = 1 , 2 . . . . ,K~
w h e r e V~k is the k t h b a t t e r y storage level and A V~ is the b a t t e r y energy storage discretization step. As discussed in §4.1, the wind velocity w(t) is a random variable t a k i n g values in the interval [_w(t), w(t)] given in (17). The wind velocity w(t) is discretized into A(t) discrete wind levels: w j ( t ) = _w(t) + (~ - 1) Aw
of the scheduling period, given that the energy stored in the b a t t e r y at the beginning of hou{ t is
A(t)
The global insolation and the load d e m a n d as well as their p r o b a b i l i t y distributions are also discretized with steps As and AL, respectively: s,,(t) = s_(t) + (p - 1) As
IL = 1, 2 . . . . .
M(t)
Pr~(t) = Pr{s(t) = s,,(t)}
p = 1, 2 . . . . .
M(t)
L,,(t) = L(t) + (u - 1) AL
u = 1, 2 . . . . .
Y(t)
Pr)'(t) = Pr{L(t) = L,(t)}
u = 1, 2 . . . . .
Y(t)
With the discretization of the b a t t e r y storage, the wind velocity, the global insolation and the load demand, the stochastic dynamic programming state space is formed. Each state in the state space is defined by five indices (t, k, .~, It, u) specifying time, b a t t e r y storage level, wind velocity level, global insolation level and load dem a n d level, respectively. The state probability Pr~(t) Pr~(t) Pr~.(t) is a s s o c i a t e d with each state of the state space (t, k, ),, p, u). This is the probability of o c c u r r e n c e of the specific levels of the f o r e c a s t e d quantities, assuming their statistical independence. (b) T h e r e c u r s i v e f o r m u l a . The stochastic dynamic p r o g r a m m i n g recursive formula is derived with the help of the following definitions: (1) RC(t, k, )., IL, u): minimum expected production cost from the beginning of h o u r t to the end
(2) PC(t, k, ~, g, u; t + 1, m): p r o d u c t i o n cost during h o u r t, given t h a t the b a t t e r y stored energy at the beginning of hour t is VB(t) = V~k
the average wind velocity, global insolation and load demand during hour t are w(t) = w~(t),
s(t) = s,,(t)
and L(t) = L,,(t)
and the b a t t e r y stored energy at the beginning of hour t + 1 is VB (t + 1) = V~,,
The o p t i m u m b a t t e r y storage policy is comp u t e d by applying the stochastic dynamic programming recursive formula RC(t, k, )~, p, u)
E
=rain
[PC(t,k,i,l~, u;t + l,m)
+RC(t + 1, m, r, z, v)]
(21)
with
E [PC(t, k, )., p, u; t + 1, m) + RC(t + 1, m, r, z, v)] 'tr, u. u{
=PC(t, k, 2, p, u; t + 1, m) +
~,
RC(t + 1, m, r, u, v)Pr[,~(t)Pr~(t)Pr~'(t)
{r, u, vl
where the first term is taken out of the expectation since it is independent of r, z and v. That is, the production cost during hour t does not depend on the wind velocity, the global insolation and the load level that will be observed during hour t ÷ 1 (Wr(t T 1), sz(t ÷ 1), and L~(t + 1)). The production cost PC(t, k, )., p, u; t + 1, m) is computed as follows. - The power delivered by the battery is computed using (9):
99
" 1 [VB(t) - VB(t + 1)]
PB (t) =
=l(k -m)AVB ~/B [VB(t) -- Vn(t + 1)1 =(k - m ) A V B
if m > k
ifm
- The unconventional generation is computed assuming w(t)= w~(t) and s(t)=s~(t). The load demand is L(t) = Lu(t). - The diesel plant output PD(t) is computed using (12) and if the minimum diesel plant output requirement is violated the unconventional generation is reduced. - If any of the constraints (10) and (14) is violated, the corresponding cost is set to infinity: PC(t, k, it, It, u; t + 1, m) = oo For the computation of PC(t, k, 2, ~, u; t + 1, m) the required diesel plant output PD(t) is allocated to the ND units of the diesel plant according to the unit commitment determined in step 1. (c) The dynamic programming results. The results of the dynamic p r o g r a m m i n g algorithm will be in the following form. For every state in the dynamic p r o g r a m m i n g state space defined by (t, k, 2, p, u) the optimum b a t t e r y discrete storage level m* for the next time interval t + 1 is computed and stored in the form of a 'look-up table' (m* is the value of m t h a t minimizes the expression in (21)). In other words, the optimal battery storage policy is computed in the form
V*m(t + 1) = fB(Vnk(t), w~.(t), s,(t), Lu(t), t) t=0 .... ,T-1
(22)
The optimal b a t t e r y storage policy (22), computed at the beginning of the day, is in the form of a 'look-up table' t h a t contains the optimum operating decision (V*m(t + 1)) for every h o u r and for all the possible operating states of the next day. The m e a n i n g of the b a t t e r y storage policy (22) can be better explained if we drop the discrete level indices (assuming very fine discretization) so t h a t (22) becomes
V*(t + 1) = fB(VB(t), w(t), s(t), L(t), t)
V*(t + 1) can only be d e t e r m i n e d during h o u r t w h e n the m e a s u r e m e n t s of w(t), s(t) and L(t) are available. This is a feedback control policy in which m e a s u r e m e n t s of the c u r r e n t operating state are used in order to t a k e the optimal decision. The decision is not t a k e n once at the beginning of the day, but decisions are t a k e n every h o u r during the system on-line operation, w h e n new m e a s u r e m e n t s are available. This is described next. 4.3. On-line operation The results of the system's operation scheduling, performed at the beginning of the day, are: (a) the optimal diesel unit commitment, derived as an optimal sequence of decisions (18) and (b) the optimal b a t t e r y storage policy, derived in terms of a 'look-up table' (22). During the tth h o u r of the system operation, when m e a s u r e m e n t s of VB(t), w(t), s(t) and L(t) are available, the optimum b a t t e r y storage level at the end of h o u r t, V*(t + 1), is read from the 'look-up table' formed during the operation scheduling phase. V*(t + 1), t o g e t h e r with the b a t t e r y storage level at the beginning of the hour, Vn(t), specify the desired b a t t e r y charging or discharging during h o u r t, PB(t). Measurements of the wind park and solar plant o u t p u t Pw(t) and Ps(t) are also available. Therefore, from (13) and (15), the desired inverter output PsB(t) and diesel plant output PD(t) are determined and appropriate reference signals are sent to the inverter and the diesel plant. The desired diesel plant o u t p u t PD(t) is allocated to the diesel units committed during h o u r t.
5.
The performance of the m e t h o d is demonstrated by a case study with the power system of T A B L E 1. Diesel u n i t d a t a i
_PDi (kW)
/~i (kW)
NLCi (S/h)
ICi (S/kWh)
1 2 3 4 5 6 7 8
50 50 30 30 30 30 30 30
530 530 200 200 100 100 100 100
3.57 3.57 3.57 3.57 1.07 1.07 1.07 1.07
0.0615 0.0615 0.0661 0.0661 0.0657 0.0657 0.0657 0.0657
(23)
T h a t is, the optimum b a t t e r y storage level at the beginning of the next h o u r is a function of the b a t t e r y storage level at the beginning of the c u r r e n t h o u r and the wind velocity, the global insolation and the load d e m a n d m e a s u r e d during the c u r r e n t hour. Since w(t), s(t) and L(t) are not k n o w n at the beginning of the scheduling period,
Results
Power demand (kW)
203.6 173.4 184.7 145.1 173.7 174.7 168.6 183.2 217.8 281.8 283.6 286.2 306.6 275.1 228.9 204.3 229.7 257.8 229.1 320.1 409.7 385.5 321.6 241.2
Hour
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24
95.8 73.4 84.7 45.1 73.7 74.7 93.3 70.3 99.9 274.2 199.1 176.4 262.7 182.7 96.7 95.8 194.9 194.4 152.7 274.7 338.6 327.6 266.3 175.4
Diesel plant output (kW)
100.0 100.0 100.0 100.0 100.0 100.0 65.8 88.2 59.2 9.2 6.6 42.1 5.3 28.9 36.8 48.7 32.9 52.6 76.3 75.0 71.1 57.9 55.3 65.8
Wind park output (kW)
0.0 0.0 0.0 0.0 0.0 0.1 11.8 27.7 44.9 59.0 69.3 74.0 75.2 69.7 59.8 46.1 29.5 13.1 0.1 0.0 0.0 0.0 0.0 0.0
Solar plant output (kW)
TABLE 2. O p t i m u m system o p e r a t i o n r e s u l t s
10.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 20.0 60.0 15.0 0.0 -33.3 0.0 45.0 20.0 26.7 0.0 0.0 -26.7 0.0 0.0 0.0 0.0
Battery output (kW)
7.8 0.0 0.0 0.0 0.0 0.0 9.6 24.8 58.7 1.6 78.0 67.7 38.6 63.4 95.4 59.8 1.9 10.7 0.1 -29.6 0.0 0.0 0.0 0.0
Inverter output (kW)
400.0 390.0 390.0 390.0 390.0 390.0 390.0 390.0 390.0 370.0 415.0 400.0 400.0 425.0 425.0 380.0 360.0 380.0 380.0 380.0 400.0 400.0 400.0 400.0
Battery stored energy (kWh) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 274.2 0.0 0.0 262.7 0.0 0.0 0.0 0.0 0.0 0.0 274.7 338.6 327.6 266.3 175.4
Unit 1 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 2 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 3 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 4 (kW) 95.8 73.4 84.7 45.1 73.7 74.7 93.3 70.3 99.9 0.0 99.1 76.4 0.0 82.7 96.7 95.8 94.9 94.4 52.7 0.0 0.0 0.0 0.0 0.0
Unit 5 (kW)
Diesel p l a n t u n i t c o m m i t m e n t and dispatch
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 100.0 100.0 0.0 100.0 0.0 0.0 i00.0 I00.0 100.0 0.0 0.0 0.0 0.0 0.0
Unit 6 (kW) 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 7 (kW)
0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0
Unit 8 (kW)
($) (S/h) 7.4 5.9 6.6 4.0 5.9 6.0 7.2 5.7 7.6 20.4 15.2 13.7 19.7 14.2 7.4 7.4 15.0 14.9 12.2 20.5 24.4 23.7 19.9 14.4
304.6 294.1 286.9 277.1 271.5 264.0 258.9 250.9 246.5 242.7 223.9 208.2 198.2 180.4 168.3 161.2 158.1 145.3 126.7 114.8 92.8 67.2 40.4 14.4
Total cost cost
Fuel
101 TABLE 3. Summary of system operation results Per cent of total production Demand (kWh) Production (kWh) Diesel plant Wind park Solar plant Total Losses (kWh) Battery Inverter Total
5950.0
97.5
4140.0 1396.0 563.6 6099.6
67.9 22.9 9.2 100.0
63.3 86.3 149.6
1.0 1.5 2.5
Cost ($)
313.1
the Greek island of Cythnos. The system consists of a diesel plant, a wind park and a solar plant with storage battery. The diesel plant consists of eight units (2 × 630 kVA, 2 × 250 kVA, 4 × 125 kVA) with a total installed capacity of 2.26 MVA. Table 1 contains economic data for the diesel units. The wind park consists of five identical wind turbine generators (20kW each). The nominal park power output is reached at wind velocities of about 11 m/s. The solar plant with peak power of 100 kW feeds a battery unit (600 kWh). The battery is not allowed to discharge below 180kWh and its efficiency is 75%. The inverter efficiency as a function of the inverter output is given in Fig. 2. The solar plant transformer rating is 100 kW. The amount of energy stored in the battery at the end of the scheduling period must be equal to the initial battery energy (400 kWh). A minimum diesel plant output requirement of 30 kW is imposed. The optimal operation of the system for a typical summer day is derived. The forecasts and the measured values of the load demand, the wind velocity and the global insolation are shown in Figs. 4-6. The optimum operation of the island's energy resources is given in Table 2. Table 3 gives a summary of the contribution of the various energy resources used in order to cover the power demand and system losses. It is observed in Table 2 that the battery usage obeys the following two rules: (a) battery discharging takes place during hours when it helps to avoid the commitment of an additional diesel unit;
(b) battery charging takes place during hours when there is no need to commit an additional diesel unit for this purpose. The above two rules for the battery usage, first proposed in ref. 2 based on engineering judgement, are fully supported by the results of our method.
6. Conclusions A method for the determination of the optimum operation of an autonomous system has been presented. The method gives a feedback control of the system operation that can follow unpredictable changes in the weather conditions. The proposed method can be used for the optimization of the on-line operation of an autonomous system with conventional and unconventional energy sources.
Acknowledgements This research was supported financially by the Greek Ministry of Industry, Energy and Technology and the National Fellowship Foundation. The authors wish to thank the Public Power Corporation in Greece for the information given.
Nomenclature i L(t)
L(t) No
PB(t) PDi(t) PD(t) P_s(t) Ps(t) Pw(t) Pw(t)
s(t) ~(t) T t
V~(t) w(t) &(t)
diesel unit index, i = 1, 2. . . . , ND load demand during hour t load demand forecast for hour t number of diesel units = VB(t) -- VB(t + 1), battery discharging (PB > 0) or charging (PB < 0) during hour t output of ith diesel unit during hour t diesel plant output during hour t solar plant output during hour t available solar plant output during hour t wind park output during hour t available wind park output during hour t global insolation during hour t global insolation forecast for hour t scheduling horizon (T =24, for 24-hour scheduling) time index, t = 1, 2. . . . . T energy stored in battery at beginning of hour t wind velocity during hour t wind velocity forecast for hour t
102
~B
GL(t)
a~(t) am(t)
b a t t e r y efficiency i n v e r t e r efficiency s t a n d a r d d e v i a t i o n o f l o a d d e m a n d forec a s t e r r o r for h o u r t s t a n d a r d d e v i a t i o n of g l o b a l i n s o l a t i o n f o r e c a s t e r r o r for h o u r t s t a n d a r d d e v i a t i o n of w i n d v e l o c i t y forec a s t e r r o r for h o u r t
A bar above a variable denotes an upper bound while a bar below a variable denotes a lower bound.
References 1 A. G. B a k i r t z i s a n d P. S. Dokopoulos. S h o r t t e r m g e n e r a t i o n s c h e d u l i n g in a s m a l l a u t o n o m o u s s y s t e m w i t h u n c o n v e n t i o n a l e n e r g y sources, IEEE Trans., P W R S - 3 (3) (1988) 1230 1236. 2 J. C h a d j i v a s i l i a d i s , G. H a c k e n b e r g , W. K t e i n k a u f a n d F. Raptis, P o w e r m a n a g e m e n t fbr t h e c o m p o u n d o p e r a t i o n of diesel g e n e r a t o r sets with wind e n e r g y a n d p h o t o v o l t a i c p l a n t s .
Europ. Wind Energy Conf. (EWEC), Rome, Italy, 1986. 3 (I. C. C o n t a x i s , J. K a b o u r i s a n d J. C h a d j i v a s i l i a d i s , O p t i m u m o p e r a t i o n of' a n a u t o n o m o u s e n e r g y system, Europ. Wind
Energy Conf. (EWEC), Rome, Italy, 1986. 4 G. P. Box a n d G. M. J e n k i n s , Time Series Analysis bbreeasting and Control, Holden-Day, S a n F r a n c i s c o , CA. 1970.